Global well-posedness for one-dimensional compressible Navier--Stokes system in dynamic combustion with small $BV\cap L^1$ initial data

GLOBAL WELL-POSEDNESS F OR ONE-DIMENSIONAL COMPRESSIBLE NA VIER–STOKES SYSTEM IN D YNAMIC COMBUSTION WITH SMALL B V ∩ L 1 INITIAL D A T A SIRAN LI, HAIT A O W ANG, AND JIANING Y ANG Abstra ct. W e establish the global w ell-p osedness theory of small BV w eak solutions to a one- dimensional compressible Na vier–Stokes model for reacting gas mixtures in dynamic combustion. The unkno wns of the PDE system consist of the sp eciﬁc volume, v elo city , temp erature, and mass fraction of the reactan t. F or initial data that are small perturbations around the constant equilibrium state (1 , 0 , 1 , 0) in the L 1 ( R ) ∩ BV( R ) -norm, we establish the lo cal-in-time existence of w eak solutions via an iterativ e sc heme, show the stabilit y and uniqueness of lo cal w eak solutions, and prov e the global-in-time existence of solutions for initial data with small BV- norm via an analysis of the Green’s function of the linearised system. The large-time b ehaviour of the global BV w eak solutions is also characterised. This work is motiv ated by and extends the recen t global well-posedness theory for BV w eak solutions to the one-dimensional isen tropic Na vier–Stokes and Na vier–Stokes–F ourier systems developed in [T.-P . Liu, S.-H. Y u, Commun. Pure Appl. Math. 75 (2022), 223–348] and [H. W ang, S.-H. Y u, X. Zhang, Arc h. Ration. Mech. Anal. 245 (2022), 375–477]. Contents 0. In tro duction 2 1. Preliminaries 7 1.1. Heat kernel with constan t conductivit y coeﬃcient 7 1.2. BV functions on R 7 1.3. Heat kernel with BV-conductivit y co eﬃcien t 7 1.4. A lemma from F ourier anla ysis 11 2. Lo cal solution 11 2.1. Iteration scheme 12 2.2. Con v ergence of the iteration sc heme 18 2.3. Impro v ed regularit y in time 30 2.4. Stabilit y and Uniqueness 36 3. Green’s F unction 40 3.1. High-F requency Analysis ( η → ∞ ), Singular P art 44 3.2. Lo w-F requency Analysis ( η → 0 ), Regular P art 50 4. Global W ell-p osedness 52 4.1. Represen tation by Green’s function 53 4.2. Global existence, uniqueness and large time b ehaviour 61 App endix A. 84 References 88 Date : F ebruary 10, 2026. 1 0. Intr oduction W e are concerned with the global well-posedness theory of a one-dimensional (1D) com- pressible Navier–Stok es mo del for a reacting gas mixture in dynamic combustion. The system of partial diﬀerential equations (PDE) in the Lagrangian co ordinates reads as follows:            v t − u x = 0 , u t + p x =  µu x v  x , E t + ( pu ) x =  µuu x v  x + Ä ν θ x v ä x + Ä q Dz x v 2 ä x , z t + K ϕ ( θ ) z =  D v 2 z x  x , (0.1) See, e.g. , G.-Q. Chen [4]. Throughout this w ork, the ph ysical v ariables v ≡ 1 ρ , u , θ , and z denote the sp eciﬁc volume (namely , the in v erse of density), velocity , temp erature, and the mass fraction of the reactan t, resp ectively . Let us ﬁrst describ e the physics of the PDE system (0.1). The reacting gas mixture in consideration has total sp eciﬁc energy E = e + u 2 2 + q z , (0.2) where e is the sp eciﬁc internal energy , and q is the diﬀerence b etw een the heat of the reactant and the product. F or ideal gases, we ha ve the constitutive relations: e = c v θ , p = aθ v (0.3) where c v is the sp eciﬁc heat capacity at constan t v olume, and a equals the Boltzmann’s constan t times the molecular w eight. The comp onen ts of the gas mixture in consideration is assumed to ob ey the same γ -law with γ = c p c v > 1 , where c p is the speciﬁc heat capacity tak en at constan t pressure. The function ϕ = ϕ ( θ ) : [0 , ∞ ) → [0 , ∞ ) describ es the rate of c hemical reaction at temp erature θ . Throughout this pap er, it is assumed to b e Lipsc hitz contin uous. The positive constants µ , ν , D , and K are the bulk viscosity , heat conductivit y , diﬀusivit y , and reactan t rate coeﬃcient, resp ectiv ely . In view of the discussions in the previous paragraph, w e ma y recast Eq. (0.1) as follo ws:            v t − u x = 0 , u t + p x =  µu x v  x , θ t + p c v u x − µ c v v u 2 x = Ä ν θ x c v v ä x + q c v K ϕ ( θ ) z , z t + K ϕ ( θ ) z =  D v 2 z x  x . (0.4) The pressure p is giv en b y the constitutiv e relations in Eq. (0.3). There is abundant literature on well-posedness and large-time b ehaviour of solutions to Eq. (0.1), or equiv alently , Eq. (0.4). Our list of references here is b y no means exhaustiv e. Chen [4] established the global existence of weak solutions for (0.1) on a b ounded spatial domain x ∈ [0 , 1] with more general reaction rate functions ϕ ( e.g. , for ϕ satisfying the Arrenhius’ la w, with a jump discontin uity on (0 , ∞ ) ); large-time b eha viour of w eak solutions has also been obtained. Moreo v er, in the case of Lipschitz con tin uous reaction rate functions, Chen–Hoﬀ– T rivisa [5, 6] pro v ed the existence for large discontin uous initial data. Li [19] extended the ab ov e results to un b ounded domains R or R + with suitable b oundary conditions. See also Ducomet [10], W ang [28], Jiang–Zheng [18], and man y others on the well-posedness theory and asymptotic 2 b eha viour of classical solutions to PDEs for 1D reactive and/or radiativ e gas dynamics. In addition, for the 1D com bustion dynamic PDE mo dels, detailed asymptotic analysis for the large-time behaviour has b een carried out for the interactions of sp eciﬁc t yp es of elementary w a ves. See [12, 24, 25, 33] among other references. W e also mention Jenssen–Lyng–Williams [17], F eng–Hong–Zhu [11], W ang–W en [29], W ang– W u [30] and Gao–Huang–Kuang–W ang–Xiang for the study of dynamic com bustion PDE in m ulti-dimensions, as well as Qin–Zhang–Su–Cao [26], Zhang [35], Liao–W ang–Zhao [20], W an– Zhang [27], and Zh u [36], etc. for reﬁned results under assumptions of spherical and cylindrical symmetry of solutions. It is interesting to inv estigate whether the results and metho ds in our w ork can be extended to the ab ov e settings. In the abov e, by we ak solutions we mean the functions ( v , u, θ , z ) satisfying        v − 1 ∈ L ∞ t H 1 x , ( u, θ − 1 , z ) ∈ L ∞ t H 1 x ∩ L 2 t H 2 x , ( u t , θ t , z t ) ∈ L 2 t L 2 x . (0.5) The w ell-p osedness theory for weak solutions has b een established on b ounded or unbounded 1D domains for initial data ( v 0 , u 0 , θ 0 , z 0 ) ≡ ( u, v , θ, z )   { t =0 } satisfying    0 < m 0 ≤ v 0 ( x ) , θ 0 ( x ) ≤ M 0 , | u 0 ( x ) | ≤ M 1 < ∞ , 0 ≤ z 0 ( x ) ≤ 1 for a.e. x ; ( v 0 − 1 , u, θ 0 − 1 , z 0 ) ∈ H 1 x , z 0 ∈ L 1 x , (0.6) where m 0 , M 0 , and M 1 are p ositive constants. See [4 – 6, 19]. In con trast, this pap er aims to establish the weak solution theory in a weak er top ology than that in Eq. (0.5). In particular, the speciﬁc density v = ρ − 1 is no w taken to b e a p erturbation of the constant background state: v − 1 ∈ C t BV x in R 2 + = R × [0 , ∞ ) . (See Deﬁnition 0.1 for precise form ulation.) This shall be established for initial data satisfying ∥ v 0 − 1 ∥ ( L 1 ∩ BV)( R ) ≪ 1 together with similar conditions on u 0 , θ 0 , and z 0 . Our w ork is primarily motiv ated by the recent seminal work [22] b y T.-P . Liu and S.-H. Y u, which establishes the global existence theory of 1D compressible Navier–Stok es equations for small BV initial data, through a delicate study on the Green’s function for 1D heat equation in div ergence-form with BV-co eﬃcients. It echoes the classical theory for the system of conserv ation la ws in one spatial dimension via Glimm’s scheme of random choice ( [14]; also cf. Dafermos [8] and Bressan [2] for comprehensive treatmen t of hyperb olic conserv ation laws). More precisely , Liu–Y u [22] considered the isen tropic Navier–Stok es equations    v t − u x = 0 , u t + p ( v ) x =  µu x v  x , (0.7) with initial data small in the L 1 ∩ BV -norm, provided that p ′ ( v ) < 0 . One of the ke y ingredien ts of the pro of in [22] is the construction of the fundamental solution H ( x, t ; y ; f ) to the heat equation 3 with BV coeﬃcients ( ( x, t ) ∈ R 2 + are the spacetime v ariables and y , f are parameters):    ∂ t H ( x, t ; y ; f ) − ∂ x  f ∂ x H ( x, t ; y ; f )  = 0 , H ( x, 0; y ; f ) = δ ( x − y ) , (0.8) where δ ( z ) is the Dirac delta measure on R supp orted at z = 0 , and f is a B V function with inf x ∈ R f ( x ) > 0 , ∥ f ∥ B V ≪ 1 . The analytical properties of H ( x, t ; y ; f ) together with an iteration sc heme lead to the lo cal existence and contin uous dep endence on initial data of the w eak solutions. In addition, b y in tro ducing an “ eﬀe ctive kernel function ” that interpolates b etw een the short-time heat kernel and long-time Green’s function, Liu-Y u derived integral represen tations of w eak solutions, and th us prov ed that weak solutions exist globally in time and decay to the constan t equilibrium state at the optimal rate of t − 1 / 2 for p olytropic gases, i.e. , when p ( v ) = Av − γ with 1 ≤ γ < e . The p oint wise construction of the Green’s function in 1D w as initiated in Zeng [34]. It has b een combined with the classical time-asymptotic analysis of the p oint wise con vergence of smo oth solutions to nonlinear w a ves in Liu–Zeng [23]. The particular construction of the Green’s function here and in [22, 32] is motiv ated by that for the Boltzmann equation, esp ecially the “particlelik e” and “wa v elik e” decomp osition in Liu–Y u [21]. See Deng–Y u [9] for related dev elopmen ts. W ang–Y u–Zhang [32] extended the w ell-p osedness theory from the 1D isen tropic Na vier– Stok es Eq. (0.7) to the full Navier–Stok es–F ourier system:        v t − u x = 0 , u t + p ( v , θ ) x =  µu x v  x ,  e + 1 2 u 2  t + ( pu ) x =  κ v θ x + µ v uu x  x , (0.9) also under the small B V ∩ L 1 x assumption for the initial data. Based on the heat kernel analysis in [22], the authors dev elop ed new Hölder-in-time estimates for the heat kernel and Lipsc hitz in space estimates for the ﬂuxes of u and θ . In this w ork, we extend the global w ell-p osedness theory for small BV-solutions laid down in [22] and further dev elop ed in [32] to the 1D Navier–Stok es combustion mo del for γ -la w gas mixtures, namely Eqs. (0.1) or (0.4). W e ﬁrst state our framew ork of w eak solutions. Deﬁnition 0.1. The quadruplet ( v , u, θ, z ) := [0 , t ♯ ) × R → R 4 is a we ak solution to Eqs. (0.1) or (0.4) if the fol lowing holds: (1) The PDE holds in the distributional sense: for any test function φ ∈ C ∞ 0 ([0 , t ♯ ) × R ) ,                                      Z + ∞ 0 Z R [ φ x u − φ t v ] d x d t = Z R φ ( x, 0) v ( x, 0) d x, Z + ∞ 0 Z R h φ x  µu x v − p  − φ t u i d x d t = Z R φ ( x, 0) u ( x, 0) d x, Z + ∞ 0 Z R ï φ x Å ν c v v θ x ã + φ Å p c v u x − µ c v v ( u x ) 2 − q c v K ϕ ( θ ) z ã − φ t θ ò d x d t = Z R φ ( x, 0) θ ( x, 0) d x, Z + ∞ 0 Z R ï φ x Å D z x v 2 ã + φK ϕ ( θ ) z − φ t z ò d x d t = Z R φ ( x, 0) z ( x, 0) d x ; (0.10) 4 (2) The fol lowing r e gularity c onditions ar e veriﬁe d:                        v ( x, t ) − 1 ∈ C  [0 , t ♯ ) ; L 1 ( R ) ∩ BV( R )  ; u ( x, t ) ∈ L ∞  0 , t ♯ ; W 1 , 1 ( R ) ∩ L ∞ ( R )  , √ tu x ( x, t ) ∈ L ∞ (0 , t ♯ ; L ∞ ( R )) , √ tu t ( x, t ) ∈ L ∞  0 , t ♯ ; L 1 ( R )  , tu t ( x, t ) ∈ L ∞ (0 , t ♯ ; L ∞ ( R )) ; θ ( x, t ) − 1 ∈ L ∞  0 , t ♯ ; W 1 , 1 ( R ) ∩ L ∞ ( R )  , √ tθ x ( x, t ) ∈ L ∞ (0 , t ♯ ; L ∞ ( R )) , √ tθ t ( x, t ) ∈ L ∞  0 , t ♯ ; L 1 ( R )  , tθ t ( x, t ) ∈ L ∞ (0 , t ♯ ; L ∞ ( R )) ; z ( x, t ) ∈ L ∞  0 , t ♯ ; W 1 , 1 ( R ) ∩ L ∞ ( R )  , √ tz x ( x, t ) ∈ L ∞ (0 , t ♯ ; L ∞ ( R )) , √ tz t ( x, t ) ∈ L ∞  0 , t ♯ ; L 1 ( R )  , tz t ( x, t ) ∈ L ∞ (0 , t ♯ ; L ∞ ( R )) . (0.11) The main results of our current work are to establish the global existence, stability , unique- ness, and large-time b eha viour of solutions to Eqs. (0.1) or (0.4) in the sense of Deﬁnition 0.1, pro vided that the initial data is a small L 1 ∩ BV -p erturbation of the constant bac kground state [1 , 0 , 1 , 0] ⊤ . Recall that ∥ w ∥ BV := ∥ w ∥ L ∞ x + T ot . V ar . ( w ; R ) , where T ot . V ar . ( w ; R ) is the total v ariation of w o ver x . Unless otherwise sp eciﬁed, all the norms with resp ect to the spatial v ariable x are taken ov er R . The key smallness condition for the initial data in this paper reads as follows: Ther e exists a smal l p ositive c onstant δ ≪ 1 such that ∥ v 0 − 1 ∥ L 1 x + ∥ v 0 ∥ B V + ∥ u 0 ∥ L 1 x + ∥ u 0 ∥ B V + ∥ θ 0 − 1 ∥ L 1 x + ∥ θ 0 ∥ B V + ∥ z 0 ∥ L 1 x + ∥ z 0 ∥ B V < δ. (0.12) Our main theorems are as follo ws: Theorem 0.2 (Local existence and regularit y) . Ther e exists a universal c onstant δ > 0 such that the fol lowing holds. Supp ose that the initial data ( v 0 , u 0 , θ 0 , z 0 ) ≡ ( u, v , θ, z )   { t =0 } satisfy (0.12) . Then ther e exists t ♯ > 0 such that Eq. (0.4) (with Lipschitz c ontinuous r e acting r ate function ϕ ) admits a we ak solution ( v , u, θ , z ) in the sense of Deﬁnition 0.1 over R × [0 , t ♯ ) . Mor e over, for some c onstant C ♯ > 0 we have the fol lowing: • R e gularity estimates:                                        max ® ∥ u ( · , t ) ∥ L 1 x , ∥ u ( · , t ) ∥ L ∞ x , ∥ u x ( · , t ) ∥ L 1 x , √ t ∥ u x ( · , t ) ∥ L ∞ x , √ t ∥ u t ( · , t ) ∥ L 1 x , t ∥ u t ( · , t ) ∥ L ∞ x ´ ≤ 2 C ♯ δ, max ® ∥ θ ( · , t ) − 1 ∥ L 1 x , ∥ θ ( · , t ) − 1 ∥ L ∞ x , ∥ θ x ( · , t ) ∥ L 1 x , √ t ∥ θ x ( · , t ) ∥ L ∞ x , √ t ∥ θ t ( · , t ) ∥ L 1 x , t ∥ θ t ( · , t ) ∥ L ∞ x ´ ≤ 2 C ♯ δ, max    ∥ v ( · , t ) ∥ B V , ∥ v ( · , t ) − 1 ∥ L 1 x , ∥ v ( · , t ) − 1 ∥ L ∞ x , √ t ∥ v t ( · , t ) ∥ L ∞ x © ≤ 2 C ♯ δ, v − 1 = v ∗ ˜ a + v ∗ j , v ∗ j ( x, t ) = P ω 0 (p ossibly smal ler than that in The or em 0.2 ab ove) such that the fol lowing holds. L et ( v ϵ 0 , u ϵ 0 , θ ϵ 0 , z ϵ 0 ) and ( v ι 0 , u ι 0 , θ ι 0 , z ι 0 ) b e two initial data, b oth verifying the smal lness c ondition (0.12) . Then, ther e exists t ∗ ≪ 1 such that the c orr esp onding we ak solutions ( v ϵ , u ϵ , θ ϵ , z ϵ ) and ( v ι , u ι , θ ι , z ι ) of Eq. (0.4) b oth exist in R × [0 , t ∗ ) , and they satisfy the stability estimate: ∥ v ϵ − v ι ∥ L 1 x + ∥ u ϵ − u ι ∥ L 1 x + ∥ θ ϵ − θ ι ∥ L 1 x + ∥ z ϵ − z ι ∥ L 1 x ≤ C 2 Ä ∥ v ϵ 0 − v ι 0 ∥ L 1 x + ∥ v ϵ 0 − v ι 0 ∥ L ∞ x + ∥ v ϵ 0 − v ι 0 ∥ B V + ∥ u ϵ 0 − u ι 0 ∥ L ∞ x + ∥ u ϵ 0 − u ι 0 ∥ L 1 x + ∥ θ ϵ 0 − θ ι 0 ∥ L ∞ x + ∥ θ ϵ 0 − θ ι 0 ∥ L 1 x + ∥ z ϵ 0 − z ι 0 ∥ L ∞ x + ∥ z ϵ 0 − z ι 0 ∥ L 1 x ä (0.15) for some C 2 > 0 . In p articular, for a ﬁxe d initial datum satisfying the c ondition (0.12) , we ak solutions to Eq. (0.4) in the sense of Deﬁnition 0.1 ar e unique. Theorem 0.4 (Global existence and large-time b ehaviour) . Ther e exists a universal c onstant δ > 0 (p ossibly smal ler than that in The or em 0.3 ab ove) such that the fol lowing holds. Supp ose that the initial datum ( v 0 , u 0 , θ 0 , z 0 ) satisﬁes the smal lness c ondition (0.12) . Then the unique lo c al solution to Eq. (0.4) c onstructe d in The or ems 0.2 and 0.3 extends glob al ly in time. Mor e over, ther e exists a p ositive c onstant C 3 such that ∥ √ t + 1( v ( · , t ) − 1) ∥ L ∞ x + ∥ √ t + 1 u ( · , t ) ∥ L ∞ x + ∥ √ t + 1( θ ( · , t ) − 1) ∥ L ∞ x + ∥ √ t + 1 z ( · , t ) ∥ L ∞ x +    √ tu x ( · , t )    L ∞ x +    √ tθ x ( · , t )    L ∞ x +    √ tz x ( · , t )    L ∞ x ≤ C 3 δ for any t ∈ (0 , + ∞ ) . The O ( t − 1 / 2 ) -deca y rate as t → ∞ of the w eak solution established in Theorem 0.4 is optimal, since it agrees with the decay rate for the heat k ernel. In this w ork, building on the framew ork in [22, 32], w e o vercome additional analytical diﬃculties caused b y the reactan t mass fraction term z . On the one hand, for the local existence of weak solutions, we adopt several tec hnical estimates for z to ensure the conv ergence of the relev ant iteration scheme. See the pro of of Lemmata 2.2 and 2.6. On the other hand, to extend the lo cal solution to the global one, one needs to further develop the Green’s matrix analysis in [22, 32]. In particular, the presence of z mak es Eq. (3.2), the linearisation of Eq. (0.4), a 4 × 4 partially degenerate h yp erb olic system. Hence, the Green’s matrix G is a 4 × 4 matrix, with a mo de λ 4 arisen from the z -equation. The b ound for z requires sev eral delicate no v el estimates for G 4 j , 1 ≤ j ≤ 4 , as well as ∂ x G 4 k and ∂ xx G 4 k , k = 2 , 3 . See, e.g. , Lemma 4.1 for details. F or future inv estigations, an important question is to determine the b ehaviour of the weak and/or strong solutions in the singular limit when all or some of the parameters µ, ν, D tend to zero. W e exp ect that the w eak solutions are unstable in the limit when all the parameters µ, ν, D → 0 , as it has been observ ed that in dimension one, steady planar detonation w av es are unstable and may ev olve in to oscillating wa v es, known as pulsating denotation w a v es. See [7, 1. In tro duction] by Chen–W agner and the references therein. On the other hand, the analysis in D. W ang [28] suggests that sho c k wa v e, turbulence, v acuum, mass or heat concen tration shall not dev elop in ﬁnite time for the partially dissipativ e system with D = 0 while µ, ν remain positive. 6 1. Preliminaries This section collects several analytical to ols that will be used throughout the paper. 1.1. Heat kernel with constant conductivit y co eﬃcien t. Deﬁnition 1.1. W e denote by K ( x, t ) the standar d he at kernel on R 2 + , namely the solution to ( ∂ t K ( x, t ) − ∂ xx K ( x, t ) = 0 , lim t → 0+ K ( x, t ) = δ ( x ) , (1.1) wher e δ ( x ) is the Dir ac delta me asur e supp orte d at x = 0 . The following prop erty of K ( x, t ) is relev ant to our later developmen ts. Lemma 1.1 (Lemma 2.1 [3]) . F or any m, j ∈ { 0 , 1 , 2 } , t ≥ 0 , and p ∈ [1 , + ∞ ] , one has that         ∂ j t ∂ m x K ( t, x )    ≤ O (1)  t 1 2 + | x |  1+2 j + m ,  R R | ∂ m x K ( t, x ) | p d x  1 p ≤ O (1) t 1 2 Ä 1 p − 1 − m ä . 1.2. BV functions on R . W e recall some prop erties of BV functions in dimension one; cf . [1]. F or an y lo cal Borel measure λ on an open set Ω ⊂ R , the Radon–Nikodým theorem guaran tees a unique decomp osition λ = λ a + λ s , where λ a and λ s are the absolutely contin uous and singular parts of λ with resp ect to the Leb esgue measure on R , resp ectively . The singular part λ s can b e further decomp osed in to a purely atomic one λ j and a diﬀuse singular one λ c . Th us, for f ∈ BV ( R ) one has the decomposition D f = D a f + D c f + D j f for the distributional deriv ativ e D f , where D a f , D c f , and D j f are the absolutely con tinuous part, singularly contin uous part, and purely atomic part of D f , resp ectively . Here D j f con- tributes to the jump p art of f , denoted as f j ( x ) := X ω ∈D ,ω 0 and h ( t ) = 0 for t ≤ 0 . The total v ariation of f ∈ B V ( R ) is th us ∥ f ∥ B V = | D a f | ( R ) + | D c f | ( R ) + | D j f | ( R ) = Z R | ∂ x f a | d x + Z R | d f c | + X ω ∈D    f j ( x ) | x = ω + x = ω −    . (1.3) F or our purp ose of establishing BV -estimates for f , it has b een sho wn in [32, Lemma 3.4] that the estimates for the absolutely con tin uous and singular parts are similar. Hence w e shall fo cus only on the absolutely con tin uous part in what follo ws. W rite f ˜ a ≡ f a + f c . Note that for f ∈ B V ∩ L 1 x ( R ) , one has that f ( x ) → 0 as | x | → ∞ . Also, ∥ f ∥ L ∞ x ( R ) = sup x ∈ R     f ( x ) − lim | x |→ R f ( x )     ≤ ∥ f ∥ B V ( R ) . (1.4) 1.3. Heat k ernel with BV-conductivity co eﬃcien t. Now, let us deﬁne the fundamental solution H ( x, t ; y , t 0 ; f ) for the heat equation with time-dep endent BV conductivit y co eﬃcients. Here x, t are v ariables and y , t 0 , f are parameters for the function H . 7 Deﬁnition 1.2 (F undamental solution) . Assume a function f ( x, t ) satisﬁes the fol lowing c on- ditions: ther e exist p ositive c onstants ¯ f and δ ∗ ≪ 1 such that ( ∥ f ( · ) − ¯ f ∥ L 1 x ≤ δ ∗ , ∥ f ( · , t ) ∥ B V ≤ δ ∗ , ∥ f t ( · , t ) ∥ L ∞ x ≤ δ ∗ max Ä 1 √ t , 1 ä , D ≡ { ω | f ( ω , t ) is not c ontinuous at ω } is invariant with r esp e ct to t. (1.5) The fundamental solution H ( x, t ; y , t 0 ; f ) is the we ak solution to the initial value pr oblem ( ∂ t H ( x, t ; y , t 0 ; f ) − ∂ x  f ( x, t ) ∂ x H ( x, t ; y , t 0 ; f )  = 0 for t 0 < t, H ( x, t 0 ; y , t 0 ; f ) = δ ( x − y ) . (1.6) That is, for any smo oth test function ψ ∈ C ∞ c ( R × [ t 0 , ∞ )) , Z ∞ t 0 Z R H ( x, t ; y , t 0 ; f ) n − ∂ t ψ ( x, t ) − ∂ x ( f ( x, t ) ∂ x ψ ( x, t )) o d x d t = ψ ( y , t 0 ) . (1.7) Lemma 1.2 (Lemma 2.6 in [22]) . The fundamental solution H of the system (1.6) satisﬁes                R R H ( x, t ; y , τ ; f ) d x = R R H ( x, t ; y , τ ; f ) d y = 1 , R R H x ( x, t ; y , τ ; f ) d x = R R H x ( x, t ; y , τ ; f ) d y = 0 , R R H y ( x, t ; y , τ ; f ) d x = R R H y ( x, t ; y , τ ; f ) d y = 0 , R R H t ( x, t ; y , τ ; f ) d x = R R H t ( x, t ; y , τ ; f ) d y = 0 , R R H τ ( x, t ; y , τ ; f ) d x = R R H τ ( x, t ; y , τ ; f ) d y = 0 . (1.8) The following lemmas pro vide p oint wise, deriv ative, and integral b ounds for the funda- men tal solution H ( x, t ; y , t 0 ; f ) . Lemma 1.3 (Liu–Y u [22]) . Supp ose f satisﬁes the c onditions in (1.5) . Then ther e exist p ositive c onstants C ∗ and t ♯ ≪ 1 such that the we ak solution to Eq. (1.6) exists and satisﬁes | H ( x, t ; y , t 0 ; f ) | ≤ C ∗ e − ( x − y ) 2 C ∗ ( t − t 0 ) √ t − t 0 , | H x ( x, t ; y , t 0 ; f ) | + | H y ( x, t ; y , t 0 ; f ) | ≤ C ∗ e − ( x − y ) 2 C ∗ ( t − t 0 ) t − t 0 ,     Z t t 0 H x ( x, τ ; y , t 0 ; f ) d τ     ,     Z t t 0 H x ( x, t ; y , s ; f ) d s     ≤ C ∗ e − ( x − y ) 2 C ∗ ( t − t 0 ) for t ∈ ( t 0 , t 0 + t ♯ ) . Lemma 1.4 (Lemma 2.2 in W ang–Y u–Zhang [32]) . Under the same assumptions in L emma 1.3, ther e exists a c onstant C ∗ > 0 such that for any t ∈ ( t 0 , t 0 + t ♯ ) , the we ak solution H of Eq. (1.6) satisﬁes the fol lowing: | H xy ( x, t ; y , t 0 ; f ) | + | H t ( x, t ; y , t 0 ; f ) | ≤ C ∗ e − ( x − y ) 2 C ∗ ( t − t 0 ) ( t − t 0 ) 3 2 , (1.9) | H ty ( x, t ; y , t 0 ; f ) | ≤ C ∗ e − ( x − y ) 2 C ∗ ( t − t 0 ) ( t − t 0 ) 2 ,     Z t t 0 H xy ( x, τ ; y , t 0 ; f )d τ − δ ( x − y ) f ( x, t 0 ) − Z t t 0 f ( x, t 0 ) − f ( x, τ ) f ( x, t 0 ) H xy ( x, τ ; y , t 0 ; f ) d τ     ≤ C ∗ e − ( x − y ) 2 C ∗ ( t − t 0 ) √ t − t 0 ,     Z t t 0 H xy ( x, t ; y , s ; f ) d s + δ ( x − y ) f ( y , t ) 8 − Z t t 0 f ( y , t ) − f ( y , s ) f ( y , t ) H xy ( x, t ; y , s ; f ) d s     ≤ C ∗ e − ( x − y ) 2 C ∗ ( t − t 0 ) √ t − t 0 , Z t t 0 H xx ( x, τ ; y , t 0 ; f ) d τ = − δ ( x − y ) f ( x, t 0 ) − 1 f ( x, t 0 ) ∂ x ï Z t t 0 ( f ( x, τ ) − f ( x, t 0 )) H x ( x, τ ; y , t 0 ; f ) d τ ò + O (1) Ñ | ∂ x f ( x, t 0 ) | e − ( x − y ) 2 C ∗ ( t − t 0 ) + e − ( x − y ) 2 C ∗ ( t − t 0 ) √ t − t 0 é , for x / ∈ D , Z t t 0 H xxy ( x, τ ; y , t 0 ; f ) d τ = 1 f ( x, t 0 ) × ï δ ′ ( x − y ) − Z t t 0 ∂ x [( f ( x, τ ) − f ( x, t 0 )) H xy ( x, τ ; y , t 0 ; f )] d τ ò − ∂ x f ( x, t 0 ) f 2 ( x, t 0 ) ï δ ( x − y ) − Z t t 0 ( f ( x, τ ) − f ( x, t 0 )) H xy ( x, τ ; y , t 0 ; f ) d τ ò + O (1) Ñ | ∂ x f ( x, t 0 ) | e − ( x − y ) 2 C ∗ ( t − t 0 ) √ t − t 0 + e − ( x − y ) 2 C ∗ ( t − t 0 ) t − t 0 é , for x / ∈ D , Z t t 0 H t ( x, t ; y , s ; f ) d s = H ( x, t − t 0 ; y ; f ( · , t )) − δ ( x − y ) + O (1) δ ∗ e − ( x − y ) 2 C ∗ ( t − t 0 ) . (1.10) Next, we set the follo wing norms ∥| f |∥ ∞ ≡ sup t ∈ ( 0 ,t ♯ ) ∥ f ( · , t ) ∥ L ∞ x , ∥| f |∥ 1 ≡ sup t ∈ ( 0 ,t ♯ ) ∥ f ( · , t ) ∥ L 1 x , (1.11) ∥| f |∥ B V ≡ sup t ∈ ( 0 ,t ♯ ) ∥ f ( · , t ) ∥ B V , and the lik e. W e hav e the follo wing comparison estimates for heat k ernels with diﬀeren t conduc- tivit y co eﬃcien ts. Lemma 1.5 (Corollaries 4.4 and 4.5 in [22]) . Supp ose that f a and f b b oth satisfy the c onditions in (1.5) . Then ther e ar e p ositive c onstants t ♯ ≪ 1 and C ∗ such that for any t ∈ ( t 0 , t 0 + t ♯ ) ,    H ( x, t ; y , t 0 ; f b ) − H ( x, t ; y , t 0 ; f a )    ≤ C ∗ e − ( x − y ) 2 C ∗ ( t − t 0 ) √ t − t 0       f a − f b       ∞ ,    H x ( x, t ; y , t 0 ; f a ) − H x ( x, t ; y , t 0 ; f b )    +    H y ( x, t ; y , t 0 ; f a ) − H y ( x, t ; y , t 0 ; f b )    ≤ C ∗ e − ( x − y ) 2 C ∗ ( t − t 0 ) t − t 0 h | log( t − t 0 )       f a − f b       ∞ +       f a − f b       B V + √ t − t 0 Å       f a − f b       1 + | log t |         √ τ | log τ | ∂ τ î f a − f b ó         ∞ ãò ,     Z t t 0 î H x ( x, τ ; y , t 0 ; f a ) − H x ( x, τ ; y , t 0 ; f b ) ó d τ     9 ≤ C ∗ e − ( x − y ) 2 C ∗ ( t − t 0 ) h       f a − f b       ∞ +       f a − f b       B V +       f a − f b       1 +         √ τ | log τ | ∂ τ Ä f a − f b ä         ∞ ò . Lemma 1.6 (Lemma 2.5 in [32]) . Under the assumptions in L emma 1.5, ther e exist p ositive c onstants t ♯ ≪ 1 and C ∗ such that for any t ∈ ( t 0 , t 0 + t ♯ ) , it holds that    H xy ( x, t ; y , t 0 ; f a ) − H xy ( x, t ; y , t 0 ; f b )    +    H t ( x, t ; y , t 0 ; f a ) − H t ( x, t ; y , t 0 ; f b )    ≤ C ∗ e − ( x − y ) 2 C ∗ ( t − t 0 ) ( t − t 0 ) 3 / 2 h | log ( t − t 0 ) |       f a − f b       ∞ +       f a − f b       B V + √ t − t 0 Å       f a − f b       1 + | log t |         √ τ | log τ | ∂ τ î f a − f b ó         ∞ ãò ,     Z t t 0 î H y ( x, t ; y , s ; f a ) − H y ( x, t ; y , s ; f b ) ó d s     ≤ C ∗ e − ( x − y ) 2 C ∗ ( t − t 0 ) h       f a − f b       ∞ +       f a − f b       B V +       f a − f b       1 +         √ τ | log τ | ∂ τ ( f a − f b )         ∞ ò ,     Z t t 0 î H x ( x, t ; y , s ; f a ) − H x ( x, t ; y , s ; f b ) ó d s     ,     Z t t 0 î H y ( x, τ ; y , t 0 ; f a ) − H y ( x, τ ; y , t 0 ; f b ) ó d τ     ≤ C ∗ e − ( x − y ) 2 C ∗ ( t − t 0 ) h       f a − f b       ∞ +       f a − f b       B V +       f a − f b       1 +         √ τ | log τ | ∂ τ ( f a − f b )         ∞ ò , Z t t 0 î H xy ( x, τ ; y , t 0 ; f a ) − H xy ( x, τ ; y , t 0 ; f b ) ó d τ = ï 1 f a ( x, t 0 ) − 1 f b ( x, t 0 ) ò δ ( x − y ) − Z t t 0 ï f a ( x, τ ) − f a ( x, t 0 ) f a ( x, t 0 ) H xy ( x, τ ; y , t 0 ; f a ) − f b ( x, τ ) − f b ( x, t 0 ) f b ( x, t 0 ) H xy ( x, τ ; y , t 0 ; f b ) ô d τ + O (1) e − ( x − y ) 2 C ∗ ( t − t 0 ) √ t − t 0 h | log( t − t 0 )       f a − f b       ∞ +       f a − f b       B V +       f a − f b       1 +         √ τ | log τ | ∂ τ ( f a − f b )         ∞ ò , Z t t 0 î H xy ( x, t ; y , s ; f a ) − H xy ( x, t ; y , s ; f b ) ó d s = ï 1 f a ( y , t ) − 1 f b ( y , t ) ò δ ( x − y ) + Z t t 0 ï f a ( y , t ) − f a ( y , s ) f a ( y , t ) H xy ( x, t ; y , s ; f a ) − f b ( y , t ) − f b ( y , s ) f b ( y , t ) H xy ( x, t ; y , s ; f b ) ô d s + O (1) e − ( x − y ) 2 C ∗ ( t − t 0 ) √ t − t 0 h | log( t − t 0 ) |       f a − f b       ∞ 10 +       f a − f b       B V +       f a − f b       1 +         √ τ | log τ | ∂ τ Ä f a − f b ä         ∞ ò . Remark 1.3. The fol lowing useful observations ar e state d in [22, 32]. (1) F or the b ackwar d he at e quation ® ( ∂ τ + ∂ y f ( y , τ ) ∂ y ) H ( x, t ; y , τ ; f ) = 0 , τ < t, H ( x, t ; y , t ; f ) = δ ( x − y ) , (1.12) the pr o duct f ( y , τ ) H y ( x, t ; y , τ ; f ) is c ontinuous with r esp e ct to y . (2) The we ak solution to the he at e quation (1.6) c an b e deﬁne d similarly as p er Deﬁnition 1.2. Consider the e quation (1.6) with a sour c e term c onservative form, u t ( x, t ) = ( f ( t, x ) u x ( x, t ) + g ( x, t )) x , wher e g ( x, t ) ∈ B V with r esp e ct to x . The mild solution c onstructe d by he at kernel and Duhamel principle is also a we ak solution to the ab ove e quation in the distributional sense. F urthermor e, the ﬂux term ( f ( t, x ) u x ( x, t ) + g ( x, t )) is c ontinuous with r esp e ct to x if one of the fol lowing two c onditions holds: • g ( x, t ) is Lipschitz c ontinuous in x , i.e., ∥ g x ( x, t ) ∥ L ∞ x < + ∞ ; • g ( x, t ) is Hölder c ontinuous in t , i.e., ther e exist C > 0 and 0 < α < 1 such that | g ( x, t ) − g ( x, s ) | ≤ C ( t − s ) α s α , 0 < s < t. 1.4. A lemma from F ourier anlaysis. The follo wing lemma will be used in Section 3. Lemma 1.7 (Prop osition 2.2 in [22]) . L et ˆ f ( η ) b e the F ourier tr ansform of f ( x ) . Supp ose that ˆ f ( η ) is analytic in the r e gion | Im( η ) | < σ 0 and ˆ f ( η ) has the asymptotic pr op erty: ˆ f ( η ) = O (1) · η (1 + | η | ) 2 m +2 + O (1) · 1 (1 + | η | ) 2 m +2 , m ≥ 0 , η → ∞ . Then f ( x ) ∈ H 2 m ( R ) and 2 m X j =0     d j d x j f ( x )     = O (1) e − σ 0 | x | , x ∈ R . 2. Local solution This section is dev oted to the construction of lo cal-in-time w eak solutions to Eq. (0.4). F or con v enience of the reader, w e repro duce Eq. (0.4) as follo ws.                v t − u x = 0 , u t + p x =  µu x v  x , θ t + p c v u x − µ c v v ( u x ) 2 = Ä ν c v v θ x ä x + q c v K ϕ ( θ ) z , z t + K ϕ ( θ ) z =  D v 2 z x  x , ( v , u, θ, z ) | t =0 = ( v 0 , u 0 , θ 0 , z 0 ) , (2.1) where the initial datum is a p erturbation around the constan t state [1 , 0 , 1 , 0] ⊤ . W e set v 0 = 1 + v ∗ 0 , u 0 = u ∗ 0 , θ 0 = 1 + θ ∗ 0 , z 0 = z ∗ 0 , 11 and requires as in (0.12) that ∥ v ∗ 0 ∥ B V ∩ L 1 x + ∥ u ∗ 0 ∥ B V ∩ L 1 x + ∥ θ ∗ 0 ∥ B V ∩ L 1 x + ∥ z ∗ 0 ∥ B V ∩ L 1 x ≤ δ ≪ 1 . (2.2) The existence of lo cal solutions shall b e established in a standard w ay . One ﬁrst constructs a sequence of approximate solutions via an iterativ e scheme, then derive uniform a priori estimates for this sequence using Lemmas 1.3 – 1.6, and ﬁnally prov e the con vergence of the approximate sequence in suitable top ologies via compactness arguments. 2.1. Iteration sc heme. F ollowing Itay a [16], we construct a sequence of approximate solutions ( V n , U n , Θ n , Z n ) via an iteration sc heme. At each step n + 1 , w e solve the system                        V n +1 t − U n +1 x = 0 , U n +1 t −  µU n +1 x 1+ V n  x = − p (1 + V n , 1 + Θ n ) x , Θ n +1 t −  ν Θ n +1 x c v (1+ V n )  x = − p (1+ V n , 1+Θ n ) c v U n x + µ c v (1+ V n ) ( U n x ) 2 + q c v K ϕ (1 + Θ n ) Z n , Z n +1 −  DZ n +1 x (1+ V n ) 2  x = − K ϕ (1 + Θ n ) Z n ,  V n +1 , U n +1 , Θ n +1 , Z n +1    t =0 = ( v ∗ 0 , u ∗ 0 , θ ∗ 0 , z ∗ 0 ) ,  V 0 , U 0 , Θ 0 , Z 0  = (0 , 0 , 0 , 0) , (2.3) The base step ( n = 0 ) for this iteration is the following linear problem:                V 1 t − U 1 x = 0 , U 1 t −  µU 1 x  x = 0 , Θ 1 t −  ν Θ 1 x b  x = 0 , Z 1 −  D Z 1 x  x = 0 ,  V 1 , U 1 , Θ 1 , Z 1    t =0 = ( v ∗ 0 , u ∗ 0 , θ ∗ 0 , z ∗ 0 ) , (2.4) By using Lemma 1.1, w e ha ve the following estimates for  V 1 , U 1 , Θ 1 , Z 1  . Lemma 2.1. Supp ose that the initial datum ( v ∗ 0 , u ∗ 0 , θ ∗ 0 , z ∗ 0 ) satisﬁes the smal lness c ondition (2.2) . Ther e exists a p ositive c onstant C ♯ such that for any 0 < t < t ♯ , the solution to Eq. (2.4) satisﬁes                                              max ¶   U 1 ( · , t )   L 1 x ,   U 1 ( · , t )   L ∞ x ,   U 1 x ( · , t )   L 1 x , √ t   U 1 x ( · , t )   L ∞ x , t   U 1 t ( · , t )   L ∞ x © ≤ C ♯ δ, 0 < t < t ♯ , max ¶   Θ 1 ( · , t )   L 1 x ,   Θ 1 ( · , t )   L ∞ x ,   Θ 1 x ( · , t )   L 1 x , √ t   Θ 1 x ( · , t )   L ∞ x , t   Θ 1 t ( · , t )   L ∞ x © ≤ C ♯ δ, 0 < t < t ♯ , max ¶ √ t   V 1 t ( · , t )   L ∞ x ,   V 1 ( · , t )   B V ,   V 1 ( x, t )   L 1 x ,   V 1 ( x, t )   L ∞ x © ≤ C ♯ δ, 0 < t < t ♯ , max ¶   Z 1 ( · , t )   L 1 x ,   Z 1 ( · , t )   L ∞ x ,   Z 1 x ( · , t )   L 1 x , √ t   Z 1 x ( · , t )   L ∞ x , t   Z 1 t ( · , t )   L ∞ x © ≤ C ♯ δ, 0 < t < t ♯ , ∥ V 1 ( · , t ) − V 1 ( · , s ) ∥ B V ≤ C ♯ t − s √ t δ, 0 ≤ s ≤ t < t ♯ ,    V 1 ( · , t )   x = z + x = z −    =    v ∗ 0 ( · ) | x = z + x = z −    , z ∈ D , 0 < t < t ♯ , wher e t ♯ > 0 is suﬃciently smal l c onstructe d in L emma 1.3, and D is the disc ontinuity set of v ∗ 0 . 12 With Lemma 2.1 at hand, we no w pro ceed with induction. Fix k ∈ N . Assume for eac h n ≤ k that Eq. (2.3) has a solution ( V n , U n , Θ n , Z n ) that satisﬁes the follo wing:                            0 < δ, t ♯ ≪ 1 , 1 ≤ n ≤ k , 0 < t < t ♯ , max ¶ ∥ U n ( · , t ) ∥ L 1 x , ∥ U n ( · , t ) ∥ L ∞ x , ∥ U n x ( · , t ) ∥ L 1 x , √ t ∥ U n x ( · , t ) ∥ L ∞ x © ≤ 2 C ♯ δ, max ¶ ∥ Θ n ( · , t ) ∥ L 1 x , ∥ Θ n ( · , t ) ∥ L ∞ x , ∥ Θ n x ( · , t ) ∥ L 1 x , √ t ∥ Θ n x ( · , t ) ∥ L ∞ x © ≤ 2 C ♯ δ, max ¶ ∥ Z n ( · , t ) ∥ L 1 x , ∥ Z n ( · , t ) ∥ L ∞ x , ∥ Z n x ( · , t ) ∥ L 1 x , √ t ∥ Z n x ( · , t ) ∥ L ∞ x © ≤ 2 C ♯ δ, max ¶ ∥ V n ( · , t ) ∥ B V , ∥ V n ( · , t ) ∥ L 1 x , ∥ V n ( · , t ) ∥ L ∞ x , √ t ∥ V n t ( · , t ) ∥ L ∞ x © ≤ 2 C ♯ δ, ∥ V n ( · , t ) − V n ( · , s ) ∥ B V ≤ 2 C ♯ δ ( t − s ) | log( t − s ) | √ t , 0 ≤ s < t,    V n ( · , t ) | x = ω + x = ω −    ≤ 2    v ∗ 0 ( · ) | x = ω + x = ω −    , ω ∈ D . (2.5) The case n = 1 follo ws from Lemma 2.1. W e will sho w (2.5) for  V k +1 , U k +1 , Θ k +1 , Z k +1  . T o this end, let us ﬁrst in troduce sev eral notations: µ k ≡ µ 1 + V k , N k 1 ( x, t ) ≡ − ∂ x p Ä 1 + V k , 1 + Θ k ä , ν k ≡ ν c v (1 + V k ) , D k ≡ D (1 + V k ) 2 , (2.6) N k 2 ( x, t ) ≡ − p  1 + V k , 1 + Θ k  c v U k x + µ c v (1 + V k ) Ä U k x ä 2 + q c v K ϕ (1 + Θ k ) Z k . Then, in view of the Duhamel principle, w e hav e the expressions: V k +1 ( t, x ) = v ∗ 0 ( x ) + Z t 0 U k +1 x ( x, s ) d s, (2.7) U k +1 ( x, t ) = Z R H ( x, t ; y , 0; µ k ) u ∗ 0 ( y ) d y + Z t 0 Z R \D H y Ä x, t ; y , s ; µ k ä p Ä 1 + V k , 1 + Θ k ä d y d s (2.8) = : I u 1 + I u 2 , Θ k +1 ( x, t ) = Z R H ( x, t ; y , 0; ν k ) θ ∗ 0 ( y ) d y + Z t 0 Z R H ( x, t ; y , s ; ν k ) N k 2 ( y , s ) d y d s (2.9) = : I θ 1 + I θ 2 , Z k +1 ( x, t ) = Z R H ( x, t ; y , 0; D k ) z ∗ 0 ( y ) d y − Z t 0 Z R H ( x, t ; y , s ; D k ) K ϕ (1 + Θ k ) Z k d y d s (2.10) = : I z 1 + I z 2 . By adapting the arguments in W ang–Y u–Zhang [32], we readily obtain the follo wing: Lemma 2.2. F or δ, t ♯ > 0 suﬃciently smal l, Eq. (2.5) holds for ( V k +1 , U k +1 , Θ k +1 , Z k +1 ) when- ever 0 < t < t ♯ . Sektch of pr o of for L emma 2.2. Our arguments are an adaptation of [32], so only the diﬀerences are emphasised. Ke y ingredients are the fundamental solution estimates in Lemmas 1.3 – 1.6. Throughout the proof, constan ts O (1) and C ♯ are indep endent of δ , k , and t whenever t < t ♯ . 13 T o b egin with, in view of Eq. (2.5), w e ﬁnd that D k satisﬁes Eq. (1.5). Thus, b y applying Lemma 1.3 one obtains the follo wing: (1) Estimate of ∥ Z k +1 ∥ L 1 x : In tegrating the fourth equation in (2.3) o ver [0 , t ] × R , we obtain Z R Z k +1 ( x, t ) d x + Z t 0 Z R K ϕ (1 + Θ k ) Z k ( y , τ ) d y d τ ≤ Z R z ∗ 0 ( x ) d x. (2.11) Since ϕ is Lipsc hitz contin uous and satisﬁes ϕ ( θ ) ≥ 0 , it follo ws from the smallness condition on the initial data (2.2) that Z t 0 Z R K ϕ (1 + Θ k ) Z k ( y , τ ) d y d τ ≤ δ, (2.12) and    Z k +1    L 1 x ≤ ∥ z ∗ 0 ∥ L 1 x ≤ C ♯ δ. (2.13) (2) Estimate of ∥ Z k +1 ∥ L ∞ x : F rom (2.10) and the estimate | H ( x, t ; y , s ; D k ) | ≤ O (1) e − ( x − y ) 2 C ∗ ( t − s ) √ t − s in Lemma 1.3, for suﬃcien tly small δ ∥ Z k +1 ∥ L ∞ x ≤ Z R    H ( x, t ; y , 0; D k )    | z ∗ 0 ( y ) | d y + Z t 0 Z R    H ( x, t ; y , s ; D k )    K ϕ (1 + Θ k ) Z k ( y , s ) d y d s ≤ O (1) Z R e − ( x − y ) 2 C ∗ t √ t d y · ∥ z ∗ 0 ∥ L ∞ x + O (1) Z t 0 Z R e − ( x − y ) 2 C ∗ ( t − s ) √ t − s    Z k    L ∞ x d y d s ≤ O (1) δ + O (1) tδ ≤ 2 C ♯ δ. (2.14) (3) Estimate of ∥ Z k +1 x ∥ L 1 x : Diﬀeren tiating (2.10) with resp ect to x gives Z k +1 x ( x, t ) = Z R H x ( x, t ; y , 0; D k ) z ∗ 0 ( y ) d y − Z t 0 Z R H x ( x, t ; y , s ; D k ) K ϕ (1 + Θ k ) Z k d y d s. (2.15) Since R R H x ( x, t ; ω , 0; D k ) d ω = 0 , w e deﬁne the an ti-deriv ative W ( x, t ; y , 0; D k ) =    R y −∞ H x ( x, t ; w , 0; D k ) d w , for y < x, − R ∞ y H x ( x, t ; w , 0; D k ) d w , for y ≥ x. (2.16) Since z ∗ 0 is a BV function, in tegration b y parts for the Stieltjes integral yields Z R     Z R d W ( x, t ; y , 0; D k ) z ∗ 0 ( y )     d x = Z R     Z R W ( x, t ; y , 0; D k ) d z ∗ 0 ( y )     d x ≤ Z R Z R    W ( x, t ; y , 0; D k )    | d z ∗ 0 ( y ) | d x ≤ O (1) Z R e − ( x − y ) 2 C ∗ t √ t d x · ∥ z ∗ 0 ∥ B V ≤ C ♯ δ. 14 Using the estimate | H x ( x, t ; y , s ; D k ) | ≤ O (1) e − ( x − y ) 2 C ∗ ( t − s ) t − s from Lemma 1.3 and the ansatz (2.5), we obtain for suﬃcien tly small δ ∥ Z k +1 x ∥ L 1 x ≤ Z R Z R    H x ( x, t ; y , 0; D k )    | z ∗ 0 ( y ) | d y d x + Z R Z t 0 Z R    H x ( x, t ; y , s ; D k )    K ϕ (1 + Θ k ) Z k ( y , s ) d y d s d x ≤ O (1) Z R e − ( x − y ) 2 C ∗ t √ t d x · ∥ z ∗ 0 ∥ B V + O (1) Z R Z t 0 Z R e − ( x − y ) 2 C ∗ ( t − s ) t − s K ϕ (1 + Θ k ) Z k ( y , s ) d y d s d x ≤ O (1) δ + O (1) Z t 0       e − x 2 C ∗ ( t − s ) t − s       L 1 x    Z k    L 1 x d s ≤ O (1) δ + O (1) Z t 0 1 √ t − s δ d s ≤ O (1)( δ + √ tδ ) ≤ 2 C ♯ δ. (2.17) (4) Estimate of ∥ Z k +1 x ∥ L ∞ x : Similarly , for suﬃcien tly small δ ∥ Z k +1 x ∥ L ∞ x ≤ Z R    H x ( x, t ; y , 0; D k )    | z ∗ 0 ( y ) | d y + Z t 0 Z R    H x ( x, t ; y , s ; D k )    K ϕ (1 + Θ k ) Z k d y d s ≤ O (1) Z R e − ( x − y ) 2 C ∗ t t d y ∥ z ∗ 0 ∥ L ∞ x + O (1) Z t 0 Z R e − ( x − y ) 2 C ∗ ( t − s ) t − s    Z k    L ∞ x d y d s ≤ O (1) δ √ t + O (1) δ Z t 0 1 √ t − s d s ≤ O (1) Å δ √ t + δ √ t ã ≤ 2 C ♯ δ √ t . (2.18) Com bining the estimates in steps (1)–(4), we conclude that Z k +1 satisﬁes the b ounds in (2.5) for suﬃcien tly small δ and t ♯ . Similar estimates hold for V k +1 , U k +1 , and Θ k +1 analogously . This completes the induction. □ W e next sho w the Lipsc hitz contin uit y of V k +1 in time across the jump set. Lemma 2.3. F or suﬃciently smal l δ > 0 and t ♯ > 0 , V k +1 satisﬁes the fol lowing Lipschitz c ontinuity estimate for 0 ≤ s < t < t ♯ : X ω ∈D     V k +1 x ( · , t )    ω + ω − − V k +1 x ( · , s )    ω + ω −     ≤ O (1) Z t s Å 1 + 1 √ τ ã d τ X ω ∈D    v ∗ 0 ( · ) | ω + ω −    ≤ O (1) t − s √ t . Pr o of. Our argumen ts are directly adapted from [32, Lemma 3.4]. Since µU k x 1 + V k − 1 − p (1 + V k − 1 , 1 + Θ k − 1 ) 15 is contin uous in x b y Remark 1.3, we ha ve that V k +1 t ( · , t )    ω + ω − = U k +1 x ( · , t )    ω + ω − = V k µ ( · , t )      ω + ω − Ç µU k x 1 + V k − 1 − p (1 + V k − 1 , 1 + Θ k − 1 ) å + a (1 + Θ k ) µ ( · , t )      ω + ω − . Th us, integrating this iden tit y ov er [0 , t ] and applying Lemma 2.2, we conclude that     V k +1 ( · , t )    ω + ω −     ≤    v ∗ 0 ( · ) | ω + ω −    + 1 µ Z t 0 Ñ µ   U k +1 x   ( · , s ) 1 − ∥ V k ∥ L ∞ x + 2 a Ä 1 +   Θ k   L ∞ x ä 1 − ∥ V k ∥ L ∞ x é d s × sup 0 ≤ s ≤ t     V k ( · , s )    ω + ω −     ≤ O (1) Ä 1 + √ t + t ä    v ∗ 0 ( · ) | ω + ω −    ≤ 2    v ∗ 0 ( · ) | ω + ω −    . (2.19) □ The following result is a v arian t of [32, Lemma 3.2], but the presence of Z k in the N 2 term (see Eq. (2.6)) brings about essen tial diﬀerences. W e th us provide a detailed proof below. Lemma 2.4. F or suﬃciently smal l δ and t ♯ , the fol lowing time diﬀer enc e estimates hold for Θ k +1 , whenever 0 < s ≤ t < t ♯ :    Θ k +1 ( · , t ) − Θ k +1 ( · , s )    L ∞ x ≤ O (1) Ç ( δ + δ 2 ) t − s √ s √ t + δ t − s √ t + δ √ t − s + δ 2 √ t − s √ s å , (2.20)    Θ k +1 ( · , t ) − Θ k +1 ( · , s )    L 1 x ≤ O (1) Ç δ t − s √ s √ t + δ ( t − s ) + δ 2 t − s √ t + δ √ t − s √ s + δ 2 √ t − s å    Θ k +1 x ( · , t ) − Θ k +1 x ( · , s )    L 1 x ≤ O (1) Ç √ t − s | log ( t − s ) | √ t √ s δ + √ t − sδ + √ t − s √ t δ 2 + √ t − s | log ( t − s ) | δ 2 å . Pr o of. In view of Eq. (2.5), ν k satisﬁes Eq. (1.5), so we can apply Lemmas 1.3 and 1.4 to derive the required estimates. Indeed, b y the represen tation formula for Θ k +1 in Eq. (2.9), we ha v e Θ k +1 ( y , t ) − Θ k +1 ( y , s ) = Z R Ä H ( y, t ; ω , 0; ν k ) − H ( y, s ; ω , 0; ν k ) ä θ ∗ 0 ( ω ) d ω + Z t s Z R H ( y, t ; ω , τ ; ν k ) N k 2 ( ω , τ ) d ω d τ + Z s 0 Z R Ä H ( y, t ; ω , τ ; ν k ) − H ( y, s ; ω , τ ; ν k ) ä N k 2 ( ω , τ ) d ω d τ = I 1 + I 2 + I 3 . 16 (1) F or I 1 , we use the estimate of H t from (1.9) in Lemma 1.4 to obtain |I 1 | =     Z R Z t s H σ ( y , σ ; ω , 0; ν k ) θ ∗ 0 ( ω ) d σ d ω     ≤ Z R Z t s    H σ ( y , σ ; ω , 0; ν k )    d σ d ω ∥ θ ∗ 0 ∥ L ∞ x ≤ O (1) δ Z R Z t s e − ( y − ω ) 2 C ∗ σ σ 3 2 d σ d ω ≤ O (1) δ Z t s 1 σ d σ ≤ O (1) δ Z t s 1 √ s √ σ d σ ≤ O (1) δ ( t − s ) √ s √ t . (2) F or I 2 , using the expression for N k 2 in (2.6), the estimate for H from Lemma 1.3, and the ansatz (2.5), we ha v e: |I 2 | ≤ Z t s Z R    H ( y, t ; ω , τ ; ν k )    Ç      a (1 + Θ k ) c v (1 + V k ) U k y ( ω , τ )      +      µ  U k y ( ω , τ ) 2  c v (1 + V k )      + q c v K ϕ (1 + Θ k ) Z k ( ω , τ ) ã d ω d τ ≤ O (1) Z t s Z R e − ( y − ω ) 2 C ∗ ( t − τ )) √ t − τ Å δ √ τ + δ 2 τ ã d ω d τ + O (1) Z t s 1 √ t − τ Z R Z k ( ω , τ ) d ω d τ ≤ O (1) Z t s Å δ √ τ + δ 2 τ + δ √ t − τ ã d τ ≤ O (1) Ç δ t − s √ t + δ 2 t − s √ t √ s + δ √ t − s å . (3) Similarly , for I 3 , comb ining the expression of N k 2 in (2.6), the estimate for H t from Lemma 1.4, and the ansatz (2.5), we obtain |I 3 | =     Z s 0 Z R Z t s H σ ( y , σ ; ω , τ ; ν k ) d σ N k 2 ( ω , τ ) d ω d τ     ≤ O (1) Z s 0 Z R Z t s e − ( y − ω ) 2 C ∗ ( σ − τ )) ( σ − τ ) 3 / 2 d σ ï    U k ω    Å δ + δ √ τ ã + Z k ( ω , τ ) ò d ω d τ ≤ O (1) Z s 0 Z R Å 1 √ s − τ − 1 √ t − τ ã h   U k ω    δ + Z k ( ω , τ ) i d ω d τ + O (1) Z s 2 0 Z R Å 1 √ s − τ − 1 √ t − τ ã    U k ω    δ √ τ d ω d τ + Z s s 2 Z t s 1 σ − τ δ 2 τ d σ d τ ≤ O (1)( δ + δ 2 ) Z s 0 Å 1 √ s − τ − 1 √ t − τ ã d τ + O (1) δ 2 Z s 2 0 Å 1 √ s − τ − 1 √ t − τ ã 1 √ τ d τ + O (1) δ 2 Z s s 2 1 √ s − τ Z t s 1 √ σ − τ 1 τ d σ d τ ≤ O (1)( δ + δ 2 )( t − s √ t + √ t − s ) + O (1) δ 2 √ t − s √ s + O (1) δ 2 √ s Z s s 2 1 √ s − τ t − s √ t − τ 1 √ τ d τ ≤ O (1)( δ + δ 2 )( t − s √ t + √ t − s ) + O (1) δ 2 √ t − s √ s . 17 W e com bining the estimates for I 1 , I 2 , and I 3 to conclude Eq. (2.20). □ 2.2. Con v ergence of the iteration scheme. Our goal is to establish the conv ergence of the iteration scheme in Eq. (2.3) for t ∈ (0 , t ♯ ) . T o this end, w e need to estimate the diﬀerences b et ween adjacen t terms in the sequence of approximate solutions ( V n , U n , Θ n , Z n ) , with resp ect to the norms ( ∥|·|∥ ∞ , ∥|·|∥ 1 , ∥|·|∥ B V ) deﬁned in Eq. (1.11). Observ e that the PDE for ( V n +1 − V n , U n +1 − U n , Θ n +1 − Θ n , Z n +1 − Z n ) is as follows:                        ∂ t ( V n +1 − V n ) − ∂ x ( U n +1 − U n ) = 0 , ∂ t ( U n +1 − U n ) − ∂ x  µ ( U n +1 − U n ) x 1+ V n  = − ∂ x  µU n x ( V n − V n − 1 ) (1+ V n )(1+ V n − 1 )  + N n 1 − N n − 1 1 , ∂ t (Θ n +1 − Θ n ) − ∂ x  ν (Θ n +1 − Θ n ) x c v (1+ V n )  = − ∂ x  ν Θ n x ( V n − V n − 1 ) c v (1+ V n )(1+ V n − 1 )  + N n 2 − N n − 1 2 , ∂ t  Z n +1 − Z n  − ∂ x  D ( Z n +1 − Z n ) x (1+ V n ) 2  = − ∂ x  DZ n x ( V n − V n − 1 )( V n + V n − 1 +2) (1+ V n ) 2 (1+ V n − 1 ) 2  + N n 3 − N n − 1 3 , V n +1 ( x, 0) − V n ( x, 0) = U n +1 ( x, 0) − U n ( x, 0) = Θ n +1 ( x, 0) − Θ n ( x, 0) = 0 , Z n +1 ( x, 0) − Z n ( x, 0) = 0 . (2.21) Here, N n 1 − N n − 1 1 = − ∂ x Å a (Θ n − Θ n − 1 ) 1 + V n − a ( V n − 1 − V n )(1 + Θ n − 1 ) (1 + V n − 1 )(1 + V n ) ã , (2.22) N n 2 − N n − 1 2 = − U n x c v Å p n − µU n x 1 + V n ã + U n − 1 x c v Å p n − 1 − µU n − 1 x 1 + V n − 1 ã + q c v K ϕ (1 + Θ n ) Z n − q c v K ϕ (1 + Θ n − 1 ) Z n − 1 = O (1) î    V n − V n − 1   +   Θ n − Θ n − 1    | U n x | +   V n − V n − 1   ( U n x ) 2 +  1 +   U n x + U n − 1 x      U n x − U n − 1 x   +   Z n − Z n − 1   +   Θ n − Θ n − 1   Z n − 1  , (2.23) N n 3 − N n − 1 3 = − K ϕ (1 + Θ n ) Z n + K ϕ (1 + Θ n − 1 ) Z n − 1 = O (1)    Z n − Z n − 1   +   Θ n − Θ n − 1   Z n − 1  . (2.24) Note that w e hav e used the Lipsc hitz con tin uit y of ϕ , whic h accoun ts for the term   Θ n − Θ n − 1   on the righ t-hand side. An application of the Duhamel principle to Eq. (2.21) yields that  V n +1 − V n  ( x, t ) = Z t 0 ∂ x ( U n +1 − U n )d τ , (2.25)  U n +1 − U n  ( x, t ) = Z t 0 Z R H y ( x, t ; y , τ ; µ n ) µU n y ( V n − V n − 1 ) (1 + V n )(1 + V n − 1 ) ( y , τ ) d y d τ + Z t 0 Z R H ( x, t ; y , τ ; µ n )  N n 1 − N n − 1 1  ( y , τ ) d y d τ , (2.26)  Θ n +1 − Θ n  ( x, t ) = Z t 0 Z R H y ( x, t ; y , τ ; ν n ) ν Θ n y ( V n − V n − 1 ) c v (1 + V n )(1 + V n − 1 ) ( y , τ ) d y d τ + Z t 0 Z R H ( x, t ; y , τ ; ν n )  N n 2 − N n − 1 2  ( y , τ ) d y d τ , (2.27)  Z n +1 − Z n  ( x, t ) = Z t 0 Z R H y ( x, t ; y , τ ; D n ) D Z n y ( V n − V n − 1 )( V n + V n − 1 + 2) (1 + V n ) 2 (1 + V n − 1 ) 2 ( y , τ ) d y d τ 18 + Z t 0 Z R H ( x, t ; y , τ ; D n )  N n 3 − N n − 1 3  ( y , τ ) d y d τ , (2.28) where µ n , ν n and D n are deﬁned as in (2.6). W e shall establish the follo wing log-Lipschitz in-time bounds for Θ n +1 − Θ n and Θ n +1 x − Θ n x . Lemma 2.5. F or suﬃciently smal l δ and t ♯ , ther e exists a p ositive c onstant C 2 such that for any 0 < t < t ♯ , we have the estimates:     Θ n +1 ( · , t ) − Θ n ( · , t ) | log t |     L ∞ x ≤ C 2  p t ♯ + δ  Å     V n − V n − 1     ∞ +         Θ n − Θ n − 1 | log τ |         ∞ +         U n x − U n − 1 x | log τ |         1 + 1 | log t |     Z n − Z n − 1     ∞ ã ,   Θ n +1 ( · , t ) − Θ n ( · , t )   L 1 x ≤ C 2  p t ♯ + δ       V n − V n − 1     1 +     V n − V n − 1     ∞ +     Θ n − Θ n − 1     1 + √ t | log t |         U n x − U n − 1 x | log τ |         1 + √ t     Z n − Z n − 1     1 ã , √ t | log t |   Θ n +1 x ( · , t ) − Θ n x ( · , t )   L ∞ x ≤ C 2  p t ♯ + δ  Å     V n − V n − 1     ∞ + 1 | log t |     V n − V n − 1     B V + 1 | log t |     V n − V n − 1     1 +         √ τ | log τ |  U n x − U n − 1 x          ∞ +         U n x − U n − 1 x | log τ |         1 +         Θ n − Θ n − 1 | log τ |         ∞ + 1 | log t |     Z n − Z n − 1     ∞ ã ,   Θ n +1 x ( · , t ) − Θ n x ( · , t )   L 1 x | log t | ≤ C 2  p t ♯ + δ  Å     V n − V n − 1     ∞ + 1 | log t |     V n − V n − 1     B V + 1 | log t |     V n − V n − 1     1 +         √ τ | log τ |  U n x − U n − 1 x          ∞ +         U n x − U n − 1 x | log τ |         1 + 1 | log t |     Θ n − Θ n − 1     1 + 1 | log t |     Z n − Z n − 1     1 ã . Recall the norms ∥|·|∥ ∞ , ∥|·|∥ 1 and ∥|·|∥ B V from Eq. (1.11). Pr o of. Assume the estimates in Eq. (2.5). Then b oth ν n and ν n − 1 satisfy Eq. (1.5). W e may th us apply Lemmas 1.3 – 1.6 to corresp onding fundamental solution H ( x, t ; y , 0; · ) . First, thanks to the represen tation form ula (2.27), w e hav e that | (Θ n +1 − Θ n )( x, t ) | ≤ Z t 0 Z R | H y ( x, t ; y , τ ; ν n ) | ν   Θ n y     V n − V n − 1   c v | (1 + V n )(1 + V n − 1 ) | ( y , τ ) d y d τ + Z t 0 Z R | H ( x, t ; y , τ ; ν n ) |   N n 2 − N n − 1 2   ( y , τ ) d y d τ . F or the ﬁrst term on the righ t-hand side, w e apply the estimate for H y in Lemma 1.3 and that for | Θ n y | in Eq. (2.5). F or the second term, we tak e the expression for N n 2 − N n − 1 2 from (2.23), and apply the b ound on H in Lemma 1.3 and Eq. (2.5). In this w a y , one obtains that | (Θ n +1 − Θ n )( x, t ) | 19 ≤ O (1) Z t 0 Z R e − ( x − y ) 2 C ∗ ( t − τ ) t − τ   Θ n y   ( y , τ ) d y d τ     V n − V n − 1     ∞ + O (1) Z t 0 Z R e − ( x − y ) 2 C ∗ ( t − τ ) √ t − τ ïÅ     V n − V n − 1     ∞ + | log τ |         Θ n − Θ n − 1 | log τ |         ∞ ã | U n x | +     V n − V n − 1     ∞ | U n x | 2 ó d y d τ + O (1) Z t 0 1 √ t − τ | log τ | Å 1 + δ √ τ ã d τ         U n x − U n − 1 x | log τ |         1 + O (1) Z t 0 Z R e − ( x − y ) 2 C ∗ ( t − τ ) √ t − τ d y d τ     Z n − Z n − 1     ∞ + O (1) Z t 0 1 √ t − τ ∥ Z n − 1 ∥ L 1 x | log τ | d τ         Θ n − Θ n − 1 | log τ |         ∞ ≤ O (1) Z t 0 1 √ t − τ δ √ τ d τ     V n − V n − 1     ∞ + O (1) Z t 0      V n − V n − 1     ∞ + | log τ |         Θ n − Θ n − 1 | log τ |         ∞ ã δ √ τ d τ + O (1) Z t 0 1 √ t − τ δ 2 √ τ d τ     V n − V n − 1     ∞ + O (1) Z t 0 1 √ t − τ | log τ | Å 1 + δ √ τ ã d τ         U n x − U n − 1 x | log τ |         1 + O (1) t     Z n − Z n − 1     ∞ + O (1) Z t 0 1 √ t − τ δ | log τ | d τ         Θ n − Θ n − 1 | log τ |         ∞ ≤ O (1)( √ t + δ ) Å     V n − V n − 1     ∞ + | log t |         Θ n − Θ n − 1 | log τ |         ∞ + | log t |         U n x − U n − 1 x | log τ |         1 +     Z n − Z n − 1     ∞  . Next, we derive the L 1 -estimate for  Θ n +1 − Θ n  . Z R   (Θ n +1 − Θ n )( x, t )   d x ≤ Z R Z t 0 Z R | H y ( x, t ; y , τ ; ν n ) | ν   Θ n y     V n − V n − 1   c v | (1 + V n )(1 + V n − 1 ) | ( y , τ ) d y d τ d x + Z R Z t 0 Z R | H ( x, t ; y , τ ; ν n ) |   N n 2 − N n − 1 2   ( y , τ ) d y d τ d x ≤ O (1) Z t 0 1 √ t − τ δ √ τ d τ     V n − V n − 1     1 + O (1) Z t 0 δ √ τ d τ      V n − V n − 1     1 +     Θ n − Θ n − 1     1  + O (1) Z t 0 δ 2 √ τ d τ     V n − V n − 1     ∞ + O (1) Z t 0 | log τ | Å 1 + δ √ τ ã d τ         U n x − U n − 1 x | log τ |         1 + O (1) t     Z n − Z n − 1     1 + O (1) Z t 0   Z n − 1   L 1 x d τ     Θ n − Θ n − 1     1 ≤ O (1)( √ t + δ )      V n − V n − 1     1 +     V n − V n − 1     ∞ +     Θ n − Θ n − 1     1 + √ t | log t |         U n x − U n − 1 x | log τ |         1 + √ t     Z n − Z n − 1     1 ã . 20 T o con trol Θ n +1 x − Θ n x , we diﬀerentiate Eq. (2.9) with resp ect to x to deduce (Θ n +1 x − Θ n x )( x, t ) = Z R  H x ( x, t ; y , 0; ν n ) − H x ( x, t ; y , 0; ν n − 1 )  θ ∗ 0 ( y ) d y + Z t 0 Z R H x ( x, t ; y , τ ; ν n − 1 )  N n 2 ( y , τ ) − N n − 1 2 ( y , τ )  d y d τ + Z t 0 Z R  H x ( x, t ; y , τ ; ν n ) − H x ( x, t ; y , τ ; ν n − 1 )  N n 2 ( y , τ ) d y d τ = : I 1 + I 2 + I 3 . W e estimate I 1 b y using the L ∞ -b ound for θ ∗ 0 and the comparison estimates for H x in Lemma 1.5: |I 1 | ≤ Z R    H x ( x, t ; y , 0; ν n ) − H x ( x, t ; y , 0; ν n − 1 )    d y ∥ θ ∗ 0 ∥ L ∞ x ≤ O (1) δ Z R e − ( x − y ) 2 C ∗ t t d y î | log t |     V n − V n − 1     ∞ +     V n − V n − 1     B V + √ t     V n − V n − 1     1 + √ t | log t |         √ τ | log τ |  U n x − U n − 1 x          ∞ ò ≤ O (1) δ √ t î | log t |     V n − V n − 1     ∞ +     V n − V n − 1     B V + √ t     V n − V n − 1     1 + √ t | log t |         √ τ | log τ |  U n x − U n − 1 x          ∞ ò . F or I 2 , we apply the estimate of H x in Lemma 1.3 to deduce that |I 2 | ≤ Z t 0 Z R   H x ( x, t ; y , τ ; ν n − 1 )     N n 2 ( y , τ ) − N n − 1 2 ( y , τ )   d y d τ ≤ O (1) Z t 0 Z R e − ( x − y ) 2 C ∗ ( t − τ ) t − τ î    V n − V n − 1   +   Θ n − Θ n − 1    | U n x | +   V n − V n − 1   ( U n x ) 2 +  1 +   U n x + U n − 1 x      U n x − U n − 1 x   +   Z n − Z n − 1   +   Θ n − Θ n − 1   Z n − 1  d y d τ ≤ O (1) Z t 0 1 √ t − τ δ √ τ d τ     V n − V n − 1     ∞ + O (1) Z t 0 1 √ t − τ δ √ τ | log τ | d τ         Θ n − Θ n − 1 | log τ |         ∞ + O (1) Ç Z t 2 0 1 t − τ δ 2 √ τ d τ + Z t t 2 1 √ t − τ δ 2 τ d τ å     V n − V n − 1     ∞ + O (1) Z t 2 0 Å 1 + δ √ τ ã 1 t − τ | log τ | d τ         U n x − U n − 1 x | log τ |         1 + O (1) Z t t 2 Å 1 + δ √ τ ã 1 √ t − τ | log τ | √ τ d τ           √ τ  U n x − U n − 1 x  | log τ |           ∞ + O (1) Z t 0 1 √ t − τ d τ     Z n − Z n − 1     ∞ + O (1) Z t 0 1 √ t − τ | log τ | d τ         Θ n − Θ n − 1 | log τ |         ∞   Z n − 1   L ∞ x ≤ O (1) δ Å     V n − V n − 1     ∞ + | log t |         Θ n − Θ n − 1 | log τ |         ∞ + δ √ t     V n − V n − 1     ∞ ã + O (1)( δ + √ t ) | log t | √ t         U n x − U n − 1 x | log τ |         1 +           √ τ  U n x − U n − 1 x  | log τ |           ∞ ! 21 + O (1) √ t     Z n − Z n − 1     ∞ + O (1) δ √ t | log t |         Θ n − Θ n − 1 | log τ |         ∞ , where we split the time integral at t 2 to handle the singularity at τ = t . Finally , we estimate I 3 b y combining the comparison estimates for H x in Lemma 1.5 with b ounds for N n 2 . W e again split the in tegral at t 2 . |I 3 | ≤ O (1) Z t 0 Z R    H x ( x, t ; y , τ ; ν n ) − H x ( x, t ; y , τ ; ν n − 1 )    ï   U n y   Å 1 + δ √ τ ã + Z n ò ( y , τ ) d y d τ ≤ O (1) Z t 0 Z R e − ( x − y ) 2 C ∗ ( t − τ ) t − τ  | log( t − τ ) |     V n − V n − 1     ∞ +     V n − V n − 1     B V + √ t − τ     V n − V n − 1     1 + √ t − τ | log t |         √ τ | log τ |  U n x − U n − 1 x          ∞ ò ï   U n y   Å 1 + δ √ τ ã + Z n ò d y d τ ≤ O (1) ñ Z t 0 1 √ t − τ | log( t − τ ) | δ √ τ d τ + Z t 2 0 1 t − τ | log( t − τ ) | δ 2 √ τ d τ + Z t t 2 1 √ t − τ | log( t − τ ) | δ 2 τ d τ ô ×     V n − V n − 1     ∞ + O (1) ñ Z t 0 1 √ t − τ δ √ τ d τ + Z t 2 0 1 t − τ δ 2 √ τ d τ + Z t t 2 1 √ t − τ δ 2 τ d τ ô ×     V n − V n − 1     B V + O (1) ñ Z t 0 δ √ τ d τ + Z t 2 0 1 √ t − τ δ 2 √ τ d τ + Z t t 2 δ 2 τ d τ ô     V n − V n − 1     1 + O (1) ñ Z t 0 δ √ τ d τ + Z t 2 0 1 √ t − τ δ 2 √ τ d τ + Z t t 2 δ 2 τ d τ ô | log t |         √ τ | log τ |  U n x − U n − 1 x          ∞ + O (1) ∥ Z n ∥ L ∞ x Z t 0 1 √ t − τ  | log( t − τ ) |     V n − V n − 1     ∞ +     V n − V n − 1     B V + √ t − τ     V n − V n − 1     1 + √ t − τ | log t |         √ τ | log τ |  U n x − U n − 1 x          ∞ ò d τ ≤ O (1) ( √ t + δ ) δ √ t î | log t |     V n − V n − 1     ∞ +     V n − V n − 1     B V + √ t     V n − V n − 1     1 + √ t | log t |         √ τ | log τ |  U n x − U n − 1 x          ∞ ò . Com bining the estimates for I 1 , I 2 and I 3 and dividing b oth sides by | log t | √ t , we obtain that √ t | log t |    Θ n +1 x − Θ n x    ≤ C 2 ( √ t + δ )      V n − V n − 1     ∞ +     V n − V n − 1     B V +     V n − V n − 1     1 +         √ τ | log τ |  U n x − U n − 1 x          ∞ +         U n x − U n − 1 x | log τ |         1 +         Θ n − Θ n − 1 | log τ |         ∞ +     Z n − Z n − 1     ∞  . (2.29) Finally , via a similar argumen t, we inte grate I 1 , I 2 , and I 3 o v er x ∈ R to obtain that   Θ n +1 x − Θ n x   L 1 x | log t | ≤ C 2 ( √ t + δ )      V n − V n − 1     ∞ +     V n − V n − 1     B V +     V n − V n − 1     1 +         √ τ | log τ |  U n x − U n − 1 x          ∞ +         U n x − U n − 1 x | log τ |         1 +     Θ n − Θ n − 1     1 +     Z n − Z n − 1     1  . (2.30) 22 This completes the pro of. □ Lemma 2.6. F or suﬃciently smal l δ and t ♯ and for 0 < t < t ♯ , ther e exists a p ositive c onstant C 2 such that the diﬀer enc es Z n +1 − Z n and their derivatives satisfy the fol lowing:   Z n +1 ( · , t ) − Z n ( · , t )   L ∞ x ≤ C 2  p t ♯ | log t | + δ  Å     V n − V n − 1     ∞ +         Θ n − Θ n − 1 | log τ |         ∞ +     Z n − Z n − 1     ∞ ã ,   Z n +1 ( · , t ) − Z n ( · , t )   L 1 x ≤ C 2  p t ♯ + δ       V n − V n − 1     1 +     Θ n − Θ n − 1     1 +     Z n − Z n − 1     1  , √ t | log t |   Z n +1 x ( · , t ) − Z n x ( · , t )   L ∞ x ≤ C 2  p t ♯ + δ  Å     V n − V n − 1     ∞ + 1 | log t |     V n − V n − 1     B V + 1 | log t |     V n − V n − 1     1 +         √ τ | log τ |  U n x − U n − 1 x          ∞ +         Θ n − Θ n − 1 | log τ |         ∞ + 1 | log t |     Z n − Z n − 1     ∞ ã ,   Z n +1 x ( · , t ) − Z n x ( · , t )   L 1 x | log t | ≤ C 2  p t ♯ + δ  Å     V n − V n − 1     ∞ + 1 | log t |     V n − V n − 1     B V + 1 | log t |     V n − V n − 1     1 +         √ τ | log τ |  U n x − U n − 1 x          ∞ + 1 | log t |     Θ n − Θ n − 1     1 + 1 | log t |     Z n − Z n − 1     1 ã . (2.31) Pr o of. The proof follo ws a similar structure to that of Lemma 2.5. Assume the b ounds in Eq. (2.5). Then, b oth D n and D n − 1 satisfy (1.5), whic h enables us to apply Lemmas 1.3 – 1.6 to the fundamen tal solutions H ( x, t ; y , τ ; D n ) and H ( x, t ; y , τ ; D n − 1 ) . Indeed, we hav e the p oint wise b ound | ( Z n +1 − Z n )( x, t ) | ≤ Z t 0 Z R | H y ( x, t ; y , τ ; D n ) |   Z n y     ( V n − V n − 1 )( V n + V n − 1 + 2)   (1 + V n ) 2 (1 + V n − 1 ) 2 ( y , τ ) d y d τ + O (1) Z t 0 Z R | H ( x, t ; y , τ ; D n ) |    Z n − Z n − 1   +   Θ n − Θ n − 1   Z n − 1  d y d τ , thanks to the represen tation formula (2.28) and the b ound for N n 3 − N n − 1 3 in Eq. (2.24). F rom here, we apply the estimates for H y ( x, t ; y , τ ; D n ) and H ( x, t ; y , τ ; D n ) in Lemma 1.3, together with the L ∞ -estimates for Z n y in Eq. (2.5), to obtain that | ( Z n +1 − Z n )( x, t ) | ≤ O (1)    Z t 0 Z R e − ( x − y ) 2 C ∗ ( t − τ ) t − τ   Z n y   d y d τ        V n − V n − 1     ∞ + O (1)    Z t 0 Z R e − ( x − y ) 2 C ∗ ( t − τ ) √ t − τ d y d τ        Z n − Z n − 1     ∞ + O (1)    Z t 0 Z R e − ( x − y ) 2 C ∗ ( t − τ ) √ t − τ | log τ | d y d τ            Θ n − Θ n − 1 | log τ |         ∞   Z n − 1   L ∞ x ≤ O (1) ß Z t 0 1 √ t − τ δ √ τ d τ ™     V n − V n − 1     ∞ + O (1) t     Z n − Z n − 1     ∞ 23 + O (1) δ ß Z t 0 | log τ | d τ ™         Θ n − Θ n − 1 | log τ |         ∞ . W e th us arrive at | ( Z n +1 − Z n )( x, t ) | ≤ O (1) Å δ     V n − V n − 1     ∞ + δ t | log t |         Θ n − Θ n − 1 | log τ |         ∞ + t     Z n − Z n − 1     ∞ ã . The L 1 -estimate for Z n +1 − Z n follo ws easily from the point wise b ound abov e: Z R | ( Z n +1 − Z n ) | ( x, t ) d x ≤ O (1) ß Z t 0 1 √ t − τ δ √ τ d τ ™     V n − V n − 1     1 + O (1) t     Z n − Z n − 1     1 + O (1) ß Z t 0 1 √ t − τ δ d τ ™     Θ n − Θ n − 1     1 ≤ O (1) Ä δ     V n − V n − 1     1 + √ tδ     Θ n − Θ n − 1     1 + t     Z n − Z n − 1     1 ä . T o estimate Z n +1 x − Z n x , we diﬀerentiate the representation form ula (2.10) of Z n +1 with resp ect to x , and then decomp ose into three terms:  Z n +1 x − Z n x  ( x, t ) = Z R  H x ( x, t ; y , 0; D n ) − H x ( x, t ; y , 0; D n − 1 )  z ∗ 0 ( y ) d y + Z t 0 Z R H x ( x, t ; y , τ ; D n )  N n 3 − N n − 1 3  ( y , τ ) d y d τ − Z t 0 Z R  H x ( x, t ; y , τ ; D n ) − H x ( x, t ; y , τ ; D n − 1 )  × K ϕ (1 + Θ n − 1 ) Z n − 1 d y d τ = : I 1 + I 2 + I 3 . W e shall b ound each term I 1 , I 2 , and I 3 separately . • F or I 1 , by the L ∞ -b ound for z ∗ 0 and the comparison estimate for H x ( x, t ; y , τ ; D n ) in Lemma 1.5, one arrives at |I 1 | ≤ O (1) δ    Z R e − ( x − y ) 2 C ∗ t t d y    î | log t |     V n − V n − 1     ∞ +     V n − V n − 1     B V + √ t     V n − V n − 1     1 + √ t | log t |         √ τ | log τ |  U n x − U n − 1 x          ∞ ò ∥ z ∗ 0 ∥ L ∞ x ≤ O (1) δ √ t î | log t |     V n − V n − 1     ∞ +     V n − V n − 1     B V + √ t     V n − V n − 1     1 + √ t | log t |         √ τ | log τ |  U n x − U n − 1 x          ∞ ò . • F or I 2 , we directly estimate that |I 2 | ≤ O (1) Z t 0 Z R e − ( x − y ) 2 C ∗ ( t − τ ) t − τ    Z n − Z n − 1   +   Θ n − Θ n − 1   Z n − 1  d y d τ ≤ O (1) ß Z t 0 1 √ t − τ d τ ™     Z n − Z n − 1     ∞ + O (1) ß Z t 0 1 √ t − τ | log τ | d τ ™         Θ n − Θ n − 1 | log τ |         ∞   Z n − 1   L ∞ x 24 ≤ O (1) √ t     Z n − Z n − 1     ∞ + O (1) δ √ t | log t |         Θ n − Θ n − 1 | log τ |         ∞ . • F or I 3 , we use the Lipschitz contin uit y of ϕ , combined with the estimate for H x ( x, t ; y , τ ; D n ) and the bound for ∥ Z n − 1 ∥ L ∞ x in Eq. (2.5), to obtain that |I 3 | ≤ O (1) Z t 0 Z R e − ( x − y ) 2 C ∗ ( t − τ ) t − τ  | log( t − τ ) |     V n − V n − 1     ∞ +     V n − V n − 1     B V + √ t − τ     V n − V n − 1     1 + √ t − τ | log t |         √ τ | log τ |  U n x − U n − 1 x          ∞ ò   Z n − 1   L ∞ x d y d τ ≤ O (1) δ ï Z t 0 1 √ t − τ | log( t − τ ) | d τ     V n − V n − 1     ∞ + Z t 0 1 √ t − τ d τ     V n − V n − 1     B V + t     V n − V n − 1     1 + t | log t |         √ τ | log τ |  U n x − U n − 1 x          ∞ ò ≤ O (1) δ ï 1 √ t | log t |     V n − V n − 1     ∞ + √ t     V n − V n − 1     B V + t     V n − V n − 1     1 + t | log t |         √ τ | log τ |  U n x − U n − 1 x          ∞ ò . Summarising that abov e p oint wise b ounds for I 1 , I 2 , and I 3 , we arrive at √ t | log t |   Z n +1 x − Z n x   ≤ C 2 ( √ t + δ ) ß     V n − V n − 1     ∞ + 1 | log t |     V n − V n − 1     B V + 1 | log t |     V n − V n − 1     1 +         √ τ | log τ |  U n x − U n − 1 x          ∞ +         Θ n − Θ n − 1 | log τ |         ∞ + 1 | log t |     Z n − Z n − 1     ∞ ™ . Next, let us derive the L 1 -estimate for Z n +1 x − Z n x = I 1 + I 2 + I 3 . • F or I 1 , combining the deﬁnition of W in (2.16), the estimates in Lemma 1.5, and the BV-norm b ound for z ∗ 0 , we obtain that Z R |I 1 | d x ≤ O (1)    Z R e − ( x − y ) 2 C ∗ t √ t d y    ∥ z ∗ 0 ∥ B V ï | log t |     V n − V n − 1     ∞ +     V n − V n − 1     B V + √ t     V n − V n − 1     1 + √ t | log t |         √ τ | log τ |  U n x − U n − 1 x          ∞ ò ≤ O (1) δ ï | log t |     V n − V n − 1     ∞ +     V n − V n − 1     B V + √ t     V n − V n − 1     1 + √ t | log t |         √ τ | log τ |  U n x − U n − 1 x          ∞ ò . • F or I 2 , a direct computation leads to Z R |I 2 | d x ≤ O (1) ß Z t 0 1 √ t − τ d τ ™     Z n − Z n − 1     1 + O (1) ß Z t 0 1 √ t − τ d τ     Θ n − Θ n − 1     1   Z n − 1   L ∞ x ™ ≤ O (1) √ t     Z n − Z n − 1     1 + O (1) δ √ t     Θ n − Θ n − 1     1 . 25 F or I 3 , a similar argument as that for I 1 leads to Z R |I 3 | d x ≤ O (1) ß Z t 0 1 √ t − τ | log( t − τ ) | δ d τ ™     V n − V n − 1     ∞ + O (1) ß Z t 0 1 √ t − τ δ d τ ™     V n − V n − 1     B V + O (1) ß Z t 0 δ d τ ™     V n − V n − 1     1 + O (1) ß Z t 0 | log t | δ d τ ™         √ τ | log τ |  U n x − U n − 1 x          ∞ ≤ O (1) δ " √ t | log t |     V n − V n − 1     ∞ + √ t     V n − V n − 1     B V + t     V n − V n − 1     1 + t | log t |         √ τ | log τ |  U n x − U n − 1 x          ∞ # . (2.32) Summarising the abov e estimates, w e infer that for suﬃciently small δ and t ♯ ,   Z n +1 x − Z n x   L 1 x | log t | ≤ C 2 ( √ t + δ ) "     V n − V n − 1     ∞ + 1 | log t |     V n − V n − 1     B V + 1 | log t |     V n − V n − 1     1 +         √ τ | log τ |  U n x − U n − 1 x          ∞ + 1 | log t |     Θ n − Θ n − 1     1 + 1 | log t |     Z n − Z n − 1     1 # . The pro of of Lemma 2.6 is now complete. □ The following tw o lemmas can be found in [32]. W e safely omit the proof here. Lemma 2.7. F or suﬃciently smal l δ and t ♯ and for 0 < t < t ♯ , ther e exists a p ositive c onstant C 2 such that the diﬀer enc es U n +1 − U n and their derivatives satisfy the fol lowing estimates:   U n +1 ( · , t ) − U n ( · , t )   L ∞ x ≤ C 2  p t ♯ | log t | + δ  Å     V n − V n − 1     ∞ +         Θ n − Θ n − 1 | log τ |         ∞ ã ,   U n +1 ( · , t ) − U n ( · , t )   L 1 x ≤ C 2  p t ♯ + δ       V n − V n − 1     1 +     Θ n − Θ n − 1     1  , √ t | log t |   U n +1 x ( · , t ) − U n x ( · , t )   L ∞ x ≤ C 2  p t ♯ + δ  Å     V n − V n − 1     ∞ + 1 | log t |     V n − V n − 1     B V + 1 | log t |     V n − V n − 1     1 +         √ τ | log τ |  U n x − U n − 1 x          ∞ +         Θ n − Θ n − 1 | log τ |         ∞ ã ,   U n +1 x ( · , t ) − U n x ( · , t )   L 1 x | log t | ≤ C 2  p t ♯ + δ  Å     V n − V n − 1     ∞ + 1 | log t |     V n − V n − 1     B V + 1 | log t |     V n − V n − 1     1 26 +         √ τ | log τ |  U n x − U n − 1 x          ∞ +         U n x − U n − 1 x | log τ |         1 +         Θ n − Θ n − 1 | log τ |         ∞ + 1 | log t |     Θ n − Θ n − 1     1 ã . (2.33) Pr o of. See [32]. □ Lemma 2.8. F or suﬃciently smal l δ and t ♯ and for 0 < t < t ♯ , ther e exists a p ositive c onstant C 2 such that the diﬀer enc es V n +1 − V n and their derivatives satisfy the fol lowing estimates:   V n +1 ( · , t ) − V n ( · , t )   L ∞ x ≤ C 2  p t ♯ + δ  Å     V n − V n − 1     ∞ +         Θ n − Θ n − 1 | log τ |         ∞ ã ,   V n +1 ( · , t ) − V n ( · , t )   L 1 x ≤ C 2  p t ♯ + δ       V n − V n − 1     1 +     Θ n − Θ n − 1     1  ,     V n +1 ( · , t ) − V n ( · , t )    ω + ω −    (2.34) ≤ C 2  p t ♯ + δ  √ t sup 0 <τ 0 such that the fol lowing holds. Supp ose that the initial datum ( v ∗ 0 , u ∗ 0 , θ ∗ 0 , z ∗ 0 ) satisﬁes the smal lness c ondition (2.2) with δ . Then ther e exists a p ositive c onstant t ♯ such that system (2.1) admits a we ak solution ( v , u, θ, z ) = ( v ∗ + 1 , u ∗ , θ ∗ + 1 , z ∗ ) , t < t ♯ ≪ 1 , with the estimates                          max ¶ ∥ u ( · , t ) ∥ L 1 x , ∥ u ( · , t ) ∥ L ∞ x , ∥ u x ( · , t ) ∥ L 1 x , √ t ∥ u x ( · , t ) ∥ L ∞ x © ≤ 2 C ♯ δ, max ¶ ∥ θ ( · , t ) − 1 ∥ L 1 x , ∥ θ ( · , t ) − 1 ∥ L ∞ x , ∥ θ x ( · , t ) ∥ L 1 x , √ t ∥ θ x ( · , t ) ∥ L ∞ x © ≤ 2 C ♯ δ, max ¶ ∥ z ( · , t ) ∥ L 1 x , ∥ z ( · , t ) ∥ L ∞ x , ∥ z x ( · , t ) ∥ L 1 x , √ t ∥ z x ( · , t ) ∥ L ∞ x © ≤ 2 C ♯ δ, max ¶ ∥ v ( · , t ) ∥ B V , ∥ v ( · , t ) − 1 ∥ L 1 x , ∥ v − 1( · , t ) ∥ L ∞ x , √ t ∥ v t ( · , t ) ∥ L ∞ x © ≤ 2 C ♯ δ, v ∗ = v ∗ ˜ a + v ∗ j , v ∗ j ( x, t ) = P ω 0 such that the fol lowing holds. Supp ose that the initial datum ( v ∗ 0 , u ∗ 0 , θ ∗ 0 , z ∗ 0 ) satisﬁes the smal lness c ondition in Eq. (2.2) with this δ . L et ( v , u, θ, z ) b e the c orr esp onding lo c al we ak solution c onstructe d in The or em 2.1. Then (1) Ther e exists a p ositive c onstant C ♯ such that max ¶ √ t ∥ u t ( · , t ) ∥ L 1 x , t ∥ u t ( · , t ) ∥ L ∞ x , √ t ∥ θ t ( · , t ) ∥ L 1 x , t ∥ θ t ( · , t ) ∥ L ∞ x , √ t ∥ z t ( · , t ) ∥ L 1 x , t ∥ z t ( · , t ) ∥ L ∞ x © ≤ 2 C ♯ δ, (2.46) in addition to the estimates in Eq. (2.35) . (2) The ﬂuxes of u , θ , and z (deﬁne d in The or em 2.1) ar e glob al ly Lipschitz c ontinuous with r esp e ct to x , pr ovide d that 0 < t < t ♯ . (3) The sp e ciﬁc volume v ( x, t ) satisﬁes      ∥ v ( · , t ) − v ( · , s ) ∥ B V ≤ O (1) δ ( t − s ) | log( t − s ) | √ t , ∥ v ( · , t ) − v ( · , s ) ∥ L ∞ x ≤ O (1) δ t − s √ t , ∥ v ( · , t ) − v ( · , s ) ∥ L 1 x ≤ O (1) δ ( t − s ) (2.47) for any 0 ≤ s < t < t ♯ . Pr o of. (1) follows directly from the Lemma 2.11. Mean while, the L ∞ x -norm of u t ( x, t ) , θ t ( x, t ) , and z t ( x, t ) are ﬁnite whenever 0 < t < t ♯ , thanks to Lemma 2.11. This pro v es (2). F or (3), from the representation form ula (2.7) for V k +1 and the bound   U k +1 x   L 1 x ≤ 2 C ♯ δ , w e deduce that    V k +1 ( x, t ) − V k +1 ( x, s )    L 1 x =     Z t 0 U k +1 x ( x, τ ) d τ − Z s 0 U k +1 x ( x, τ ) d τ     L 1 x ≤ Z t s    U k +1 x ( x, τ )    L 1 x d τ ≤ O (1) δ ( t − s ) . 35 F urthermore, from Eq. (2.5) and Lemma 2.3, w e obtain the ﬁrst tw o prop erties in Eq. (2.47) for V k +1 . W e th us conclude by applying the strong conv ergence result in Theorem 2.1. □ Recall the total sp eciﬁc energy E = c v θ + u 2 2 + q z . W e ma y deduce from the ab ov e dev elopmen ts the existence of local w eak solutions to Eq. (0.1). Corollary 2.3. Supp ose that the initial datum ( v ∗ 0 , u ∗ 0 , θ ∗ 0 , z ∗ 0 ) satisfy the smal lness c ondition (2.2) for a suﬃciently smal l δ . L et ( v , u, θ , z ) b e the c orr esp onding we ak solution to Eq. (0.4) on R × [0 , t ♯ ) c onstructe d in The or ems 2.1 and 2.2, wher e t ♯ ≪ 1 is suﬃciently smal l. Then ( v , u, E , z ) is a we ak solution to the original system (0.1) with initial datum ( v 0 , u 0 , E 0 , z 0 ) = Å 1 + v ∗ 0 , u ∗ 0 , c v (1 + θ ∗ 0 ) + ( u ∗ 0 ) 2 2 + q z ∗ 0 , z ∗ 0 ã . Pr o of. As ( v , u, θ , z ) is the weak solution constructed in Theorems 2.1 and 2.2, it also satisﬁes Eq. (0.4) in the distributional sense. Since u t is deﬁned in the strong sense, w e can tak e φu as the test function in the second identit y of Eq. (0.10). Additionally , we ma y tak e c v φ as the test function in the third identit y of Eq. (0.10) and q φ as the test function in the fourth equation of Eq. (2.1), respectively . Th us E satisﬁes Eq. (0.1) in the distributional sense. □ Remark 2.4. The r e gularity r esult in Eq. (2.47) for v (obtaine d fr om The or em 2.2) is the same as in that in Eq. (0.11) . Thus, for e ach 0 ≤ t < t ♯ , the mapping x 7→ v ( t, x ) is c ontinuous on L 1 x ( R ) ∩ L ∞ x ( R ) ∩ B V . 2.4. Stabilit y and Uniqueness. This subsection fo cuses on the stability of w eak solution con- structed in Theorems 2.1 and 2.2. F rom this one may conclude the contin uous dep endence of w eak solution on the initial data, as w ell as the uniqueness of w eak solution. Lemma 2.12. Ther e exists a universal c onstant 0 < δ ∗ ≪ 1 such that the fol lowing holds. Supp ose that the initial data satisfy ∥ v 0 − 1 ∥ L 1 x + ∥ v 0 ∥ B V + ∥ u 0 ∥ L 1 x + ∥ u 0 ∥ B V + ∥ θ 0 − 1 ∥ L 1 x + ∥ θ 0 ∥ B V + ∥ z 0 ∥ L 1 x + ∥ z 0 ∥ B V < δ ∗ ≪ 1 . (2.48) L et ( v , u, θ , z ) b e any we ak solution to Eq. (0.4) as in Deﬁnition 0.1 with the ab ove initial data. L et C ♯ and δ b e the p ar ameters given in The or ems 2.1 and 2.2. Then ther e exists a smal l p ositive c onstant t ∗ such that for any t ∈ (0 , t ∗ ) , one has that max ¶ ∥ u ( · , t ) ∥ L 1 x , ∥ u ( · , t ) ∥ L ∞ x , ∥ u x ( · , t ) ∥ L 1 x , √ t ∥ u x ( · , t ) ∥ L ∞ x © ≤ 2 C ♯ δ, max ¶ ∥ θ ( · , t ) ∥ L 1 x , ∥ θ ( · , t ) ∥ L ∞ x , ∥ θ x ( · , t ) ∥ L 1 x , √ t ∥ θ x ( · , t ) ∥ L ∞ x © ≤ 2 C ♯ δ, max ¶ ∥ z ( · , t ) ∥ L 1 x , ∥ z ( · , t ) ∥ L ∞ x , ∥ z x ( · , t ) ∥ L 1 x , √ t ∥ z x ( · , t ) ∥ L ∞ x © ≤ 2 C ♯ δ. (2.49) Pr o of. By Theorems 2.1 and 2.2, under the assumption (2.48) for the initial data, Eq. (0.4) has at least one weak solution. Eq. (2.49) follo ws directly from Eqs. (2.35) and (2.46). Next, by Remark 2.4, there exists a small t ∗ suc h that ∥ v ( · , t ) − 1 ∥ L 1 x ≤ C ♯ δ ∗ , ∥ v ( · , t ) − 1 ∥ L ∞ x ≤ C ♯ δ ∗ , ∥ v ( · , t ) − 1 ∥ B V ≤ C ♯ δ ∗ (2.50) whenev er t ∈ (0 , t ∗ ) . F or δ ∗ and t ∗ suﬃcien tly small, the terms µ v , ν c v v , and D v 2 satisfy the condition in Eq. (1.5). This allows us to construct the corresp onding fundamen tal solution H ( x, t ; y , t 0 ; · ) . 36 Applying Duhamel’s principle and in tegration b y parts, w e arriv e at the representation formulae: u ( x, t ) = Z R H ( x, t ; y , 0; µ v ) u ( y , 0) d y + Z t 0 Z R H y ( x, t ; y , τ ; µ v ) p ( y , τ ) d y d τ , (2.51) θ ( x, t ) = Z R H ( x, t ; y , 0; ν c v v ) θ ( y , 0) d y + Z t 0 Z R H ( x, t ; y , τ ; ν c v v ) Å − pu y b + µ c v v ( u y ) 2 + q c v K ϕ ( θ ) z ã ( y , τ ) d y d τ , (2.52) z ( x, t ) = Z R H ( x, t ; y , 0; D v 2 ) z ( y , 0) d y + Z t 0 Z R H ( x, t ; y , τ ; D v 2 ) K ϕ ( θ ) z ( y , τ ) d y d τ . (2.53) Note that the smallness of u and θ without the z -term can be established as in [32, pro of of Lemma 5.1], so here w e fo cus on pro ving the smallness of z ( x, t ) . Using the smallness condition on the initial data as in (2.48), we integrate b oth sides of the fourth equation in (0.4) o v er [0 , t ] × R to obtain Z R z ( x, t ) d x + Z t 0 Z R K ϕ ( θ ) z ( y , τ ) d y d τ ≤ Z R z ∗ 0 ( x ) d x. Since K ϕ ( θ ) ≥ 0 , the second in tegral on the left-hand side is non-negativ e, which implies ∥ z ∥ L 1 x ≤ ∥ z ∗ 0 ∥ L 1 x ≤ O (1) δ ∗ . Then, using the estimate for | H ( x, t ; y , τ ; D k ) | in Lemma 1.3 and the ab o ve L 1 -norm of z , for suﬃcien tly small δ ∥ z ∥ L ∞ x ≤ Z R    H ( x, t ; y , 0; D k )    | z ∗ 0 ( y ) | d y + Z t 0 Z R    H ( x, t ; y , τ ; D k )    K ϕ ( θ ) z ( y , τ ) d y d s ≤ O (1) Z R e − ( x − y ) 2 C ∗ t √ t d y · ∥ z ∗ 0 ∥ L ∞ x + O (1) Z t 0 Z R e − ( x − y ) 2 C ∗ ( t − τ ) √ t − τ z ( y , τ ) d y d τ ≤ O (1) δ ∗ + O (1) Z t 0 1 √ t − τ ∥ z ∥ L 1 x d τ ≤ O (1) δ ∗ + O (1) δ ∗ √ t ≤ O (1) δ ∗ . (2.54) Next, we diﬀerentiate (2.53) with respect to x to get the representation of z x and then apply the smallness condition for initial data (2.48), the estimate for H x ( x, t ; y , τ ; D v 2 ) in Lemma 1.3, the Lipsc hitz contin uit y of ϕ , and the regularit y in (0.11) to deriv e the L ∞ -estimate for z x ( x, t ) | z x ( x, t ) | ≤ Z R     H x ( x, t ; y , 0; D v 2 )     | z ( y , 0) | d y + Z t 0 Z R     H x ( x, t ; y , τ ; D v 2 )     | K ϕ ( θ ) z ( y , τ ) | d y d τ ≤ O (1)    Z R e − ( x − y ) 2 C ∗ t t d y    ∥ z 0 ∥ L ∞ x + O (1)    Z t 0 Z R e − ( x − y ) 2 C ∗ ( t − τ t − τ d y d τ    ∥ z ∥ L ∞ x ≤ O (1) δ ∗ √ t + O (1) δ ∗ Z t 0 1 √ t − τ d τ ≤ O (1) δ ∗ ( 1 √ t + √ t ) . 37 F or the L 1 -norm of z x , similarly to Eq. (2.16), w e deﬁne an ti-deriv ative of H x ( x, t ; y , 0; D v 2 ) : W Å x, t ; y , 0; D v 2 ã =    R y −∞ H x  x, t ; w , 0; D v 2  d w for y < x, − R ∞ y H x  x, t ; w , 0; D v 2  d w for y ≥ x. Using a similar argument as for   Z k +1 x   L 1 x in Lemma 2.2, we ﬁnd that Z R | z x ( x, t ) | d x ≤ Z R Z R     H x ( x, t ; y , 0; D v 2 )     | z ( y , 0) | d y d x + Z R Z t 0 Z R     H x ( x, t ; y , τ ; D v 2 )     K ϕ ( θ ) z ( y , τ ) d y d τ d x ≤ Z R Z R     W Å x, t ; y , 0; D v 2 ã     | d z 0 ( y ) | d x + O (1) Z t 0 Z R Z R e − ( x − y ) 2 C ∗ ( t − τ ) t − τ z ( y , τ ) d y d x d τ ≤ O (1) Z R e − ( x − y ) 2 C ∗ t √ t d y ∥ z 0 ∥ B V + O (1) Z t 0 1 √ t − τ d τ ∥ z ∥ L 1 x ≤ O (1) δ ∗ (1 + √ t ) . Therefore, for suﬃcien tly small δ ∗ and t ∗ suc h that δ ∗ (1 + √ t ∗ ) ≤ δ, w e conclude that max ¶ ∥ z ( · , t ) ∥ L 1 x , ∥ z ( · , t ) ∥ L ∞ x , ∥ z x ( · , t ) ∥ L 1 x , √ t ∥ z x ( · , t ) ∥ L ∞ x © ≤ 2 C ♯ δ whenev er 0 < t < t ∗ . This completes the pro of. □ Lemma 2.13. Ther e is a univer al p ositive c onstant δ 0 such that the fol lowing holds. Supp ose that the initial data ( v ϵ 0 , u ϵ 0 , θ ϵ 0 , z ϵ 0 ) and ( v ι 0 , u ι 0 , θ ι 0 , z ι 0 ) b oth satisfy the smal lness c ondition: ∥ v 0 − 1 ∥ L 1 x + ∥ v 0 ∥ B V + ∥ u 0 ∥ L 1 x + ∥ u 0 ∥ B V + ∥ θ 0 − 1 ∥ L 1 x + ∥ θ 0 ∥ B V + ∥ z 0 ∥ L 1 x + ∥ z 0 ∥ B V ≤ δ ∗ . L et ( v ϵ , u ϵ , θ ϵ , z ϵ ) and ( v ι , u ι , θ ι , z ι ) b e the c orr esp onding we ak solutions to the system (2.1) as c onstructe d in The or em 2.1. Then ther e exists a p ositive c onstant C 2 such that F [ v ϵ − v ι , u ϵ − u ι , θ ϵ − θ ι , z ϵ − z ι ] ≤ C 2 Ä ∥ v ϵ 0 − v ι 0 ∥ L 1 x + ∥ v ϵ 0 − v ι 0 ∥ L ∞ x + ∥ v ϵ 0 − v ι 0 ∥ B V + ∥ u ϵ 0 − u ι 0 ∥ L ∞ x + ∥ u ϵ 0 − u ι 0 ∥ L 1 x + ∥ θ ϵ 0 − θ ι 0 ∥ L ∞ x + ∥ θ ϵ 0 − θ ι 0 ∥ L 1 x + ∥ z ϵ 0 − z ι 0 ∥ L ∞ x + ∥ z ϵ 0 − z ι 0 ∥ L 1 x ä , wher e F is a functional deﬁne d in (2.36) . In p articular, we have that ∥ v ϵ − v ι ∥ L 1 x + ∥ u ϵ − u ι ∥ L 1 x + ∥ θ ϵ − θ ι ∥ L 1 x + ∥ z ϵ − z ι ∥ L 1 x ≤ C 2 Ä ∥ v ϵ 0 − v ι 0 ∥ L 1 x + ∥ v ϵ 0 − v ι 0 ∥ L ∞ x + ∥ v ϵ 0 − v ι 0 ∥ B V + ∥ u ϵ 0 − u ι 0 ∥ L ∞ x + ∥ u ϵ 0 − u ι 0 ∥ L 1 x + ∥ θ ϵ 0 − θ ι 0 ∥ L ∞ x + ∥ θ ϵ 0 − θ ι 0 ∥ L 1 x + ∥ z ϵ 0 − z ι 0 ∥ L ∞ x + ∥ z ϵ 0 − z ι 0 ∥ L 1 x ä . Pr o of. The existence and contin uit y of the ﬂuxes of the weak solutions ( v ϵ , u ϵ , θ ϵ , z ϵ ) and ( v ι , u ι , θ ι , z ι ) follo ws from Theorem 2.1. In addition, the integral representation form ulae for b oth solutions can also be deriv ed via Duhamel’s principle. 38 F ollowing the pro of of conv ergence for approximate solutions (Lemmas 2.5 – 2.8), we hav e ( z ϵ x − z ι x ) ( x, t ) = Z R [ H x ( x, t ; y , 0; D ϵ ) − H x ( x, t ; y , 0; D ι )] z ϵ 0 ( y ) d y + Z R H x ( x, t ; y , 0; D ι ) ( z ϵ 0 − z ι 0 ) ( y ) d y − Z t 0 Z R [ H x ( x, t ; y , τ ; D ϵ ) − H x ( x, t ; y , τ ; D ι )] K ϕ ( θ ϵ ) z ϵ ( y , τ ) d y d τ − Z t 0 Z R H x ( x, t ; y , τ ; D ι ) [ K ϕ ( θ ϵ ) z ϵ ( y , τ ) − K ϕ ( θ ι ) z ι ( y , τ )] d y d τ Note, how ev er, that the set of discontin uities D ϵ and D ι ma y not coincide; compare with Lemma 2.6. W e deﬁne instead D = D ϵ ∪ D ι , and note that all the previous estimates remain v alid on D . Using the L ∞ -b ound for H x from Lemma 1.3, we estimate that     Z R H x ( x, t ; y , 0; D ι ) ( z ϵ 0 − z ι 0 ) ( y ) d y     ≤ O (1) Z R | H x ( x, t ; y , 0; D ι ) | d y ∥ z ϵ 0 − z ι 0 ∥ L ∞ x ≤ O (1) 1 √ t ∥ z ϵ 0 − z ι 0 ∥ L ∞ x . This together with Lemma 2.6 implies that, for suﬃciently small δ and t ♯ , √ t | log t | ∥ z ϵ x ( · , t ) − z ι x ( · , t ) ∥ L ∞ x ≤ O (1) | log t | ∥ z ϵ 0 − z ι 0 ∥ L ∞ x + C 2  p t ♯ + δ  Å ∥| v ϵ − v ι |∥ ∞ + 1 | log t | ∥| v ϵ − v ι |∥ B V + 1 | log t | ∥| v ϵ − v ι |∥ 1 +         √ τ | log τ | ( u ϵ x − u ι x )         ∞ +         θ ϵ − θ ι | log τ |         ∞ + 1 | log t | ∥| z ϵ − z ι |∥ ∞ ã . F ollowing the similar argumen ts as in Lemmas 2.5, 2.6, 2.7, and 2.8, we ﬁnd that F [ v ϵ − v ι , u ϵ − u ι , θ ϵ − θ ι , z ϵ − z ι ] ≤ 15 C 2  δ + p t ♯ | log t ♯ |  F [ v ϵ − v ι , u ϵ − u ι , θ ϵ − θ ι , z ϵ − z ι ] + O (1) | log ( t ) | ∥ θ ϵ 0 − θ ι 0 ∥ L ∞ x + O (1) ∥ θ ϵ 0 − θ ι 0 ∥ L 1 x + O (1) √ t | log( t ) | ∥ θ ϵ 0 − θ ι 0 ∥ L ∞ x + O (1) | log ( t ) | ∥ u ϵ 0 − u ι 0 ∥ L ∞ x + O (1) ∥ u ϵ 0 − u ι 0 ∥ L ∞ x + O (1) ∥ u ϵ 0 − u ι 0 ∥ L 1 x + ∥ v ϵ 0 − v ι 0 ∥ L 1 x + ∥ v ϵ 0 − v ι 0 ∥ L ∞ x + ∥ v ϵ 0 − v ι 0 ∥ B V + O (1) | log ( t ) | ∥ z ϵ 0 − z ι 0 ∥ L ∞ x + + O (1) ∥ z ϵ 0 − z ι 0 ∥ L ∞ x + O (1) ∥ z ϵ 0 − z ι 0 ∥ L 1 x . By choosing δ , t ♯ suﬃcien tly small suc h that 15 C 2  δ + p t ♯ | log t ♯ |  < 1 , w e obtain the desired results. This completes the proof. □ No w, by collecting the previous results in this section, we are at the stage of concluding the main theorem on stabilit y and uniqueness of weak solutions. 39 Theorem 2.5. Ther e exists a univer al p ositive c onstant δ 0 such that the fol lowing holds. Supp ose that the initial data ( v ϵ 0 , u ϵ 0 , θ ϵ 0 , z ϵ 0 ) and ( v ι 0 , u ι 0 , θ ι 0 , z ι 0 ) b oth satisfy the smal lness c onditions: ∥ v 0 − 1 ∥ L 1 x + ∥ v 0 ∥ B V + ∥ u 0 ∥ L 1 x + ∥ u 0 ∥ B V + ∥ θ 0 − 1 ∥ L 1 x + ∥ θ 0 ∥ B V + ∥ z 0 ∥ L 1 x + ∥ z 0 ∥ B V ≤ δ ∗ . L et ( v ϵ , u ϵ , θ ϵ , z ϵ ) and ( v ι , u ι , θ ι , z ι ) b e the c orr esp onding we ak solutions in the sense of Deﬁni- tion 0.1. Then ther e exists a p ositive c onstant C 2 such that F [ v ϵ − v ι , u ϵ − u ι , θ ϵ − θ ι , z ϵ − z ι ] ≤ C 2 Ä ∥ v ϵ 0 − v ι 0 ∥ L 1 x + ∥ v ϵ 0 − v ι 0 ∥ L ∞ x + ∥ v ϵ 0 − v ι 0 ∥ B V + ∥ u ϵ 0 − u ι 0 ∥ L ∞ x + ∥ u ϵ 0 − u ι 0 ∥ L 1 x + ∥ θ ϵ 0 − θ ι 0 ∥ L ∞ x + ∥ θ ϵ 0 − θ ι 0 ∥ L 1 x + ∥ z ϵ 0 − z ι 0 ∥ L ∞ x + ∥ z ϵ 0 − z ι 0 ∥ L 1 x ä , wher e F is the functional deﬁne d in Eq. (2.36) . In p articular, for δ ∗ suﬃciently smal l, ther e exists t ∗ > 0 such that Eq. (0.4) admits a unique we ak solution in the sense of Deﬁnition 0.1 for t ∈ [0 , t ∗ ) . Pr o of. The existence of w eak solutions has b een established in Theorems 2.1 and 2.2, whenev er δ ∗ < δ and t < t ♯ with δ and t ♯ in Theorem 2.1. Then, by Lemma 2.12, the w eak solution remains small ov er t ∈ (0 , t ∗ ) . W e th us deduce from Lemma 2.13 the estimate: F [ v ϵ − v ι , u ϵ − u ι , θ ϵ − θ ι , z ϵ − z ι ] ≤ C 2 Ä ∥ v ϵ 0 − v ι 0 ∥ L 1 x + ∥ v ϵ 0 − v ι 0 ∥ L ∞ x + ∥ v ϵ 0 − v ι 0 ∥ B V + ∥ u ϵ 0 − u ι 0 ∥ L ∞ x + ∥ u ϵ 0 − u ι 0 ∥ L 1 x + ∥ θ ϵ 0 − θ ι 0 ∥ L ∞ x + ∥ θ ϵ 0 − θ ι 0 ∥ L 1 x + ∥ z ϵ 0 − z ι 0 ∥ L ∞ x + ∥ z ϵ 0 − z ι 0 ∥ L 1 x ä , 0 < t < t ∗ . In particular, if the t w o solutions ha v e the same initial data, then F [ v ϵ − v ι , u ϵ − u ι , θ ϵ − θ ι , z ϵ − z ι ] = 0 , (2.55) whic h implies the t w o solutions coincide almost ev erywhere. Moreo v er, since b oth solutions b elong to the class (0.11), they satisfy the following con tin uity prop erties: • v ( x, t ) has both left and right limits at x ∈ R for 0 < t < t ∗ , • u ( x, t ) , θ ( x, t ) and z ( x, t ) are con tinuous on R for 0 < t < t ∗ . Using these con tin uit y prop erties, w e conclude that ( v ϵ , u ϵ , θ ϵ , z ϵ ) = ( v ι , u ι , θ ι , z ι ) , x ∈ R , 0 < t < t ∗ . This completes the pro of. □ 3. Green’s Function This section is dev oted to deriving the point wise estimates for the Green’s function G asso ciated to the PDE of the form Eq. (3.2), which is a balance law of div ergence form with the drift coeﬃcient in BV. In this lo w regularit y regime, G has both singular and regular parts, whose detailed estimates are crucial for the analysis of the large-time b ehaviour of weak solutions to Eq. (0.1) in §4 b elo w. The notations in the subsequen t parts of the pap er follows W ang–Y u–Zhang [31]. In par- ticular, we denote b y G , λ j , and λ ∗ j the Green’s function, the eigen v alues, and the corresp onding appro ximated eigen v alues. The sym bol ˆ M j is reserv ed for the mo des of the Green’s function at frequency λ j ; see Eq. (3.11) for the precise deﬁnition. 40 W e ﬁrst in tro duce the v ariables E = e + u 2 2 + q z , U = ( v , u, E , z ) , p ( v , e ( E , u, z )) = a c v v ( E − u 2 2 − q z ) ≃ E − u 2 2 − q z v . Note that e u = − u , e E = 1 , and e z = − q . Eq. (0.1) can b e expressed as                            v t − u x = 0 , u t + p v v x + p e e u u x + p e e E E x + p e e z z x =  µu x v  x , E t + up v v x + ( p + up e e u ) u x + up e e E E x + up e e z z x = ÅÅ µu v − ν u c v v ã u x + ν c v v E x ã x + ÅÅ − q ν c v v + q D v 2 ã z x ã x , z t + K ϕ ( θ ) z = Å D v 2 z x ã x . (3.1) This system can also be expressed in vector form as U t + F ( U ) x = ( B ( U ) U x ) x + A ⇐ ⇒ U t + F ′ ( U ) U x = ( B ( U ) U x ) x + A, (3.2) where U = á v u E z ë , F ( U ) = á − u p pu 0 ë , F ′ ( U ) = á 0 − 1 0 0 p v − p e u p e − q p e p v u p − p e u 2 p e u − q p e u 0 0 0 0 ë , B ( U ) = á 0 0 0 0 0 µ v 0 0 0 Ä µ v − ν c v v ä u ν c v v − q ν c v v + q D v 2 0 0 0 D v 2 ë , A = á 0 0 0 − K ϕ ( θ ) z ë . (3.3) Consider U = ¯ U + V , where ¯ U is a constan t state. Then, the linearization of Eq. (3.1) around ¯ U reads: V t + F ′ ( ¯ U ) V x − B ( ¯ U ) V xx =  N 1 ( V ; ¯ U ) + N 2 ( V ; ¯ U )  x + A, (3.4) where N 1 and N 2 are the nonlinear terms originating from the hyperb olic and parab olic parts, resp ectiv ely: N 1 ( V ; ¯ U ) = −  F ( ¯ U + V ) − F ( ¯ U ) − F ′ ( ¯ U ) V  , N 2 ( V ; ¯ U ) =  B ( U ) − B ( ¯ U )  V x . (3.5) Deﬁnition 3.1. W e write G ( x, t ; ¯ U ) for the Gr e en ’s function of the line arise d e quation (3.4) : ® ∂ t G ( x, t ; ¯ U ) =  − F ′ ( ¯ U ) ∂ x + B ( ¯ U ) ∂ xx  G ( x, t ; ¯ U ) , G ( x, 0; ¯ U ) = δ ( x ) I , (3.6) wher e I is the identity matrix, δ ( x ) is the Dir ac delta function, and F ′ ( U ) and B ( U ) ar e the c o eﬃcient matric es after line arization ar ound the c onstant state ¯ U . Ov er the frequency domain, the F ourier-transformed Green’s function ˆ G ( η , t ; ¯ U ) satisﬁes ® ∂ t ˆ G ( x, t ; ¯ U ) =  − iη F ′ ( ¯ U ) − η 2 B ( ¯ U )  ˆ G ( x, t ; ¯ U ) , ˆ G ( η , 0; ¯ U ) = I . (3.7) 41 The eigenv alues λ j are obtained b y solving the c haracteristic equation: 0 = det( λI + iη F ′ ( ¯ U ) + η 2 B ( ¯ U )) = Å λ + η 2 D v 2 ã ï λη 2 pp e +  λ  λ + η 2 µ v  − η 2 p v  Å λ + η 2 ν c v v ãò . (3.8) Lemma 3.1. Ther e exists a suﬃciently smal l p ositive c onstant σ 0 such that the matrix − iη F ′ ( ¯ U ) − η 2 B ( ¯ U ) has 4 distinct eigenvalues λ 1 – λ 4 whenever | η | > 0 and | Im( η ) | < σ 0 . Pr o of. The second factor on the right-most term of Eq. (3.8) coincides with the characteristic equation in [31, Lemma 3.1], wherein it is sho wn that it has 3 distinct roots λ 1 , λ 2 , λ 3 pro vided that | η | > 0 and | Im( η ) | < σ 0 for some σ 0 > 0 suﬃciently small. Clearly , λ 4 = − D v 2 η 2 is another eigen v alue. It remains to v erify that λ 4 is distinct from λ j ( j = 1 , 2 , 3) . Substituting λ 4 = − D v 2 η 2 in to the second factor of (3.8), we obtain that λ 4 η 2 pp e +  λ 4  λ 4 + η 2 µ v  − η 2 p v  Å λ 4 + η 2 ν c v v ã = − η 4 c v v 4 ï η 2 D ( D − µv )( ν v − c v D ) v 2 + aθ ( v ν − c v D − aD ) ò . (3.9) F or λ 4 to coincide with an y of λ j ( j = 1 , 2 , 3) , the right-most term in (3.9) m ust constan tly v anish. In this case, one of the following tw o conditions must hold: • [ a ] D = µv and ν v = ( a + c v ) D = γ c v D ; • [ b ] There exists a special η suc h that η 2 = − aθ ( v ν − c v D − aD ) v 2 D ( D − µv )( ν v − c v D ) . Condition [b] dep ends on a sp eciﬁc η , but w e are concerned with all η with | η | > 0 and | Im( η ) | < σ 0 . Th us [ b ] is imp ossible. On the other hand, it is kno wn that the Prandtl num ber Pr = µc p ν for ideal gas is smaller than one. Hence, µc v ν = Pr c v c p = Pr 1 γ < 1 γ , (3.10) whic h implies that ν > µc v γ . This rules out [ a ] . □ Lemma 3.2. F or any 0 < r < R , ther e exists a p ositive c onstant ˜ b > 0 such that Re( λ j ( η )) ≤ − ˜ b for al l r e al η with r < | η | < R , j = 1 , 2 , 3 , 4 . Pr o of. F or j = 1 , 2 , 3 , it follows from [31, Lemma 3.2] that there exists b > 0 such that Re( λ j ( η )) ≤ − b for all real η with r < | η | < R . F or j = 4 , since D > 0 and v > 0 , we ha v e Re( λ 4 ) = − D v 2 | η | 2 ≤ − D v 2 r 2 . T ake ˜ b = min( b, D v 2 r 2 ) to conclude. □ T o proceed, w e represent the Green’s function in the frequency domain as follo ws: ˆ G ( η , t ; ¯ U ) = 4 X j =1 e λ j t ˆ M j , ˆ M j = adj  s + iη F ′ ( ¯ U ) + η 2 B ( ¯ U )    s = λ j Q k  = j ( λ j − λ k ) , (3.11) 42 where adj( A ) is the adjugate matrix of A . The standard Green’s function G ( x, t ; ¯ U ) can b e deriv ed via the inv erse F ourier transform: G ( x, t ; ¯ U ) = F − 1 ( ˆ G ( η , t ; ¯ U )) . It has b een sho wn in [22, 31] that G ( x, t ; ¯ U ) is decomposed into singular and regular parts: G ( x, t ; ¯ U ) = G ∗ ( x, t ; ¯ U ) | {z } regular part + G † ( x, t ; ¯ U ) | {z } singular part . (3.12) W e shall estimate these t w o parts in detail below. W e ﬁrst expand the eigenv alues λ j as Lauren t p olynomials in terms of η − 1 . F or suﬃciently large η , we hav e the following asymptotic expansions (see [31]): λ 1 = v p v µ − v 3  ν θ e p 2 v + µpp e p v  ν µ 3 θ e η − 2 + v 5 p v  µ 2 p 2 p 2 e + 2 ν 2 θ 2 e p 2 v + µpp e p v (3 ν θ e + µ )  ν 2 µ 5 θ 2 e η − 4 − v 7 p v  µ 3 p 3 p 3 e + 3 µ 2 p 2 p 2 e p v (2 ν θ e + µ ) + 5 ν 3 θ 3 e p 3 v + µpp e p 2 v  10 ν 2 θ 2 e + 4 ν µθ e + µ 2  ν 3 µ 7 θ 3 e η − 6 + O (1) η − 8 , λ 2 = − η 2 µ v + v ( µpp e + ν θ e p v − µp v ) µ ( µ − ν θ e ) + η − 2 Ñ v 3 Ä µ 2 pp e − p v ( µ − ν θ e ) 2 ä ( µpp e + p v ( ν θ e − µ )) µ 3 ( µ − ν θ e ) 3 é + η − 4 Ç v 5  2 µ 5 p 3 p 3 e + µ 2 p 2 p 2 e p v  ν 3 θ 2 e − 5 ν 2 µθ 2 e + 10 ν µ 2 θ e − 6 µ 3  µ 5 ( µ − ν θ e ) 5 å + v 5 Ä µpp e p 2 v ( µ − ν θ e ) 2  3 ν 2 θ 2 e − 8 ν µθ e + 6 µ 2  − 2 p 3 v ( µ − ν θ e ) 5 ä µ 5 ( µ − ν θ e ) 5 é + η − 6 Ç v 7  5 µ 7 p 4 p 3 e − µ 3 p 3 p 3 e p v  ν 4 θ 3 e − 7 ν 3 µθ 3 e + 21 ν 2 µ 2 θ 2 e − 35 ν µ 3 θ e + 20 µ 4  µ 7 ( µ − ν θ e ) 7 å + v 7 Ä 3 µ 2 p 2 p 2 e p 2 v ( µ − ν θ e ) 2  − 2 ν 3 θ 3 e + 9 ν 2 µθ 2 e − 15 ν µ 2 θ e + 10 µ 3  ä µ 7 ( µ − ν θ e ) 7 − v 7 Ä µpp e p 3 v ( µ − ν θ e ) 3  − 10 ν 3 θ 3 e + 36 ν 2 µθ 2 e − 45 ν µ 2 θ e + 20 µ 3  + 5 p 4 v ( µ − ν θ e ) 7 ä µ 7 ( µ − ν θ e ) 7 é + O (1) η − 8 , λ 3 = − η 2 ν θ e v + pv p e ν θ e − µ + η − 2 Ç pv 3 p e ( ν θ e ( pp e + p v ) − µp v ) ν θ e ( ν θ e − µ ) 3 å + η − 4 Ñ pv 5 p e Ä 2 ν 2 p 2 θ 2 e p 2 e + pp e p v  4 ν 2 θ 2 e − 5 ν µθ e + µ 2  + p 2 v ( µ − ν θ e ) 2 ä ν 2 θ 2 e ( ν θ e − µ ) 5 é + η − 6 Ç pv 7 p e  5 ν 3 p 3 θ 3 e p 2 e − p 2 p 2 e p v  − 15 ν 3 θ 3 e + 21 ν 2 µθ 2 e − 7 ν µ 2 θ e + µ 3  ν 3 θ 3 e ( ν θ e − µ ) 7 − 3 pp e p 2 v ( µ − 3 ν θ e ) ( µ − ν θ e ) 2 ν 3 θ 3 e ( ν θ e − µ ) 7 − pv 7 p e Ä p 3 v ( µ − ν θ e ) 3 ä η 6 ν 3 θ 3 e ( ν θ e − µ ) 7 é + O (1) η − 8 , λ 4 = − η 2 D v 2 . (3.13) 43 F rom [31], the real parts of the ab o ve high-frequency eigenv alues satisfy that 0 > Re( λ 1 ) > Re( λ 2 ) > Re( λ 3 ) . On the other hand, the Lewis num ber Le is approximately 1 for ideal gases, while the Prandtl n um b er Pr < 1 . Here, Le = ν v c p D , and recall that γ = c p c v > 1 . W e thus hav e ν θ e v D = ν v c v D = ν v c p D c p c v = Le γ > 1 as well as µv D = Pr · Le < 1 . F rom this w e conclude that 0 > Re( λ 1 ) > Re( λ 2 ) > Re( λ 4 ) > Re( λ 3 ) . (3.14) W e also ha ve the following lo w-frequency asymptotic expansions ( | η | → 0 , see [31]): λ 1 = η 2 ν θ e p v v ( pp e − p v ) − η 4 ν 2 pθ 2 e p e p v ( µpp e + p v ( ν θ e − µ )) v 3 ( p v − pp e ) 4 − η 6 ν 3 pθ 3 e p e p v Ä µ 2  − p 3  p 3 e + µp 2 p 2 e p v ( µ − 3 ν θ e ) + pp e p 2 v  − 2 ν 2 θ 2 e + ν µθ e + µ 2  − p 3 v ( µ − ν θ e ) 2 ä v 5 ( pp e − p v ) 7 + O (1) η 6 , λ 2 = − iη √ pp e − p v − η 2 ( ν pθ e p e + µpp e − µp v ) 2 v ( pp e − p v ) − iη 3 Ä − p 2 p 2 e ( µ − ν θ e ) 2 + 2 pp e p v  2 ν 2 θ 2 e − ν µθ e + µ 2  − µ 2 p 2 v ä 8 v 2 ( pp e − p v ) 5 / 2 + η 4 ν 2 pθ 2 e p e p v ( µpp e + p v ( ν θ e − µ )) 2 v 3 ( p v − pp e ) 4 + O (1) η 4 , λ 3 = iη √ pp e − p v − η 2 ( ν pθ e p e + µpp e − µp v ) 2 v ( pp e − p v ) − iη 3 Ä p 2 p 2 e ( µ − ν θ e ) 2 − 2 pp e p v  2 ν 2 θ 2 e − ν µθ e + µ 2  + µ 2 p 2 v ä 8 v 2 ( pp e − p v ) 5 / 2 , + η 4 ν 2 pθ 2 e p e p v ( µpp e + p v ( ν θ e − µ )) 2 v 3 ( p v − pp e ) 4 + O (1) η 4 , λ 4 = − η 2 D v 2 . (3.15) 3.1. High-F requency Analysis ( η → ∞ ), Singular P art. In this part, w e fo cus on analyzing the singular part of Green’s function, which captures high-frequency b ehaviours and singularities of the Green’s function. Due to the inv erse p ow er of η in the asymptotic expansion (3.13) as | η | → ∞ , λ j is not analytic at η = 0 . T o resolve this, we construct analytic appro ximations λ ∗ j in the neigh b ourho o d 44 of the real axis: λ ∗ 1 = β ∗ 1 + 3 X k =1 A 1 ,k (1 + η 2 ) k − K 1 ( η 2 + 1) 4 , λ ∗ 2 = − α ∗ 2 η 2 + β ∗ 2 + 3 X k =1 A 2 ,k (1 + η 2 ) k − K 2 ( η 2 + 1) 4 , λ ∗ 3 = − α ∗ 3 η 2 + β ∗ 3 + 3 X k =1 A 3 ,k (1 + η 2 ) k − K 3 ( η 2 + 1) 4 , λ ∗ 4 = − α ∗ 4 η 2 = − D v 2 η 2 , (3.16) where the coeﬃcients α ∗ j , β ∗ j , and A j,k are listed in Appendix A. Moreo v er, K 1 , K 2 , and K 3 are suﬃcien tly large positive constants ensuring the follo wing Lemma. Lemma 3.3. Fix ¯ U . W e c an ﬁnd p ositive c onstants K 1 , K 2 , K 3 , σ 0 , σ ∗ 0 and σ ∗ 1 such that (1) The appr oximate eigenvalues λ ∗ j ; j = 1 , 2 , 3 , 4 , ar e analytic in { η : | Im( η ) | < σ 0 } . (2) λ ∗ j (j=1,2,3) is an appr oximation of λ j ac cur ate up to the p ower ( η 2 + 1) − 3 : λ ∗ j = λ j + O (1) | η | − 8 . (3) In the r e gion { η : | Im( η ) | < σ 0 } , al l the appr oximate d eigenvalues λ ∗ j (j=1,2,3,4) have distinct ne gative r e al p art. Inde e d, sup | Im( η ) | <σ 0 Re ( λ ∗ 1 ) < − σ ∗ 0 , sup | Im( η ) | <σ 0 Re  λ ∗ 2 + µ 2 v  η 2 + 1   < − σ ∗ 0 , sup | Im( η ) | <σ 0 Re Å λ ∗ 3 + ν θ e 2 v  η 2 + 1  ã < − σ ∗ 0 , sup | Im( η ) | <σ 0 Re Å λ ∗ 4 + D 2 v 2  η 2 + 1  ã < − σ ∗ 0 , min j,k inf | Im( η ) | <σ 0   Re  λ ∗ k − λ ∗ j    = σ ∗ 1 . Pr o of. • F or λ ∗ 1 , λ ∗ 2 , λ ∗ 3 , it is shown in [31, Lemma 3.3] that they satisfy (1) and (2). • Since λ ∗ 4 = − D v 2 η 2 is a polynomial in η , it is analytic on C . • F or (3), applying (3.14), w e may ﬁnd suﬃcien tly large C 1 > 0 and deﬁne σ ∗ 0 = min ß 1 2 | β ∗ 1 | , 1 2 α ∗ 2 C 2 1 , 1 2 D v 2 η 2 , 1 2 α ∗ 3 C 2 1 ™ so that Re( λ ∗ j ) < − σ ∗ 0 . Next, for | η | ≤ C 1 , we kno w that Re( λ ∗ j ) ≈ β ∗ j − K j . Then, we can choose K 1 , K 2 , K 3 suﬃcien tly large with K 1 < K 2 < K 3 to ensure 0 > β ∗ 1 − K 1 > β ∗ 2 − K 2 > β ∗ 3 − K 3 . F or λ 4 , since Re( λ 4 ) ≈ − D v 2 | η | 2 , we make additional assumptions that K 2 < β ∗ 2 + D v 2 | η | 2 and K 3 > β ∗ 3 + D v 2 | η | 2 , which ensure β ∗ 2 − K 2 > − D v 2 | η | 2 > β ∗ 3 − K 3 . Therefore, we obtain 0 > Re( λ ∗ 1 ) > Re( λ ∗ 2 ) > Re( λ ∗ 4 ) > Re( λ ∗ 3 ) , | η | ≤ C 1 . 45 On the other hand, for | η | > C 1 , we kno w that Re( λ ∗ 2 ) ≈ − α ∗ 2 | η | 2 , Re( λ ∗ 4 ) = − D v 2 | η | 2 and Re( λ ∗ 3 ) ≈ − α ∗ 3 | η | 2 are negative. W e then apply (3.14) to obtain that 0 > Re( λ ∗ 1 ) > Re( λ ∗ 2 ) > Re( λ ∗ 4 ) > Re( λ ∗ 3 ) , | η | > C 1 . Th us, there exist p ositive gaps b etw een an y tw o Re( λ ∗ j ) . Set σ ∗ 1 to b e the minim um gap σ ∗ 1 = min j,k inf | Im( η ) | <σ 0   Re  λ ∗ k − λ ∗ j    to conclude the pro of. □ No w, w e are ready to construct the singular part of Green’s function using λ j and ˆ M j . A direct computation yields the expression of ˆ M j := ˆ M j ( η ; λ ) when η tends to inﬁnity: ˆ M j = á A 11 ( λ j ) A 21 ( λ j ) A 31 ( λ j ) A 41 ( λ j ) A 12 ( λ j ) A 22 ( λ j ) A 32 ( λ j ) A 42 ( λ j ) A 13 ( λ j ) A 23 ( λ j ) A 33 ( λ j ) A 43 ( λ j ) A 14 ( λ j ) A 24 ( λ j ) A 34 ( λ j ) A 44 ( λ j ) ë Q k  = j ( λ j − λ k ) , (3.17) where A 11 ( λ j ) = Å λ j + η 2 D v 2 ã ï  λ j + η 2 µ v  Å λ j + η 2 ν c v v ã + η 2 pp e ò , A 21 ( λ j ) = iη Å λ j + η 2 ν c v v ã Å λ j + η 2 D v 2 ã , A 31 ( λ j ) = η 2 p e Å λ j + η 2 D v 2 ã , A 41 ( λ j ) = − q η 2 p e Å λ j + η 2 D v 2 ã , A 12 ( λ j ) = − iη p v Å λ j + η 2 D v 2 ã Å λ j + η 2 ν c v v ã , A 22 ( λ j ) = λ j Å λ j + η 2 D v 2 ã Å λ j + iη p e + η 2 ν c v v ã , A 32 ( λ j ) = − λ j ( iη p e ) Å λ j + η 2 D v 2 ã , A 42 ( λ j ) = q λ j ( iη p e ) Å λ j + η 2 D v 2 ã , A 13 ( λ j ) = − Å λ j + η 2 D v 2 ã η p v ï η p + iu Å λ j + η 2 ν c v v ãò , A 23 ( λ j ) = − Å λ j + η 2 D v 2 ã ï λ j Å iη p + η 2 µu v − η 2 ν u c v v ã − iη λ j p e u 2 − η 2 p v u ò , A 33 ( λ j ) = Å λ j + η 2 D v 2 ã h λ j  λ j + η 2 µ v  − iη λ j p e u − η 2 p v i , A 43 ( λ j ) = q ïÅ λ j + iη p e + η 2 ν c v v ã  λ j  λ j + η 2 µ v  − η 2 p v  + λ j η 2 pp e ò − q A 33 ( λ j ) , A 14 ( λ j ) = A 24 ( λ j ) = A 34 ( λ j ) = 0 , A 44 ( λ j ) = λη 2 pp e +  λ  λ + η 2 µ v  − η 2 p v  Å λ + η 2 ν c v v ã . 46 As ˆ M j ( η ; λ j ) ma y b e non-analytic, we consider approximate eigen v alues λ ∗ j (3.16) to obtain analytic matrix ˆ M j ( η ; λ ∗ j ) , which satisﬁes: Lemma 3.4. The matrix ˆ M j ( η ; λ ∗ j ) is analytic in the r e gion { η : | Im( η ) | < σ 0 } ar ound the r e al axis, and it has the fol lowing exp ansion at inﬁnity: ˆ M ∗ j = M ∗ , 0 j + iη − 1 M ∗ , 1 j + η − 2 M ∗ , 2 j + iη − 3 M ∗ , 3 j + η − 4 M ∗ , 4 j + O (1) η − 5 , j = 1 , 2 , 3 , 4 | η | → ∞ , (3.18) wher e M ∗ ,k j ar e liste d in App endix A. Pr o of. F or j = 1 , 2 , 3 , it follows from [31, Lemma 3.4] that ˆ M ∗ j = adj( s + iη F ′ + η 2 B ) | s = λ ∗ j Q k  = j ( λ ∗ j − λ ∗ k ) is analytic in | Im( η ) | < σ 0 and admits the asymptotic expansion in Eq. (3.18). This relies on the analyticit y of the appro ximate eigen v alues λ ∗ j and the presence of the uniform spectral gap sho wn in Lemma 3.3. F or j = 4 , the analyticit y of ˆ M ∗ 4 follo ws from that of λ ∗ 4 , as w ell as the sp ectral gap | Re( λ ∗ 4 − λ ∗ j ) | ≥ σ ∗ 1 established in Lemma 3.3. □ Observ e that the top-left 3 × 3 -minor of the matrix ˆ M ∗ j has already b een computed in [31]. The other en tries of ˆ M ∗ j ( j = 1 , 2 , 3 , 4 ) are deriv ed via the observ ation (3.19) below. By using Lemma 3.3, w e deriv e the follo wing expansions for j = 1 , 2 , 3 : ( λ ∗ j − λ 1 )( λ ∗ j − λ 2 )( λ ∗ j − λ 3 ) =  λ ∗ j − λ ∗ 1 − O (1) | η | − 8   λ ∗ j − λ ∗ 2 − O (1) | η | − 8   λ ∗ j − λ ∗ 3 − O (1) | η | − 8  = ( λ ∗ j − λ ∗ 1 )( λ ∗ j − λ ∗ 2 )( λ ∗ j − λ ∗ 3 ) + O (1) | η | − 8 ( λ ∗ j − λ ∗ 1 )( λ ∗ j − λ ∗ 2 ) + O (1) | η | − 8 ( λ ∗ j − λ ∗ 1 )( λ ∗ j − λ ∗ 3 ) + O (1) | η | − 16 ( λ ∗ j − λ ∗ 1 ) + O (1) | η | − 8 ( λ ∗ j − λ ∗ 2 )( λ ∗ j − λ ∗ 3 ) + O (1) | η | − 16 ( λ ∗ j − λ ∗ 2 ) + O (1) | η | − 16 ( λ ∗ j − λ ∗ 3 ) + O (1) | η | − 24 . By Lemma 3.3, λ ∗ j = λ j + O (1) | η | − 8 and min j,k inf | Im( η ) | <σ 0    Re Ä λ ∗ k − λ ∗ j ä    = σ ∗ 1 for a p ositive n umber σ ∗ 1 . These imply that the terms of order | η | − 16 or higher decay muc h faster than those of order | η | − 8 . Th us, ( λ ∗ j − λ 1 )( λ ∗ j − λ 2 )( λ ∗ j − λ 3 ) ≃ ( λ ∗ j − λ ∗ 1 )( λ ∗ j − λ ∗ 2 )( λ ∗ j − λ ∗ 3 ) + O (1) | η | − 8 . Moreo v er, direct computation yields that, for j = 1 , 2 , 3 , A 43 ( λ ∗ j ) Q k  = j Ä λ ∗ j − λ ∗ k ä = q det( λ ∗ j I + iη F ′ ( ¯ U ) + η 2 B ( ¯ U )) Ä λ ∗ j − λ 4 ä Q k  = j Ä λ ∗ j − λ ∗ k ä − q A 33 ( λ ∗ j ) Q k  = j Ä λ ∗ j − λ ∗ k ä = O (1) | η | − 8 − q A 33 ( λ ∗ j ) Q k  = j Ä λ ∗ j − λ ∗ k ä and A 44 ( λ ∗ j ) Q k  = j Ä λ ∗ j − λ ∗ k ä = det( λ ∗ j I + iη F ′ ( ¯ U ) + η 2 B ( ¯ U )) Ä λ ∗ j − λ 4 ä Q k  = j Ä λ ∗ j − λ k ä = O (1) | η | − 8 . 47 This term v anishes as η → ∞ , which result in t w o approximations: A 43 ( λ ∗ j ) Q k  = j ( λ ∗ j − λ ∗ k ) ≃ − q A 33 ( λ ∗ j ) Q k  = j Ä λ ∗ j − λ ∗ k ä , A 44 ( λ ∗ j ) Q k  = j Ä λ ∗ j − λ ∗ k ä ≃ 0 ( j = 1 , 2 , 3) . As a consequence, Ä ˆ M ∗ j ä k 4 = − q Ä ˆ M ∗ j ä k 3 , Ä ˆ M ∗ j ä 44 = 0 ( j, k ∈ { 1 , 2 , 3 } ) . (3.19) Similar arguments also lead to ˆ M ∗ 4 = ˆ M 4 = á 0 0 0 0 0 0 0 0 0 0 0 q 0 0 0 1 ë . (3.20) F or simplicit y , we take M ∗ ,k 4 = 0 for k = 1 , 2 , 3 , 4 . No w, substituting the previous formulae for λ ∗ j and ˆ M ∗ j in to the Green’s function (3.11), w e obtain the expression for the singular part of Green’s function as follo ws: ˆ G ∗ ( η , t ; ¯ U ) = 4 X j =1 ˆ G ∗ ,j ( η , t ; ¯ U ) , ˆ G ∗ ,j = e λ ∗ j t ˆ M ∗ j , j = 1 , 2 , 3 , 4 . (3.21) No w, we give the p oin t-wise estimates for ˆ G ∗ , 4 . A ccording to Lemma 3.3, we know that the real part of λ ∗ 4 has a negativ e upp er bound. F or t ≥ 1 , w e ha ve e λ ∗ 4 t + σ ∗ 0 t = e Ä − η 2 D v 2 + D 2 v 2 ( η 2 +1) ä t + σ ∗ 0 t e − D 2 v 2 ( η 2 +1) t = O (1) e − D 2 v 2 ( η 2 +1) t = O (1) e − t/C (1 + η 2 ) m , t ≥ 1 , m ∈ Z + , whic h is analytic in the region { η : | Im( η ) | < σ 0 } . Therefore, we apply the Lemma 1.7 to obtain that e λ ∗ 4 t = e − σ ∗ 0 t − σ 0 | x | , t ≥ 1 . On the other hand, for 0 < t < 1 , it is clear that e λ ∗ 4 t = e − η 2 D v 2 t . Similar to the argumen t in [31, Lemma 3.7], w e in tro duce the sym b ol Λ ∗ 4 ( x, t ) = F − 1  e λ ∗ 4 t  . Then Λ ∗ 4 ( x, t ) =      O (1) e − σ ∗ 0 t − σ 0 | x | , t ≥ 1 , 1 √ 4 π α ∗ 4 t e − x 2 4 α ∗ 4 t , 0 < t < 1 . Hence, we hav e G ∗ , 4 ( x, t ) =Λ ∗ 4 ( x, t ) ⋆ x M ∗ 4 ( x ) = Λ ∗ 4 ( x, t ) ⋆ x Ä δ ( x ) M ∗ , 0 4 ä =      O (1) e − σ ∗ 0 t − σ 0 | x | , t ≥ 1 , ñ 1 √ 4 π α ∗ 4 t e − x 2 4 α ∗ 4 t ô M ∗ , 0 4 , 0 < t < 1 . The symbol ⋆ x denotes the con v olution in the x -v ariable. Finally , w e combine the b ounds for G ∗ ,j ( x, t )( j = 1 , 2 , 3) given in [31, Theorem 3.1] and the ab ov e estimates for G ∗ , 4 ( x, t ) to arriv e at the following estimates of G ∗ ( x, t ; ¯ U ) . Theorem 3.2. The singular p art of the Gr e en ’s function deﬁne d in Eq. (3.21) satisﬁes the fol lowing b ounds. 48 F or G ∗ , 1 , whenever t > 0 ,        G ∗ , 1 ( x, t ) = e ν p ν µ t δ ( x ) M ∗ , 0 1 + O (1) e − σ ∗ 0 t − σ 0 | x | ∂ x G ∗ , 1 ( x, t ) = e ν p ν µ t Ä d dx δ ( x ) M ∗ , 0 1 − δ ( x ) M ∗ , 1 1 ä + O (1) e − σ ∗ 0 t − σ 0 | x | ∂ 2 x G ∗ , 1 ( x, t ) = e ν p ν µ t Ä d 2 dx 2 δ ( x ) M ∗ , 0 1 − d dx δ ( x ) M ∗ , 1 1 + δ ( x ) Ä M ∗ , 2 1 − A 1 , 1 tM ∗ , 0 1 ää + O (1) e − σ ∗ 0 t − σ 0 | x | . F or G ∗ , 2 ( x, t ) , G ∗ , 3 ( x, t ) , and G ∗ , 4 ( x, t ) , if t ≥ 1 then ∂ k x G ∗ ,j ( x, t ) = O (1) e − σ ∗ 0 t − σ 0 | x | , k = 0 , 1 , 2 , 3; j = 2 , 3 , 4 . If 0 < t < 1 , then                                                                                G ∗ ,j ( x, t ) = O (1) e − σ ∗ 0 t − σ 0 | x | + e β ∗ j t » 4 π α ∗ j t e − x 2 4 α ∗ j t M ∗ , 0 j , ∂ x G ∗ ,j ( x, t ) = O (1) e − σ ∗ 0 t − σ 0 | x | + ∂ x   e β ∗ j t » 4 π α ∗ j t e − x 2 4 α ∗ j t   M ∗ , 0 j − e β ∗ j t » 4 π α ∗ j t e − x 2 4 α ∗ j t M ∗ , 1 j , ∂ 2 x G ∗ ,j ( x, t ) = O (1) e − σ ∗ 0 t − σ 0 | x | + ∂ 2 x   e β ∗ j t » 4 π α ∗ j t e − x 2 4 α ∗ j t   M ∗ , 0 j − ∂ x   e β ∗ j t » 4 π α ∗ j t e − x 2 4 α ∗ j t   M ∗ , 1 j , + e β ∗ j t » 4 π α ∗ j t e − x 2 4 α ∗ j t M ∗ , 2 j , ∂ 3 x G ∗ ,j ( x, t ) = O (1) e − σ ∗ 0 t − σ 0 | x | + ∂ 3 x   e β ∗ j t » 4 π α ∗ j t e − x 2 4 α ∗ j t   M ∗ , 0 j − ∂ 2 x   e β ∗ j t » 4 π α ∗ j t e − x 2 4 α ∗ j t   M ∗ , 1 j + ∂ x   e β ∗ j t » 4 π α ∗ j t e − x 2 4 α ∗ j t   M ∗ , 2 j + e β ∗ j t » 4 π α ∗ j t e − x 2 4 α ∗ j t M ∗ , 3 j , ∂ k x G ∗ , 4 ( x, t ) = ∂ k x ñ 1 p 4 π α ∗ 4 t e − x 2 4 α ∗ 4 t ô M ∗ , 0 4 , k = 0 , 1 , 2 , 3 . The same approac h can be directly adapted to bound the time deriv ativ es. Note that F  ∂ t G ∗ , 4 ( x, t )  = λ ∗ 4 e λ ∗ 4 t ˆ M ∗ 4 = − η 2 D v 2 e λ ∗ 4 t ˆ M ∗ 4 , F  ∂ xt G ∗ , 4 ( x, t )  = iη λ ∗ 4 e λ ∗ 4 t ˆ M ∗ 4 = − iη 3 D v 2 e λ ∗ 4 t ˆ M ∗ 4 . T aking the in verse F ourier transform, w e deduce that ∂ t G ∗ , 4 ( x, t ) = D v 2 ∂ 2 x ñ 1 p 4 π α ∗ 4 t e − x 2 4 α ∗ 4 t ô ⋆ x Ä δ ( x ) M ∗ , 0 4 ä = α ∗ 4 ∂ 2 x ñ 1 p 4 π α ∗ 4 t e − x 2 4 α ∗ 4 t ô M ∗ , 0 4 , ∂ xt G ∗ , 4 ( x, t ) = D v 2 ∂ 3 x ñ 1 p 4 π α ∗ 4 t e − x 2 4 α ∗ 4 t ô ⋆ x Ä δ ( x ) M ∗ , 0 4 ä = α ∗ 4 ∂ 3 x ñ 1 p 4 π α ∗ 4 t e − x 2 4 α ∗ 4 t ô M ∗ , 0 4 for 0 < t < 1 . T ogether with [31, Theorem 3.2], it yields the following. 49 Theorem 3.3. The time-derivatives of the singular p art of Gr e en ’s function satisfy that ∂ t G ∗ , 1 ( x, t ) = v p v µ e vp v µ t δ ( x ) M ∗ , 0 1 + O (1) e − σ ∗ 0 t − σ 0 | x | , ∂ xt G ∗ , 1 ( x, t ) = v p v µ e vp v µ t d dx δ ( x ) M ∗ , 0 1 − v p v µ e vp w µ t δ ( x ) M ∗ , 1 1 + O (1) e − σ ∗ 0 t − σ 0 | x | , ∂ t G ∗ ,j ( x, t ) = O (1) e − σ ∗ 0 t − σ 0 | x | +              α ∗ j ∂ 2 x ñ e β ∗ j t √ 4 π α ∗ j t e − x 2 4 α ∗ j t ô M ∗ , 0 j − α ∗ j ∂ x ñ e β ∗ j t √ 4 π α ∗ j t e − x 2 4 α ∗ j t ô M ∗ , 1 j + e β ∗ j t √ 4 π α ∗ j t e − x 2 4 α ∗ j t Ä α ∗ j M ∗ , 2 j + β ∗ j M ∗ , 0 j ä , 0 < t < 1 , 0 , t ≥ 1 . ∂ xt G ∗ ,j ( x, t ) = O (1) e − σ ∗ 0 t − σ 0 | x | +                        α ∗ j ∂ 3 x ñ e β ∗ j t √ 4 π α ∗ j t e − x 2 4 α ∗ j t ô M ∗ , 0 j − α ∗ j ∂ 2 x ñ e β ∗ j t √ 4 π α ∗ j t e − x 2 4 α ∗ j t ô M ∗ , 1 j + ∂ x ñ e β ∗ j t √ 4 π α ∗ j t e − x 2 4 α ∗ j t ô Ä α ∗ j M ∗ , 2 j + β ∗ j M ∗ , 0 j ä + e β ∗ j t √ 4 π α ∗ j t e − x 2 4 α ∗ j t Ä α ∗ j M ∗ , 3 j − β ∗ j M ∗ , 1 j ä , 0 < t < 1 , 0 , t ≥ 1 . ∂ t G ∗ , 4 ( x, t ) =      α ∗ 4 ∂ 2 x ñ 1 √ 4 π α ∗ 4 t e − x 2 4 α ∗ 4 t ô M ∗ , 0 4 , 0 < t < 1 , 0 , t ≥ 1 . ∂ xt G ∗ , 4 ( x, t ) =      α ∗ 4 ∂ 3 x ñ 1 √ 4 π α ∗ 4 t e − x 2 4 α ∗ 4 t ô M ∗ , 0 4 , 0 < t < 1 , 0 , t ≥ 1 . In the ab ove, j ∈ { 2 , 3 } . 3.2. Lo w-F requency Analysis ( η → 0 ), Regular P art. This subsection is dev oted to the p oin twise estimates for the regular part of Green’s function, G † = G − G ∗ . In view of Eqs. (3.11), (3.21) and the fact that λ 4 = λ ∗ 4 , the F ourier transform of the regular part G † can b e expressed as ˆ G † ( η , t ; ¯ U ) = 4 X j =1 e λ j t ˆ M j − 4 X j =1 e λ ∗ j t ˆ M ∗ j = 3 X j =1 e λ j t ˆ M j − 3 X j =1 e λ ∗ j t ˆ M ∗ j . (3.22) As λ 4 do es not contribute to the regular part, Eq. (3.22) is consistent with Eq. (3.35) in [31]. W e ma y thus directly adapt the arguments therein. The estimates we need for ˆ G † in the low frequency regime are summarised b elo w. W e refer the reader to [31] for the pro ofs. (1) W eighted Energy Estimates: W eigh ted energy estimates are used to c haracterise G † in tw o k ey regimes: the initial lay er 0 < t < 1 and the region outside the wa v e cone t ≥ 1 and | x | > 2 M t , where M is a constan t related to the Mac h num b er. • Estimates for 0 < t < 1 : F or the initial lay er, m ultiplying the equation of G † b y a p ositive deﬁnite matrix and applying integration by parts, one obtains (see [31, Lemma 3.9]) that 50 Lemma 3.5. Under the assumption ¯ U = ( ¯ v , ¯ u, ¯ E , ¯ z ) , ∥ ¯ U − (1 , 0 , c v , 0) ∥ < ε, 0 < ε ≪ 1 , (3.23) ther e exists a p ositive c onstant σ such that for 0 < t < 1 , one has that    e σ | x | G † ( x, t )    H 4 ( R ) ≤ O (1) t, 3 X k =0    ∂ k x G † ( x, t )    ≤ O (1) te − σ | x | . • Estimates for t ≥ 1 and | x | > 2 M t : F or large t outside the wa ve cone, b y choosing M suﬃciently larger than the sound sp eed and σ > 0 suﬃcien tly small, the following deca y estimates are established as Lemma 3.10 in [31]: Lemma 3.6. Under the assumption (3.23) , ther e exists a p ositive c onstant σ and M such that the fol lowing estimates hold when t ≥ 1 and | x | > 2 M t :    e ± σ ( x − M t ) G † ( x, t )    H 4 ( R ) ≤ O (1) e − σ ∗ 0 t , 3 X k =0    ∂ k x G † ( x, t )    ≤ O (1) e − σ 2 | x |− σ ∗ 0 t . (2) Long-W av e and Short-W a v e Decomp osition: F or large time ( t ≥ 1) within the w a ve cone ( | x | < 2 M t ) , the regular part is decomposed into long w av e part G † L ( | η | < δ ) and short w a v e part G † S ( | η | ≥ δ ) . • Long w a v e estimates: F or low frequencies ( | η | < δ ), eigenv alues λ j ( j = 1 , 2 , 3) ha v e the follo wing asymptotic expansions around η = 0 : λ 1 = iη β 1 − α 1 η 2 + O (1) η 3 , λ 2 = iη β 2 − α 2 η 2 + O (1) η 3 , λ 3 = iη β 3 − α 3 η 2 + O (1) η 3 , where β j are wa v e sp eeds and α j > 0 are diﬀusion co eﬃcients, deﬁned as α 1 = − ν θ e p v v ( pp e − p v ) , α 2 = α 3 = ν pθ e p e + µpp e − µp v 2 v ( pp e − p v ) , β 1 = 0 , β 2 = − √ pp e − p v , β 3 = √ pp e − p v . Using these expansions and inv erse F ourier transforms, the k ey estimate for G † L is giv en as Lemma 3.13 in [31]: Lemma 3.7. Supp ose t ≥ 1 and | x | ≤ 2 M t . Ther e exists a suﬃciently smal l p ositive c onstant σ ∗ 0 and a lar ge c onstant C such that, the long wave of the r e gular p art G † L has the fol lowing estimates         ∂ k x G † L ( x, t ) − 3 X j =1 ∂ k x Ü e − ( x + β j t ) 2 4 α j t 2 √ π α j t ê M 0 j − 3 X j =1 ∂ k +1 x Ü e − ( x + β j t ) 2 4 α j t 2 √ π α j t ê M 1 j         ≤ O (1) P 3 j =1 e − ( x + β j t ) 2 4 C t t k +2 2 M 0 j + 3 X j =1 O (1) e − ( x + β j t ) 2 4 C t t k +3 2 + O (1) e − σ ∗ 0 Ä t 2 + | x | 4 M ä , wher e M 0 j and M 1 j ar e liste d in App endix A. • F or high frequencies ( | η | ≥ δ ), applying Duhamel’s principle for the inhomogeneous equation ab out G † , the follo wing estimates are pro ven as Lemma 3.14 in [31]: 51 Lemma 3.8. Ther e exists a suﬃciently smal l p ositive c onstant σ ∗ 0 such that, when t ≥ 1 and | x | ≤ 2 M t , the short wave p arts G † S has the fol lowing estimates    ˆ G † S ( η , t )    = O (1) e − σ ∗ 0 t (1 + | η | ) 6 ,    G † S ( x, t )    H 4 ( R ) ≤ O (1) e − σ ∗ 0 t , 3 X k =0    ∂ k x G † S ( x, t )    ≤ O (1) e − σ ∗ 0 t ≤ O (1) e − σ ∗ 0 ( t 2 + | x | 4 M ) . W e deduce the following result from the four lemmas ab ov e. Theorem 3.4. Assume the c ondition (3.23) . Ther e exists a suﬃciently lar ge c onstant C and suﬃciently smal l c onstants σ 0 , σ ∗ 0 such that 3 X k =0    ∂ k x G † ( x, t )    ≤ O (1) te − σ 0 | x | for 0 < t < 1 , as wel l as for k ∈ { 0 , 1 , 2 , 3 } that         ∂ k x G † ( x, t ) − 3 X j =1 ∂ k x Ü e − ( x + β j t ) 2 4 α j t 2 √ π α j t ê M 0 j − 3 X j =1 ∂ k +1 x Ü e − ( x + β j t ) 2 4 α j t 2 √ π α j t ê M 1 j         ≤ 3 X j =1 O (1) e − ( x + β j t ) 2 4 C t t k +2 2 M 0 j + 3 X j =1 O (1) e − ( x + β j t ) 2 4 C t t k +3 2 + O (1) e − σ ∗ 0 t − σ 0 | x | , for t ≥ 1 . The explicit expr essions for M 0 j and M 1 j ar e pr esente d in App endix A. See [31, Theorem 3.3] for a pro of. 4. Global Well-posedness In this ﬁnal section, we show that the local-in-time w eak solution constructed in Theo- rem 2.1 and 2.2 can b e extended to arbitrary time, pro vided that the initial data ( v 0 , u 0 , θ 0 , z 0 ) are a small p erturbation of the constan t state (1 , 0 , 1 , 0) . (Equiv alently , ( v 0 , u 0 , E 0 , z 0 ) is a small p erturbation around ¯ U = (1 , 0 , c v , 0) , where E is the total energy .) Our pro of relies on a delicate analysis of the properties of Green’s function G constructed in §3. As in Eq. (3.12), w e consider G ( x, t ; ¯ U ) = G ∗ ( x, t ; ¯ U ) | {z } regular part + G † ( x, t ; ¯ U ) | {z } singular part , where G ∗ = á G ∗ 11 G ∗ 12 G ∗ 13 G ∗ 14 G ∗ 21 G ∗ 22 G ∗ 23 G ∗ 24 G ∗ 31 G ∗ 32 G ∗ 33 G ∗ 34 G ∗ 41 G ∗ 42 G ∗ 43 G ∗ 44 ë , G † = á G † 11 G † 12 G † 13 G † 14 G † 21 G † 22 G † 23 G † 24 G † 31 G † 32 G † 33 G † 34 G † 41 G † 42 G † 43 G † 44 ë . It satisﬁes the backw ard heat equation ∂ τ G ( x − y , t − τ ; ¯ U ) + ∂ y G ( x − y , t − τ ; ¯ U ) F ′ ( ¯ U ) + ∂ 2 y G ( x − y , t − τ ; ¯ U ) B ( ¯ U ) = 0 (4.1) 52 with the coeﬃcient matrices F ′ ( ¯ U ) = á 0 − 1 0 0 − a 0 a c v − q a c v 0 a 0 0 0 0 0 0 ë , B ( ¯ U ) = á 0 0 0 0 0 µ 0 0 0 0 ν c v − q ν c v + q D 0 0 0 D ë . (4.2) In comp onents, the PDEs for G j k are as follo ws: á ∂ τ G 11 − a∂ y G 12 ∂ τ G 12 − ∂ y G 11 + a∂ y G 13 + µ∂ 2 y G 12 ∂ τ G 13 + a c v ∂ y G 12 + ν c v ∂ 2 y G 13 ∂ τ G 21 − a∂ y G 22 ∂ τ G 22 − ∂ y G 21 + a∂ y G 23 + µ∂ 2 y G 22 ∂ τ G 23 + a c v ∂ y G 22 + ν c v ∂ 2 y G 23 ∂ τ G 31 − a∂ y G 32 ∂ τ G 32 − ∂ y G 31 + a∂ y G 33 + µ∂ 2 y G 32 ∂ τ G 33 + a c v ∂ y G 32 + ν c v ∂ 2 y G 33 ∂ τ G 41 − a∂ y G 42 ∂ τ G 42 − ∂ y G 41 + a∂ y G 43 + µ∂ 2 y G 42 ∂ τ G 43 + a c v ∂ y G 42 + ν c v ∂ 2 y G 43 ∂ τ G 14 − q a c v ∂ y G 12 + Ä − q ν c v + q D ä ∂ 2 y G 13 + D ∂ 2 y G 14 ∂ τ G 24 − q a c v ∂ y G 22 + Ä − q ν c v + q D ä ∂ 2 y G 23 + D ∂ 2 y G 24 ∂ τ G 34 − q a c v ∂ y G 32 + Ä − q ν c v + q D ä ∂ 2 y G 33 + D ∂ 2 y G 34 ∂ τ G 44 − q a c v ∂ y G 42 + Ä − q ν c v + q D ä ∂ 2 y G 43 + D ∂ 2 y G 44 í = O . (4.3) In the sequel, we shall systematically drop the background state ¯ U when there is no danger of confusion. F or instance, w e shall write G ( x − y , t − τ ) ≡ G ( x − y , t − τ ; ¯ U ) . 4.1. Represen tation by Green’s function. The heat k ernels introduced in §1 is conv enien t for constructing lo cal solutions but less suitable for analysing large-time b ehaviour. T o address this issue, as in [22, 31] we in tro duce an eﬀe ctive Gr e en ’s function G that combines the lo cal heat k ernel H and the global Green’s function G . First, ﬁx a smo oth, non-increasing cutoﬀ function X , suc h that X ( t ) ∈ C ∞ ( R + ) , X ′ ( t ) ≤ 0 ,   X ′   L ∞ ( R + ) ≤ 2 , X ( t ) =    1 , for t ∈ (0 , 1] , 0 , for t > 2 . (4.4) Next, let ν 0 b e a suﬃcien tly small p ositive constant such that the heat kernel H ( x, t ; y , τ ; 1 v ) and the lo cal solution ( v ( x, τ ) , u ( x, τ ) , E ( x, τ ) , z ( x, τ )) exist for τ ∈ ( t − 2 ν 0 , t ) . Then deﬁne      G 22 ( x, t ; y , τ ) = X Ä t − τ ν 0 ä H  x, t ; y , τ ; µ v  + Ä 1 − X Ä t − τ ν 0 ää G 22 ( x − y ; t − τ ) , G 33 ( x, t ; y , τ ) = X Ä t − τ ν 0 ä H Ä x, t ; y , τ ; ν c v v ä + Ä 1 − X Ä t − τ ν 0 ää G 33 ( x − y ; t − τ ) , G 44 ( x, t ; y , τ ) = X Ä t − τ ν 0 ä H  x, t ; y , τ ; D v 2  + Ä 1 − X Ä t − τ ν 0 ää G 44 ( x − y ; t − τ ) . (4.5) Lemma 4.1. Supp ose that the we ak solution ( v ( x, τ ) , u ( x, τ ) , E ( x, τ ) , z ( x, τ )) to Eq. (0.1) exists for τ ∈ [0 , t ] . L et ν 0 b e a suﬃciently smal l p ositive c onstant with 2 ν 0 < t such that the he at kernel H ( x, t ; y , τ ; D v 2 ) exists for τ ∈ ( t − 2 ν 0 , t ) . Then z ( x, t ) = Z R G 41 ( x − y , t )( v ( y , 0) − 1)d y + Z R G 42 ( x − y , t ) u ( y , 0) d y + Z R G 43 ( x − y , t )( E ( y, 0) − c v ) d y + Z R G 44 ( x, t ; y , 0) z ( y , 0) d y − Z t 0 Z R G 44 ( x, t ; y , τ ) K ϕ ( θ ) z ( y , τ ) d y d τ + 3 X i =1 R z i , (4.6) 53 wher e the inhomo gene ous r emainders R z i , i = 1 , 2 , 3 , ar e given by R z 1 = Z t − 2 ν 0 0 Z R ∂ y G 42 ( x − y , t − τ ) ï a ( v − 1) 2 v + a ( θ − 1)(1 − v ) v − au 2 2 c v + µu y ( v − 1) v ò d y d τ + Z t − 2 ν 0 0 Z R ∂ y G 43 ( x − y , t − τ ) ïÅ a ( θ − 1) + a (1 − v ) v ã u + ν θ y ( v − 1) v + Å ν c v − µ v ã uu y + q D ( v 2 − 1) v 2 z y ò d y d τ + Z t − 2 ν 0 0 Z R ∂ y G 44 ( x − y , t − τ ) D ( v 2 − 1) v 2 z y d y d τ , R z 2 = Z t − ν 0 t − 2 ν 0 Z R ∂ y G 42 ( x − y , t − τ ) ï a ( v − 1) 2 v + a ( θ − 1)(1 − v ) v − au 2 2 c v + µu y ( v − 1) v −X ( t − τ ν 0 ) q a c v z ò ( y , τ ) d y d τ + Z t − ν 0 t − 2 ν 0 Z R ∂ y G 43 ( x − y ; t − τ ) ï a ( θ − 1) + a (1 − v ) v u + ν ( v − 1) v θ y + Å ν c v v − µ v ã uu y + q D ( v 2 − 1) v 2 z y ò ( y , τ ) d y d τ + Z t − ν 0 t − 2 ν 0 Z R Å 1 − X ( t − τ ν 0 ) ã ∂ y G 44 ( x − y ; t − τ ) D ( v 2 − 1) v 2 z y ( y , τ ) d y d τ + Z t − ν 0 t − 2 ν 0 Z R 1 ν 0 X ′ Å t − τ ν 0 ã ï G 44 ( x − y ; t − τ ) − H Å x, t ; y , τ ; D v 2 ãò z ( y , τ ) d y d τ + Z t − ν 0 t − 2 ν 0 Z R X ( t − τ ν 0 ) ∂ y G 43 ( x − y , t − τ ) Å q ν c v − q D ã z y ( y , τ ) d y d τ , R z 3 = Z t t − ν 0 Z R ∂ y G 42 ( x − y , t − τ ) ï a ( v − 1) 2 v + a ( θ − 1)(1 − v ) v − au 2 2 c v + µu y ( v − 1) v − q a c v z ò d y d τ + Z t − ν 0 t − 2 ν 0 Z R ∂ y G 43 ( x − y ; t − τ ) ï a ( θ − 1) + a (1 − v ) v u + ν ( v − 1) v θ y + Å ν c v v − µ v ã uu y − Å q ν c v − q D ã z y ò ( y , τ ) d y d τ . Pr o of. Multiplying the vector [ G 41 ( x − y , t − τ ) , G 42 ( x, t ; y , τ ) , G 43 ( x − y , t − τ ) , G 44 ( x, t ; y , τ )] ⊤ to the system (0.1) and in tegrating ov er τ ∈ [0 , t ] and y ∈ R b y parts, we arrive at 0 = Z t 0 Z R G 41 ( x − y , t − τ ) ( v τ − u y ) ( y , τ ) d y d τ + Z t 0 Z R G 42 ( x − y , t − τ ) Å u τ + p y −  µu y v  y ã ( y , τ ) d y d τ + Z t 0 Z R G 43 ( x − y , t − τ ) Ç E τ + ( pu ) y − Å ν v θ y + µ v uu y + q D z y v 2 ã y å ( y , τ ) d y d τ + Z t 0 Z R G 44 ( x, t ; y , τ ) Ç z τ + K ϕ ( θ ) z − Å D z y v 2 ã y å ( y , τ ) d y d τ = Z t 0 Z R G 41 ( x − y , t − τ ) (( v − 1) τ − u y ) ( y , τ ) d y d τ + Z t 0 Z R G 42 ( x − y , t − τ ) Å u τ + ( p ( v , θ ) − p (1 , 1)) y −  µu y v  y ã ( y , τ ) d y d τ + Z t 0 Z R G 43 ( x − y , t − τ ) Ç ( E − c v ) τ + ( pu ) y − Å ν v θ y + µ v uu y + q D z y v 2 ã y å ( y , τ ) d y d τ + Z t 0 Z R G 44 ( x, t ; y , τ ) Ç z τ + K ϕ ( θ ) z − Å D z y v 2 ã y å ( y , τ ) d y d τ 54 = − Z R G 41 ( x − y , t ) ( v ( y , 0) − 1) dy − Z t 0 Z R ∂ τ G 41 ( x − y , t − τ )( v ( y , τ ) − 1) d y d τ + Z t 0 Z R ∂ y G 41 ( x − y , t − τ ) u ( y , τ ) d y d τ − Z R G 42 ( x − y , t ) u ( y , 0) d y − Z t 0 Z R ∂ τ G 42 ( x − y , t − τ ) u ( y , τ ) d y d τ − Z t 0 Z R ∂ y G 42 ( x − y , t − τ )  ( p − p (1 , 1)) − µu y v  d y d τ − Z R G 43 ( x − y , t ) ( E ( y , 0) − c v ) d y − Z t 0 Z R ∂ τ G 43 ( x − y , t − τ ) ( E ( y, τ ) − c v ) d y d τ − Z t 0 Z R ∂ y G 43 ( x − y , t − τ ) Å ( pu ) − Å ν v θ y + µ v uu y + q D v 2 z y ãã ( y , τ ) d y d τ + z ( x, t ) − Z R G 44 ( x, t ; y , 0) z ( y , 0) d y − Z t 0 Z R ∂ τ G 44 ( x, t ; y , τ ) z ( y , τ ) d y d τ + Z t 0 G 44 ( x, t ; y , τ ) K ϕ ( θ ) z ( y , τ ) d y d τ + Z t 0 Z R ∂ y G 44 ( x, t ; y , τ ) D v 2 z y ( y , τ ) d y d τ . In the abov e, we mak e use of G ( x − y ; 0) = δ ( x − y ) and that Z R G 44 ( x, t ; y , t ) z ( y , t ) d y = Z R G 44 ( x − y ; 0) z ( y , t ) d y = z ( x, t ) . Rearrangemen t gives us z ( x, t ) = Z R G 41 ( x − y , t )( v ( y , 0) − 1) d y + Z R G 42 ( x − y , t ) u ( y , 0) d y + Z R G 43 ( x − y , t ) ( E ( y , 0) − c v ) d y + Z R G 44 ( x, t ; y , 0) z ( y , 0) d y + Z t 0 Z R ∂ τ G 41 ( x − y , t − τ )( v ( y , τ ) − 1) d y d τ − Z t 0 Z R ∂ y G 41 ( x − y , t − τ ) u ( y , τ ) d y d τ + Z t 0 Z R ∂ τ G 42 ( x − y , t − τ ) u ( y , τ ) d y d τ + Z t 0 Z R ∂ y G 42 ( x − y , t − τ )  p − p (1 , 1) − µu y v  ( y , τ ) d y d τ + Z t 0 Z R ∂ τ G 43 ( x − y , t − τ ) ( E ( y, τ ) − c v ) d y d τ + Z t 0 Z R ∂ y G 43 ( x − y , t − τ ) Å ( pu ) − Å ν v θ y + µ v uu y + q D v 2 z y ãã ( y , τ ) d y d τ + Z t 0 Z R ∂ τ G 44 ( x, t ; y , τ ) z ( y , τ ) d y d τ − Z t 0 Z R G 44 ( x, t ; y , τ ) K ϕ ( θ ) z ( y , τ ) d y d τ − Z t 0 Z R ∂ y G 44 ( x, t ; y , τ ) D v 2 z y ( y , τ ) d y d τ =: 13 X i =1 I i . (4.7) T o proceed, w e use Eq. (4.3) to replace the term G τ b y G y . Hence, I 5 = Z t 0 Z R ∂ τ G 41 ( x − y , t − τ )( v ( y , τ ) − 1) d y d τ = Z t 0 Z R a∂ y G 42 ( x − y , t − τ )( v ( y , τ ) − 1) d y d τ , 55 I 7 = Z t 0 Z R ∂ τ G 42 ( x − y , t − τ ) u ( y , τ ) d y d τ = Z t 0 Z R  ∂ y G 41 − a∂ y G 43 − µ∂ 2 y G 42  ( x − y , t − τ ) u ( y , τ ) d y d τ , I 9 = Z t 0 Z R ∂ τ G 43 ( x − y , t − τ )( E ( y , τ ) − c v ) d y d τ = Z t 0 Z R Å − a c v ∂ y G 42 − ν c v ∂ 2 y G 43 ã ( x − y , t − τ )( E ( y , τ ) − c v ) d y d τ . F or I 11 , using the deﬁnition of G 44 ( x, t ; y , τ ) in (4.5), we split the integral into three time interv als I 11 = Z t 0 Z R ∂ τ G 44 ( x, t ; y , τ ) z ( y , τ ) d y d τ = Z t 0 Z R   − X ′ Ä t − τ ν 0 ä ν 0 H ( x, t ; y , τ ; D v 2 ) + X Å t − τ ν 0 ã H τ ( x, t ; y , τ ; D v 2 ) + X ′ Ä t − τ ν 0 ä ν 0 G 44 ( x − y , t − τ ) + Å 1 − X Å t − τ ν 0 ãã ∂ τ G 44 ( x − y , t − τ )   z ( y , τ ) d y d τ = Z t − 2 ν 0 0 Z R ∂ τ G 44 ( x − y , t − τ ) z ( y , τ ) d y d τ + Z t − ν 0 t − 2 ν 0 Z R −X ′ Ä t − τ ν 0 ä ν 0 H ( x, t ; y , τ ; D v 2 ) z ( y , τ ) d y d τ + Z t − ν 0 t − 2 ν 0 Z R X Å t − τ ν 0 ã H τ ( x, t ; y , τ ; D v 2 ) z ( y , τ ) d y d τ + Z t − ν 0 t − 2 ν 0 Z R X ′ Ä t − τ ν 0 ä ν 0 G 44 ( x − y , t − τ ) z ( y , τ ) d y d τ + Z t − ν 0 t − 2 ν 0 Z R Å 1 − X Å t − τ ν 0 ãã ∂ τ G 44 ( x − y , t − τ ) z ( y , τ ) d y d τ + Z t t − ν 0 Z R H τ ( x, t ; y , τ ; D v 2 ) z ( y , τ ) d y d τ = Z t − 2 ν 0 0 Z R Å q a c v ∂ y G 42 − Å − q ν c v + q D ã ∂ 2 y G 43 − D ∂ 2 y G 44 ã ( x − y , t − τ ) z ( y , τ ) d y d τ + Z t − ν 0 t − 2 ν 0 Z R X ′ Ä t − τ ν 0 ä ν 0 ï G 44 ( x − y , t − τ ) − H ( x, t ; y , τ ; D v 2 ) ò z ( y , τ ) d y d τ − Z t − ν 0 t − 2 ν 0 Z R X Å t − τ ν 0 ã Å D v 2 H y ( x, t ; y , τ ; D v 2 ) ã y z ( y , τ ) d y d τ + Z t − ν 0 t − 2 ν 0 Z R Å 1 − X Å t − τ ν 0 ãã Å q a c v ∂ y G 42 − Å − q ν c v + q D ã ∂ 2 y G 43 − D ∂ 2 y G 44 ã ( x − y , t − τ ) z ( y , τ ) d y d τ − Z t t − ν 0 Z R Å D v 2 H y ( x, t ; y , τ ; D v 2 ) ã y z ( y , τ ) d y d τ , in which we use the bac kw ard Eq. (1.12) for H and Eq. (4.3) for G . Finally , for I 13 , we use the deﬁnition of G 44 ( x, t ; y , τ ) to obtain I 13 = − Z t 0 Z R ∂ y G 44 ( x, t ; y , τ ) D v 2 z y ( y , τ ) d y d τ 56 = − Z t − 2 ν 0 0 Z R ∂ y G 44 ( x − y , t − τ ) D v 2 z y ( y , τ ) d y d τ − Z t − ν 0 t − 2 ν 0 Z R X Å t − τ ν 0 ã H y ( x, t ; y , τ ; D v 2 ) D v 2 z y ( y , τ ) d y d τ − Z t − ν 0 t − 2 ν 0 Z R Å 1 − X Å t − τ ν 0 ãã G 44 ( x − y , t − τ ) D v 2 z y ( y , τ ) d y d τ − Z t t − ν 0 Z R H y ( x, t ; y , τ ; D v 2 ) D v 2 z y ( y , τ ) d y d τ . Substituting the expressions for I 5 , I 7 , I 9 , I 11 and I 13 in to (4.7), we obtain the desired represen- tation formula for z ( x, t ) . □ Lemma 4.2. Supp ose the we ak solution ( v ( x, τ ) , u ( x, τ ) , E ( x, τ ) , z ( x, τ )) to Eq. (0.1) exists for τ ∈ [0 , t ] , and let ν 0 b e a suﬃciently smal l p ositive c onstant with 2 ν 0 < t such that the he at kernels H ( x, t ; y , τ ; µ v ) and H ( x, t ; y , τ ; ν c v v ) exist for τ ∈ ( t − 2 ν 0 , t ) . Then u ( x, t ) = Z R G 21 ( x − y , t )( v ( y , 0) − 1)d y + Z R G 22 ( x, t ; y , 0) u ( y , 0) d y + Z R G 23 ( x − y , t ) ( E ( y , 0) − c v ) d y + 3 X i =1 R u i , v ( x, t ) − 1 = Z R G 11 ( x − y , t )( v ( y , 0) − 1)d y + Z R G 12 ( x − y , t ) u ( y , 0) d y + Z R G 13 ( x − y , t ) ( E ( y , 0) − c v ) d y + Z t 0 Z R ∂ y G 12 ( x − y , t − τ ) ï a ( v − 1) 2 v + a ( θ − 1)(1 − v ) v − au 2 2 c v + µu y ( v − 1) v − q a c v z ò ( y , τ ) d y d τ + Z t 0 Z R ∂ y G 13 ( x − y , t − τ ) ïÅ a ( θ − 1) + a (1 − v ) v ã u + ν θ y ( v − 1) v + Å ν c v − µ v ã uu y + q ν c v z y − q D v 2 z y ò ( y , τ ) d y d τ , E ( x, t ) − c v = Z R G 31 ( x − y , t )( v ( y , 0) − 1)d y + Z R G 32 ( x − y , t ) u ( y , 0) d y + Z R G 33 ( x, t ; y , 0)( E ( y , 0) − c v ) d y + 3 X i =1 R θ i . (4.8) The inhomo gene ous r emainder terms R u i and R θ i ar e given by R u 1 = Z t − 2 ν 0 0 Z R ∂ y G 22 ( x − y , t − τ ) ï a ( v − 1) 2 v + a ( θ − 1)(1 − v ) v − au 2 2 c v + µu y ( v − 1) v − q a c v z ò ( y , τ ) d y d τ + Z t − 2 ν 0 0 Z R ∂ y G 23 ( x − y , t − τ ) ï a ( θ − 1) + a (1 − v ) v u + ν θ y ( v − 1) v + Å ν c v − µ v ã uu y + q ν c v z y − q D v 2 z y ò ( y , τ ) d y d τ , R u 2 = Z t − ν 0 t − 2 ν 0 Z R ∂ y G 22 ( x − y , t − τ ) ï a ( v − 1) 2 v + a ( θ − 1)(1 − v ) v + µu y ( v − 1) v (1 − X ) − au 2 2 c v ò ( y , τ ) d y d τ + Z t − ν 0 t − 2 ν 0 Z R ∂ y G 23 ( x − y ; t − τ ) ïÅ a ( θ − 1) + a (1 − v ) v ã u 57 + ν θ y ( v − 1) v + Å ν c v − µ v ã uu y + q ν c v z y − q D v 2 z y ò ( y , τ ) d y d τ + Z t − ν 0 t − 2 ν 0 Z R 1 ν 0 X ′ Å t − τ ν 0 ã h G 22 ( x − y ; t − τ ) − H  x, t ; y , τ ; µ v i u ( y , τ ) d y d τ + Z t − ν 0 t − 2 ν 0 Z R X Å t − τ ν 0 ã h H y  x, t ; y , τ ; µ v  − ∂ y G 22 ( x − y ; t − τ ) i a ( θ − v ) v d y d τ + Z t − ν 0 t − 2 ν 0 Z R X Å t − τ ν 0 ã [ a∂ y G 23 ( x − y , t − τ ) − ∂ y G 21 ( x − y , t − τ )] u ( y , τ ) d y d τ , R u 3 = − Z t t − ν 0 Z R ∂ y G 21 ( x − y , t − τ ) u ( y , τ ) d y d τ + Z t t − ν 0 Z R ∂ y G 22 ( x − y , t − τ ) ï a ( v − θ ) − au 2 2 c v − q a c v z ò ( y , τ ) d y d τ + Z t t − ν 0 Z R ∂ y G 23 ( x − y , t − τ ) ï pu + ν ( v − 1) v θ y + Å ν c v − µ v ã uu y + q ν c v z y − q D v 2 z y ò d y d τ + Z t t − ν 0 Z R H y ( x, t ; y , τ ; µ v ) a ( θ − v ) v ( y , τ ) d y d τ , R θ 1 = Z t − 2 ν 0 0 Z R ∂ y G 32 ( x − y , t − τ ) ï a ( v − 1) 2 v + a ( θ − 1)(1 − v ) v − au 2 2 c v + µu y ( v − 1) v − q a c v z ò d y d τ + Z t − 2 ν 0 0 Z R ∂ y G 33 ( x − y , t − τ ) ïÅ a ( θ − 1) + a (1 − v ) v ã u + ν θ y ( v − 1) v + Å ν c v − µ v ã uu y + q ν c v z y − q D v 2 z y ò ( y , τ ) d y d τ , R θ 2 = Z t − ν 0 t − 2 ν 0 Z R ∂ y G 32 ( x − y , t − τ ) ï a ( θ − 1) v + µu y ( v − 1) v + a ( v − 1) 2 v ò ( y , τ ) d y d τ − Z t − ν 0 t − 2 ν 0 Z R Å 1 − X ( t − τ ν 0 ) ã a c v ∂ y G 32 ( x − y ; t − τ )( E ( y , τ ) − c v ) d y d τ + Z t − ν 0 t − 2 ν 0 Z R 1 ν 0 X ′ Å t − τ ν 0 ã ï G 33 ( x − y ; t − τ ) − H Å x, t ; y , τ ; ν c v v ãò ( E ( y , τ ) − c v ) d y d τ + Z t − ν 0 t − 2 ν 0 Z R X ( t − τ ν 0 ) H y ( x, t ; y , τ ; ν c v v ) ï pu + Å ν c v v − µ v ã uu y + Å q ν c v v − q D v 2 z y ãò d y d τ + Z t − ν 0 t − 2 ν 0 Z R Å 1 − X ( t − τ ν 0 ) ã ∂ y G 33 ( x − y ; t − τ ) ï pu + ν ( v − 1) v θ y + Å ν c v v − µ v ã uu y + Å q ν c v v − q D v 2 ã z y ò ( y , τ ) d y d τ − Z t − ν 0 t − 2 ν 0 Z R ∂ y G 33 ( x − y ; t − τ ) au ( y , τ ) d y d τ , R θ 3 = Z t t − ν 0 Z R ∂ y G 32 ( x − y , t − τ ) ï a ( v − 1) + a ( θ − 1) + a (1 − v ) v + µu y ( v − 1) v ò ( y , τ ) d y d τ − Z t − ν 0 t − 2 ν 0 Z R ∂ y G 33 ( x − y ; t − τ ) au ( y , τ ) d y d τ + Z t t − ν 0 Z R H y ( x, t ; y , τ ; ν c v v ) ï pu + Å ν c v v − µ v ã uu y + Å q ν c v v − q D v 2 ã z y ò ( y , τ ) d y d τ . Pr o of. The pro of follows similarly to that of Lemma 4.1. There is, ho w ever, a key diﬀerence: only the ﬁrst three equations of Eq. (0.1) are used here. This is b ecause the fourth column of the Green’s function is q times the third column. 58 W e begin b y multiplying the ﬁrst three equations of Eq. (0.1) b y the v ector [ G 11 ( x − y , t − τ ) , G 12 ( x − y , t − τ ) , G 13 ( x − y , t − τ )] ⊤ , and then integrate with resp ect to τ and y . Via integration by parts, we obtain the representation form ula for v ( x, t ) . Similarly , for u ( x, t ) and E ( x, t ) we m ultiply the ﬁrst three equation of Eq. (0.1) by the v ectors [ G 21 ( x − y , t − τ ) , G 22 ( x, t ; y , τ ) , G 23 ( x − y , t − τ )] ⊤ and [ G 31 ( x − y , t − τ ) , G 32 ( x − y , t − τ ) , G 33 ( x, t ; y , τ )] ⊤ , resp ectiv ely , and then follo w the similar procedure. □ Corollary 4.1. The ﬁrst or der derivatives of R u 3 and R θ 3 ar e expr esse d b elow: ∂ x R u 3 = Z x −∞ Z y −∞ 1 µ  ∂ x G 22 ( x − ω , ν 0 ) − H x  x, t ; ω , t − ν 0 ; µ v  d ω ( av − aθ ) ( y , t − ν 0 ) d y − Z t t − ν 0 Z x −∞ Z y −∞ − 1 µ  ∂ x G 22 ( x − ω , t − τ ) − H x  x, t ; ω , τ ; µ v  d ω ( av τ − aθ τ ) ( y , τ ) d y d τ − Z + ∞ x Z + ∞ y 1 µ  ∂ x G 22 ( x − ω , ν 0 ) − H x  x, t ; ω , t − ν 0 ; µ v  d ω ( av − aθ ) ( y , t − ν 0 ) d y + Z t t − ν 0 Z + ∞ x Z + ∞ y − 1 µ  ∂ x G 22 ( x − ω , t − τ ) − H x  x, t ; ω , τ ; µ v  d ω ( av τ − aθ τ ) ( y , τ ) d y d τ + Z t t − ν 0 Z R 1 µ ( ∂ x G 21 ( x − y , t − τ ) − a∂ x G 23 ( x − y , t − τ )) ( av − aθ )( y , τ ) d y d τ + Z t t − ν 0 Z R ∂ x G 21 ( x − y , t − τ ) u y ( y , τ ) d y d τ + Z t t − ν 0 Z R ∂ x G 22 ( x − y , t − τ ) auu y c v ( y , τ ) d y d τ + Z t t − ν 0 Z R ∂ x G 22 ( x − y , t − τ ) q a c v z y ( y , τ ) d y d τ + Z t t − ν 0 Z R ∂ xy G 23 ( x − y , t − τ ) Å pu + ν ( v − 1) v θ y + Å ν c v − µ v ã uu y + q ν c v z y − q D v 2 z y ã d y d τ , and ∂ x R θ 3 = Z t t − ν 0 Z R ∂ xy G 32 ( x − y , t − τ ) ï a ( v − 1) + µu y v − 1 v + aθ − av v ò ( y , τ ) d y d τ + Z t t − ν 0 Z R Å ∂ x G 33 ( x − y , t − τ ) − H x Å x, t ; y , τ ; ν c v v ãã au y ( y , τ ) dy dτ + Z t t − ν 0 Z R H x Å x, t ; y , τ ; ν c v v ã au y ( y , τ ) dy dτ + Z t t − ν 0 Z R H x Å x, t ; y , τ ; ν c v v ã Å 1 − ν c v µ ã  uu τ − pu y + µu y v u y  ( y , τ ) dy dτ + Z x −∞ Z y −∞ H x Å x, t ; ω , t − ν 0 ; ν c v v ã d ω a µ θ ( y , t − ν 0 ) u ( y , t − ν 0 ) dy + Z t t − ν 0 Z x −∞ Z y −∞ H x Å x, t ; ω , τ ; ν c v v ã dω a µ ( θ τ u + θ u τ ) ( y , τ ) dy dτ − Z + ∞ x Z + ∞ y H x Å x, t ; ω , t − ν 0 ; ν c v v ã dω a µ θ ( y , t − ν 0 ) u ( y , t − ν 0 ) dy 59 − Z t t − ν 0 Z + ∞ x Z + ∞ y H x Å x, t ; ω , τ ; ν c v v ã dω a µ ( θ τ u + θ u τ ) ( y , τ ) dy dτ + q z x ( x, t ) − q Z R H x Å x, t ; y , t − ν 0 ; ν c v v ã z ( y , t − ν 0 ) d y + q Z t t − ν 0 Z R H x Å x, t ; y , t − ν 0 ; ν c v v ã K ϕ ( θ ) z d y d τ . Pr o of. The deriv ation for ∂ x R u 3 can b e found in [31, Corollary 4.1]. Here w e fo cus on ∂ x R θ 3 . Recall the ev olution equation for u , namely , uu τ = u  µu y v − p  y . Multiplying b oth sides by H ( x, t ; y , τ ; ν c v v ) and in tegrating ov er R × [ t − ν 0 , t ] , w e obtain that Z t t − ν 0 Z R H ( x, t ; y , τ ; ν c v v ) Å uu τ − u  µu y v − p  y ã d y d τ = 0 . Applying integration by parts, we deriv e that Z t t − ν 0 Z R H y ( x, t ; y , τ ; ν c v v ) Å ν c v v − µ v ã uu y ( y , τ ) d y d τ = − Z t t − ν 0 Z R H y ( x, t ; y , τ ; ν c v v ) Å 1 − ν c v v ã µ v uu y ( y , τ ) d y d τ = Z t t − ν 0 Z R H ( x, t ; y , τ ; ν c v v ) Å 1 − ν c v v ã h uu τ − pu y + µu y v u y i ( y , τ ) d y d τ − Z t t − ν 0 Z R H y ( x, t ; y , τ ; ν c v v ) Å 1 − ν c v v ã pu ( y , τ ) d y d τ . Using p = aθ v and the bac kw ard equation for H : ® ∂ ω Ä ν c v v ( ω , τ ) H ω ( x, t ; ω , τ ; ν c v v ) ä = − H τ ( x, t ; ω , τ ; ν c v v ) , H ( x, t ; ω , t ; ν c v v ) = δ ( x − ω ) , (4.9) w e further obtain via in tegration by parts that Z t t − ν 0 Z R H y ( x, t ; y , τ ; ν c v v ) Å 1 − ν c v v ã ν c v µ aθ v u ( y , τ ) d y d τ = Z t t − ν 0 Z x −∞ Z y −∞ ∂ ω Å ν c v v H ω ( x, t ; ω , τ ; ν c v v ) ã d ω a µ θ u ( y , τ ) d y d τ − Z t t − ν 0 Z + ∞ x Z + ∞ y ∂ ω Å ν c v v H ω ( x, t ; ω , τ ; ν c v v ) ã d ω a µ θ u ( y , τ ) d y d τ = − Z t t − ν 0 Z x −∞ Z y −∞ H τ ( x, t ; ω , τ ; ν c v v ) d ω a µ θ u ( y , τ ) d y d τ + Z t t − ν 0 Z + ∞ x Z + ∞ y H τ ( x, t ; ω , τ ; ν c v v ) d ω a µ θ u ( y , τ ) d y d τ = Z x −∞ Z y −∞ H ( x, t ; ω , t − ν 0 ; ν c v v ) d ω a µ θ ( y , t − ν 0 ) u ( y , t − ν 0 ) d y − Z + ∞ x Z + ∞ y H ( x, t ; ω , t − ν 0 ; ν c v v ) d ω a µ θ ( y , t − ν 0 ) u ( y , t − ν 0 ) d y + Z t t − ν 0 Z x −∞ Z y −∞ H ( x, t ; ω , τ ; ν c v v ) d ω a µ ( θ τ u + θ u τ ) ( y , τ ) d y d τ 60 − Z t t − ν 0 Z + ∞ x Z + ∞ y H ( x, t ; ω , τ ; ν c v v ) d ω a µ ( θ τ u + θ u τ ) ( y , τ ) d y d τ . In an analogous manner, m ultiplying z τ + K ϕ ( θ ) z = Å D v 2 z y ã y b y the k ernel H ( x, t ; y , τ ; ν c v v ) and in tegrating ov er R × [ t − ν 0 , t ] , w e deduce that Z t t − ν 0 Z R H ( x, t ; y , τ ; ν c v v ) Ç z τ + K ϕ ( θ ) z − Å D v 2 z y ã y å d y d τ = 0 . Note that Z t t − ν 0 Z R H ( x, t ; y , τ ; ν c v v ) ( z τ + K ϕ ( θ ) z ) d y d τ = − Z t t − ν 0 Z R H y ( x, t ; y , τ ; ν c v v ) D v 2 z y d y d τ . Hence, using the backw ard Eq. (4.9) for H and in tegrating by parts in τ , w e obtain that Z t t − ν 0 Z R H y ( x, t ; y , τ ; ν c v v ) ν c v v z y ( y , τ ) d y d τ = − Z t t − ν 0 Z R ∂ y Å ν c v v H y ( x, t ; y , τ ; ν c v v ) ã z ( y , τ ) d y d τ = Z t t − ν 0 Z R H τ ( x, t ; y , τ ; ν c v v ) z ( y , τ ) d y d τ = z ( x, t ) − Z R H ( x, t ; y , t − ν 0 ; ν c v v ) z ( y , t − ν 0 ) d y − Z t t − ν 0 Z R H ( x, t ; y , τ ; ν c v v ) z τ ( y , τ ) d y d τ . Finally , substituting the ab ov e results into the representation of R θ 3 giv en in Lemma 4.2 and diﬀerentiating with respect to x , we complete the pro of. □ 4.2. Global existence, uniqueness and large time b ehaviour. As in [22], we in tro duce T = sup t ≥ 0 { t : G ( τ ) < δ for all 0 < τ < t } , (4.10) view ed as the stopping time , where G ( τ ) := ∥ √ τ + 1( v ( · , τ ) − 1) ∥ L ∞ x + ∥ √ τ + 1 u ( · , τ ) ∥ L ∞ x + ∥ √ τ + 1( θ ( · , τ ) − 1) ∥ L ∞ x + ∥ √ τ + 1 z ( · , τ ) ∥ L ∞ x + ∥ v ( · , τ ) − 1 ∥ L 1 x + ∥ u ( · , τ ) ∥ L 1 x + ∥ θ ( · , τ ) − 1 ∥ L 1 x + ∥ z ( · , τ ) ∥ L 1 x + ∥ v ( · , τ ) − 1 ∥ B V + ∥ u ( · , τ ) ∥ B V + ∥ θ ( · , τ ) − 1 ∥ B V + ∥ z ( · , τ ) ∥ B V +   √ τ u x ( · , τ )   L ∞ x +   √ τ θ x ( · , τ )   L ∞ x +   √ τ z x ( · , τ )   L ∞ x . Lemma 4.3. L et ( v , u, E , z ) , C ♯ , t ♯ and δ b e the lo c al solution and c orr esp onding p ar ameters c onstructe d in The or em 2.1. W e further supp ose that ∥ v 0 − 1 ∥ L 1 x + ∥ v 0 ∥ B V + ∥ u 0 ∥ L 1 x + ∥ u 0 ∥ B V + ∥ θ 0 − 1 ∥ L 1 x + ∥ θ 0 ∥ B V + ∥ z 0 ∥ L 1 x + ∥ z 0 ∥ B V ≤ δ ∗ < δ 300 C ♯ . (4.11) Then the stopping time deﬁne d in (4.10) satisﬁes T > t ♯ . 61 Mor e over, for δ > 0 and t < T + t ♯ , the solution satisﬁes                                        max ® ∥ u ( · , t ) ∥ L 1 x , ∥ u ( · , t ) ∥ L ∞ x , ∥ u x ( · , t ) ∥ L 1 x , p min ( t, t ♯ ) ∥ u x ( · , t ) ∥ L ∞ x , p min ( t, t ♯ ) ∥ u t ( · , t ) ∥ L 1 x , min ( t, t ♯ ) ∥ u t ( · , t ) ∥ L ∞ x ´ ≤ 2 C ♯ δ, max ® ∥ θ ( · , t ) − 1 ∥ L 1 x , ∥ θ ( · , t ) − 1 ∥ L ∞ x , ∥ θ x ( · , t ) ∥ L 1 x , p min ( t, t ♯ ) ∥ θ x ( · , t ) ∥ L ∞ x , p min ( t, t ♯ ) ∥ θ t ( · , t ) ∥ L 1 x , min ( t, t ♯ ) ∥ θ t ( · , t ) ∥ L ∞ x ´ ≤ 2 C ♯ δ, max ¶ R R \D | v x ( x, t ) | d x, ∥ v ( · , t ) − 1 ∥ L 1 x , ∥ v ( · , t ) − 1 ∥ L ∞ x , p min ( t, t ♯ ) ∥ v t ( · , t ) ∥ L ∞ x © ≤ 2 C ♯ δ, max ® ∥ z ( · , τ ) ∥ L 1 x , ∥ z ( · , t ) ∥ L ∞ x , ∥ z x ( · , t ) ∥ L 1 x , p min ( t, t ♯ ) ∥ z x ( · , t ) ∥ L ∞ x , p min ( t, t ♯ ) ∥ z t ( · , t ) ∥ L 1 x , min ( t, t ♯ ) ∥ z t ( · , t ) ∥ L ∞ x ´ ≤ 2 C ♯ δ, v ∗ = v ∗ ˜ a + v ∗ j , v ∗ j ( x, t ) = P ω t ♯ . Next, we prov e the estimate (4.12) for t < T + t ♯ . By deﬁnition of the stopping time T in (4.10), for any t < T + t ♯ w e set τ 0 = max(0 , t − t ♯ ) . Then clearly τ 0 < T , which implies that G ( τ 0 ) < δ . Th us, if we take ( v ( · , τ 0 ) , u ( · , τ 0 ) , θ ( · , τ 0 ) , z ( · , τ 0 )) as the new initial data and apply the local existence and estimate results in Theorem 2.1 again, w e may contin ue the solution up to time T + t ♯ with the estimates in Eq. (4.12). This completes the pro of. □ Equipp ed with the k ey prop erty abov e for the stopping time T , now we turn to the large- time b ehaviour of the solution. It relies on reﬁned estimates for Green’s function around the constan t equilibrium state ( v , u, θ , z ) = (1 , 0 , 1 , 0) . First, ob v erse that b y using Theorems 3.2 and 3.4, we arriv e at simpliﬁed expressions for Green’s function. 62 Lemma 4.4. L et G ( x, t ) b e the Gr e en ’s function of the line arize d e quation of Eq. (0.1) ar ound the c onstant e quilibrium state ( v , u, θ , z ) = (1 , 0 , 1 , 0) . F or t ≤ 1 , we have          G ( x − y , t ) − e − K µ t δ ( x − y ) á 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ë − e β ∗ 2 t √ 4 π µt e − ( x − y ) 2 4 µt á 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 ë − e β ∗ 3 t » 4 π ν c v t e − ( x − y ) 2 4 ν c v t á 0 0 0 0 0 0 0 0 0 0 1 q 0 0 0 0 ë − 1 √ 4 π D t e − ( x − y ) 2 4 Dt á 0 0 0 0 0 0 0 0 0 0 0 q 0 0 0 1 ë          ≤ O (1) e − σ ∗ 0 t − σ 0 | x − y | + O (1) te − σ 0 | x − y | , t ≤ 1 ,          G x ( x − y , t ) − e − K µ t δ ′ ( x − y ) á 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ë + e − K µ t δ ( x − y ) á 0 1 µ 0 0 a µ 0 0 0 0 0 0 0 0 0 0 0 ë − ∂ x Ç e β ∗ 2 t √ 4 π µt e − ( x − y ) 2 4 µt å á 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 ë + e β ∗ 2 t √ 4 π µt e − ( x − y ) 2 4 µt à 0 − 1 µ 0 0 − a µ 0 a c v µ − ν − q a c v µ − ν 0 c v a c v µ − ν 0 0 0 0 0 0 í − ∂ x Ñ e β ∗ 3 t » 4 π ν c v t e − ( x − y ) 2 4 ν c v t é á 0 0 0 0 0 0 0 0 0 0 1 q 0 0 0 0 ë + e β ∗ 3 t » 4 π ν c v t e − ( x − y ) 2 4 ν c v t á 0 0 0 0 0 0 − a c v µ − ν q a c v µ − ν 0 − c v a c v µ − ν 0 0 0 0 0 0 ë − ∂ x Å 1 √ 4 π D t e − ( x − y ) 2 4 Dt ã á 0 0 0 0 0 0 0 0 0 0 0 q 0 0 0 1 ë          ≤ O (1) e − σ ∗ 0 t − σ 0 | x − y | + O (1) te − σ 0 | x − y | , t ≤ 1 ,          G xx ( x − y , t ) − e − a µ t δ ′′ ( x − y ) á 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ë + e − K µ t δ ′ ( x − y ) á 0 1 µ 0 0 a µ 0 0 0 0 0 0 0 0 0 0 0 ë − e − K µ t δ ( x − y ) àà a µ 2 0 a ν µ − q a ν µ 0 − a µ 2 0 0 c ν a 2 ν µ 0 0 0 0 0 0 0 í − t à − ( ν a 2 − µa 3 ) ν µ 3 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 íí − ∂ xx Ç e β ∗ 2 t √ 4 π µt e − ( x − y ) 2 4 µt å á 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 ë 63 + ∂ x Ç e β ∗ 2 t √ 4 π µt e − ( x − y ) 2 4 µt å à 0 − 1 µ 0 0 − a µ 0 a c v µ − ν − q a c v µ − ν 0 c v a c v µ − ν 0 0 0 0 0 0 í − e β ∗ 2 t √ 4 π µt e − ( x − y ) 2 4 µt à − a µ 2 0 a c v µ 2 − ν µ − q a c v µ 2 − ν µ 0 c v a 2 ( c v µ − ν ) 2 + a µ 2 0 0 c v a 2 c v µ 2 − ν µ 0 − c v a 2 ( c v µ − ν ) 2 q c v a 2 ( c v µ − ν ) 2 0 0 0 0 í − ∂ xx Ñ e β ∗ 3 t » 4 π ν c v t e − ( x − y ) 2 4 ν c v t é á 0 0 0 0 0 0 0 0 0 0 1 q 0 0 0 0 ë + ∂ x Ñ e β ∗ 3 t » 4 π ν c v t e − ( x − y ) 2 4 ν c v t é á 0 0 0 0 0 0 − a c v µ − ν q a c v µ − ν 0 − c v a c v µ − ν 0 0 0 0 0 0 ë + e β ∗ 3 t » 4 π ν c v t e − ( x − y ) 2 4 ν c v t à 0 0 c v a ν ( c v µ − ν ) − q c v a ν ( c v µ − ν ) 0 c v a 2 ( c v µ − ν ) 2 0 0 c 2 v a 2 ν ( c v µ − ν ) 0 − c v a 2 ( c v µ − ν ) 2 q c v a 2 ( c v µ − ν ) 2 0 0 0 0 í − ∂ xx Å 1 √ 4 π D t e − ( x − y ) 2 4 Dt ã á 0 0 0 0 0 0 0 0 0 0 0 q 0 0 0 1 ë          ≤ O (1) e − σ ∗ 0 t − σ 0 | x − y | + O (1) te − σ 0 | x − y | , t ≤ 1 . (4.14) On the other hand, for lar ge time t ≥ 1 it holds that          G ( x − y , t ) − e − a µ t δ ( x − y ) á 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ë − 4 X j =1 e − ( x − y + β j t ) 2 4 α j t 2 √ π α j t M 0 j − 3 X j =1 ∂ x Ü e − ( x − y + β j t ) 2 4 α j t 2 √ π α j t ê M 1 j         ≤ 3 X j =1 O (1) e − ( x − y + β j t ) 2 4 C t t M 0 j + 3 X j =1 O (1) e − ( x − y + β j t ) 2 4 C t t 3 2 + O (1) e − σ ∗ 0 t − σ 0 | x − y | , 1 ≤ t,          G x ( x − y , t ) − e − a µ t δ ′ ( x − y ) á 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ë + e − a µ t δ ( x − y ) á 0 1 µ 0 0 K µ 0 0 0 0 0 0 0 0 0 0 0 ë 64 − 4 X j =1 ∂ x Ü e − ( x − y + β j t ) 2 4 α j t 2 √ π α j t ê M 0 j − 3 X j =1 ∂ 2 x Ü e − ( x − y + β j t ) 2 4 α j t 2 √ π α j t ê M 1 j         ≤ 3 X j =1 O (1) e − ( x − y + β j t ) 2 4 C t t 3 2 M 0 j + 3 X j =1 O (1) e − ( x − y + β j t ) 2 4 C t t 2 + O (1) e − σ ∗ 0 t − σ 0 | x − y | , 1 ≤ t,          G xx ( x − y , t ) − e − a µ t δ ′′ ( x − y ) á 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ë + e − a µ t δ ′ ( x − y ) á 0 1 µ 0 0 K µ 0 0 0 0 0 0 0 0 0 0 0 ë − e − a µ t δ ( x − y ) àà a µ 2 0 a ν µ 0 0 − a µ 2 0 0 c v a 2 ν µ 0 0 0 0 0 0 0 í − t à − ( ν a 2 − µa 3 ) ν µ 3 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 íí − 4 X j =1 ∂ 2 x Ü e − ( x − y + β j t ) 2 4 α j t 2 √ π α j t ê M 0 j − 3 X j =1 ∂ 3 x Ü e − ( x − y + β j t ) 2 4 α j t 2 √ π α j t ê M 1 j         ≤ 3 X j =1 O (1) e − ( x − y + β j t ) 2 4 C t t 2 M 0 j + 3 X j =1 O (1) e − ( x − y + β j t ) 2 4 C t t 5 2 + O (1) e − σ ∗ 0 t − σ 0 | x − y | , 1 ≤ t. (4.15) Remark 4.2. The ab ove r esult fol lows fr om R emark 4.2 in [31]. By r esc aling time t 7→ t ν 0 for ν 0 > 0 , we may obtain analo gous estimates for t ≥ ν 0 . The only mo diﬁc ation is that al l the O (1) terms now dep end on ν 0 in the r esc aling. With the estimates for the Green’s function, we pro ceed to deriv e some a priori estimates for the solutions constructed in Theorem 2.1 and Theorem 2.2. Based on the results in [31], w e already hav e the follo wing estimates Lemma 4.5. L et ( v , u, θ , z ) b e the lo c al solution c onstructe d in The or em 2.1. F urther assume that      ∥ v 0 − 1 ∥ B V + ∥ u 0 ∥ B V + ∥ θ 0 − 1 ∥ B V + ∥ v 0 − 1 ∥ L 1 x + ∥ u 0 ∥ L 1 x + ∥ θ 0 − 1 ∥ L 1 x + ∥ z 0 ∥ L 1 x < δ ∗ , G ( τ ) < δ , ∀ τ < t, t ♯ ≥ 4 ν 0 , (4.16) Then, for any t > t ♯ we have              ∥ u ( · , t ) ∥ L 1 x ≤ C ( ν 0 ) δ ∗ + O (1)  √ ν 0 δ + δ 2  , ∥ √ 1 + t u ( · , t ) ∥ L ∞ x ≤ C ( ν 0 ) δ ∗ + O (1)  √ ν 0 δ + δ 2  , ∥ u x ( · , t ) ∥ L 1 x ≤ C ( ν 0 ) δ ∗ + O (1)  | log( ν 0 ) | δ 2 √ ν 0 + √ ν 0 δ  , ∥ √ t u x ( · , t ) ∥ L ∞ x ≤ C ( ν 0 ) δ ∗ + O (1)  | log( ν 0 ) | δ 2 √ ν 0 + √ ν 0 δ  . Using the abov e estimates for u ( x, t ) , we next deriv e estimates for θ ( x, t ) and z ( x, t ) . The argumen t for θ is similar to the pro of of Lemma 4.7 in [31], with an additional term q z ( x, t ) . Therefore, we estimate θ and z together. 65 Lemma 4.6. L et ( v , u, θ , z ) b e the lo c al solution c onstructe d in The or em 2.1, and assume the c ondition (4.16) . Then, for t > t ♯ , the fol lowing estimates hold:          ∥ z ( x, t ) ∥ L 1 x ≤ δ ∗ , ∥ √ 1 + t z ( · , t ) ∥ L ∞ x ≤ C ( ν 0 ) δ ∗ + O (1)  δ + δ 2  , ∥ θ ( · , t ) − 1 ∥ L 1 x ≤ O (1)  C ( ν 0 ) δ ∗ + √ ν 0 δ + δ 2  + O (1)  C ( ν 0 ) δ ∗ + √ ν 0 δ + δ 2  2 , ∥ √ 1 + t ( θ ( · , t ) − 1) ∥ L ∞ x ≤ O (1)  C ( ν 0 ) δ ∗ + √ ν 0 δ + δ 2  + O (1)  C ( ν 0 ) δ ∗ + √ ν 0 δ + δ 2  2 . Pr o of. By the in tegral represen tation of z ( x, t ) given in Lemma 4.1, w e ha v e z ( x, t ) = Z R G 41 ( x − y , t )( v ( y , 0) − 1)d y + Z R G 42 ( x − y , t ) u ( y , 0) d y + Z R G 43 ( x − y , t )( E ( y, 0) − c v ) d y + Z R G 44 ( x, t ; y , 0) z ( y , 0) d y − Z t 0 Z R G 44 ( x, t ; y , τ ) K ϕ ( θ ) z ( y , τ ) d y d t + 3 X i =1 R z i , where R z i are given in Lemma 4.1. Since the Green’s function G has diﬀeren t estimates for short time t ≤ 1 and large time t ≥ 1 , w e split the pro of into these tw o cases. How ev er, the large-time estimates already incorporate the main ideas of the short-time analysis, so w e only presen t the detailed pro of for t ≥ 1 . In view of Lemma 4.4, for t ≥ 1 , G 4 k has the follo wing estimates: | G 4 k ( x, t ) | ≤ 4 X j =1 O (1) e − ( x + β j t ) 2 4 C t √ t + O (1) e − σ ∗ 0 t − σ 0 | x | , k = 1 , 2 , 3 , 4 , | ∂ x G 4 k ( x, t ) | ≤ 4 X j =1 O (1) e − ( x + β j t ) 2 4 t t + O (1) e − σ ∗ 0 t − σ 0 | x | , k = 1 , 2 , 3 , 4 , (4.17) and for t ≤ 1 | G 4 k ( x, t ) | ≤ O (1) e − σ ∗ 0 t − σ 0 | x | + O (1) te − σ 0 | x | , k = 1 , 2 , 3 , | G 44 ( x, t ) | ≤ O (1) e − x 2 4 Dt √ 4 π D t + O (1) e − σ ∗ 0 t − σ 0 | x | + O (1) te − σ 0 | x | , | ∂ x G 4 k ( x, t ) | ≤ O (1) e − σ ∗ 0 t − σ 0 | x | + O (1) te − σ 0 | x | , k = 1 , 2 , 3 , | ∂ xx G 4 k ( x, t ) | ≤ O (1) e − σ ∗ 0 t − σ 0 | x | + O (1) te − σ 0 | x | , k = 1 , 2 , 3 , | ∂ x G 44 ( x, t ) | ≤ O (1)       ∂ x Ñ e − x 2 4 Dt √ 4 π D t é       + O (1) e − σ ∗ 0 t − σ 0 | x | + O (1) te − σ 0 | x | . (4. 18) As p er Remark 4.2, the O (1) terms in the abov e estimates ma y depend on ν 0 . Estimates of ∥ z ∥ L 1 x : The L 1 estimate is deriv ed directly from the fourth equation of Eq. (0.1). In tegrating b oth sides o v er (0 , t ) × R , w e hav e Z R z ( x, t ) d x + Z t 0 Z R K ϕ ( θ ) z ( y , τ ) d y d τ ≤ Z R z 0 ( x ) d x. Under the initial condition ∥ z 0 ∥ L 1 x < δ ∗ giv en in (4.16), and since K ϕ ( θ ) ≥ 0 , it follo ws that ∥ z ( · , t ) ∥ L 1 x ≤ δ ∗ , 66 and Z t 0 Z R K ϕ ( θ ) z ( y , τ ) d y d τ ≤ δ ∗ . (4.19) Then, from the pro of in [31, Lemma 4.7] and abov e estimates, we infer that ∥ θ − 1 ∥ L 1 x ≤ O (1)  C ( ν 0 ) δ ∗ + √ ν 0 δ + δ 2  + 1 2 ∥ u ( · , t ) ∥ L ∞ x ∥ u ( · , x ) ∥ L 1 x + q ∥ z ( · , t ) ∥ L 1 x ≤  C ( ν 0 ) δ ∗ + √ ν 0 δ + δ 2  + O (1)  C ( ν 0 ) δ ∗ + √ ν 0 δ + δ 2  2 √ 1 + t + O (1) δ ∗ . W e no w make an imp ortant observ ation. Deﬁne M ( τ ) = Z R K ϕ ( θ ) z ( y , τ ) d y . Then d d τ M ( τ ) = Z R K ϕ ′ ( θ ) θ τ z ( y , τ ) d y + Z R K ϕ ( θ ) z τ ( y , τ ) d y = Z R K ϕ ′ ( θ ) θ τ z ( y , τ ) d y + Z R K ϕ ( θ ) ñ Å D v 2 z y ã y − K ϕ ( θ ) z ô ( y , τ ) d y = Z R K ϕ ′ ( θ ) θ τ z ( y , τ ) d y − Z R K ϕ ′ ( θ ) θ y D v 2 z y d y − Z R K 2 ϕ 2 ( θ ) z ( y , τ ) d y =: I 1 + I 2 + I 3 . (4.20) By Lemma 4.3, we ha v e the estimate ∥ θ τ ( · , τ ) ∥ L 1 x ≤ O (1) δ p min( τ , t ♯ ) , max( τ , t ♯ ) < t. F or τ ≥ ν 0 , as ϕ is Lipsc hitz, its deriv ativ e ϕ ′ exists and is b ounded a.e.. Hence, |I 1 | ≤ O (1) δ ∥ θ τ ( · , τ ) ∥ L 1 x ≤ O (1) δ 2 √ ν 0 , |I 2 | ≤ O (1) δ ∥ θ y ( · , τ ) ∥ L ∞ x ∥ z y ( · , τ ) ∥ L 1 x ≤ O (1) δ 2 √ τ , |I 3 | ≤ O (1) δ. F or suﬃciently small δ ≪ 1 , the leading-order term in (4.20) is − O ( δ ) , while the higher-order term O ( δ 2 √ ν 0 ) is negligible. Th us, for all τ > ν 0 , M is monotone: d d τ M ( τ ) ≤ 0 . It follows that ( τ − ν 0 ) M ( τ ) = Z τ ν 0 M ( τ ) d s ≤ Z τ ν 0 M ( s ) d s ≤ δ ∗ . If τ − ν 0 ≥ 1 , since 1 τ − ν 0 ≤ 2 1+ τ − ν 0 , we hav e M ( τ ) ≤ δ ∗ τ − ν 0 ≤ O (1) δ ∗ 1 + τ , If 0 < τ − ν 0 < 1 , then M ( τ ) ≤ M ( ν 0 ) ≤ 2 M ( ν 0 ) 1 + τ − ν 0 ≤ O (1) δ ∗ 1 + τ . 67 Therefore, we conclude for all ν 0 < τ < t that M ( τ ) ≤ O (1) δ ∗ 1 + τ . (4.21) Estimate of ∥ z ( · , t ) ∥ L ∞ x : Since t ≥ t ♯ ≥ 4 ν 0 , using estimates (4.17) and the smallness condition (4.16), we hav e     Z R G 41 ( x − y , t ) ( v ( y , 0) − 1) d y     ≤ C ( ν 0 ) Z R 4 X j =1 e − ( x − y + β j t ) 2 4 C t √ t | v ( y , 0) − 1 | d y + C ( ν 0 ) Z R e − σ ∗ 0 t − σ 0 | x − y | | v ( y , 0) − 1 | d y ≤ C ( ν 0 ) δ ∗ √ t ≤ C ( ν 0 ) δ ∗ √ 1 + t . By rep eating this pro cedure, w e derive similar b ounds for the remaining initial integral terms     Z R G 42 ( x − y , t ) u ( y , 0) d y     ≤ C ( ν 0 ) δ ∗ √ 1 + t ,     Z R G 43 ( x − y , t ) ( E ( y , 0) − c v ) d y     ≤ C ( ν 0 ) δ ∗ √ 1 + t ,     Z R G 44 ( x, t ; y , 0) z ( y , 0) d y     =     Z R G 44 ( x − y , t ) z ( y , 0) d y     ≤ C ( ν 0 ) δ ∗ √ 1 + t . Next, we estimate the in tegral remainder term. Using the deﬁnition of X ( t − τ ν 0 ) , we write Z t 0 Z R G 44 ( x, t ; y , τ ) K ϕ ( θ ) z ( y , τ ) d y d τ = Z t 0 Z R X Å t − τ ν 0 ã H ( x, t ; y , τ ; D v 2 ) K ϕ ( θ ) z ( y , τ ) d y d τ + Z t 0 Z R Å 1 − X Å t − τ ν 0 ãã G 44 ( x − y , t − τ ) K ϕ ( θ ) z ( y , τ ) d y d τ = Z t − ν 0 t − 2 ν 0 Z R X Å t − τ ν 0 ã H ( x, t ; y , τ ; D v 2 ) K ϕ ( θ ) z ( y , τ ) d y d τ + Z t t − ν 0 Z R H ( x, t ; y , τ ; D v 2 ) K ϕ ( θ ) z ( y , τ ) d y d τ + Z t − 2 ν 0 0 Z R G 44 ( x − y , t − τ ) K ϕ ( θ ) z ( y , τ ) d y d τ + Z t − ν 0 t − 2 ν 0 Z R Å 1 − X Å t − τ ν 0 ãã G 44 ( x − y , t − τ ) K ϕ ( θ ) z ( y , τ ) d y d τ . T o apply the Green’s function estimates, we further split the interv al ( 0 , t − 2 ν 0 ) into ( 0 , t − 1+2 ν 0 2 ), ( t − 1+2 ν 0 2 , t − 1 ) and ( t − 1 , t − 2 ν 0 ). Using the estimates for H in Lemma 1.3, the b ounds for G 44 (4.17)(4.18), and G ( τ ) < δ , each integral part can be bounded as follows     Z t 0 Z R G 44 ( x, t ; y , τ ) K ϕ ( θ ) z ( y , τ ) d y d τ     ≤ O (1) Z t − ν 0 t − 2 ν 0 Z R X Å t − τ ν 0 ã     H ( x, t ; y , τ ; D v 2 )     z ( y , τ ) d y d τ + O (1) Z t t − ν 0 Z R     H ( x, t ; y , τ ; D v 2 )     z ( y , τ ) d y d τ 68 + O (1) Z t − 1+2 ν 0 2 0 Z R | G 44 ( x − y , t − τ ) | K ϕ ( θ ) z ( y , τ ) d y d τ + O (1) Z t − 1 t − 1+2 ν 0 2 Z R | G 44 ( x − y , t − τ ) | K ϕ ( θ ) z ( y , τ ) d y d τ + O (1) Z t − 2 ν 0 t − 1 Z R | G 44 ( x − y , t − τ ) | z ( y , τ ) d y d τ + O (1) Z t − ν 0 t − 2 ν 0 Z R Å 1 − X Å t − τ ν 0 ãã | G 44 ( x − y , t − τ ) | z ( y , τ ) d y d τ ≤ O (1) Z t − ν 0 t − 2 ν 0 Z R e − ( x − y ) 2 C ∗ ( t − τ ) √ t − τ δ √ 1 + τ d y d τ + O (1) Z t t − ν 0 Z R e − ( x − y ) 2 C ∗ ( t − τ ) √ t − τ δ √ 1 + τ d y d τ + O (1) Z t − 1+2 ν 0 2 0 Z R Ö 4 X j =1 O (1) e − ( x − y + β j ( t − τ ) ) 2 4 C ( t − τ ) √ t − τ + e − σ ∗ 0 ( t − τ ) − σ 0 | x − y | è z ( y , τ ) d y d τ + O (1) Z t − 1 t − 1+2 ν 0 2 Z R Ö 4 X j =1 O (1) e − ( x − y + β j ( t − τ ) ) 2 4 C ( t − τ ) √ t − τ + e − σ ∗ 0 ( t − τ ) − σ 0 | x − y | è K ϕ ( θ ) z ( y , τ ) d y d τ + O (1) Z t − ν 0 t − 1 Z R Ñ e − ( x − y ) 2 4 D ( t − τ ) p 4 π D ( t − τ ) + e − σ ∗ 0 ( t − τ ) − σ 0 | x − y | + ( t − τ ) e − σ 0 | x − y | é z ( y , τ ) d y d τ ≤ O (1) Z t t − 2 ν 0 δ √ 1 + τ d τ + O (1) Z t − 1+2 ν 0 2 0 Å 1 √ t − τ + e − σ ∗ 0 ( t − τ ) ã M ( τ ) d τ + O (1) Z t − 1 t − 1+2 ν 0 2 Å 1 √ t − τ + e − σ ∗ 0 ( t − τ ) ã M ( τ ) d τ + O (1) Z t − ν 0 t − 1 Ä 1 + e − σ ∗ 0 ( t − τ ) ä δ √ 1 + τ d τ ≤ O (1) ν 0 δ √ 1 + t + O (1) √ 1 + t − 2 ν 0 Z t − 1+2 ν 0 2 0 M ( τ ) d τ + O (1) Z t − 1 t − 1+2 ν 0 2 1 √ t − τ δ ∗ 1 + τ d τ ≤ O (1) Å ν 0 δ √ 1 + t + δ √ 1 + t ã , in which we use the estimates R t 0 M ( τ ) dτ < δ ∗ (4.19) and M ( τ ) ≤ O (1) δ ∗ 1+ τ (4.21). Next, w e estimate the remainder terms R z i . W e split the time in tegral at τ = t − 1 2 , τ = t − 1 and τ = t − 2 ν 0 to apply deriv ative estimates separately to the short and long time scales. Using the b ounds for ∂ y G 4 k from (4.17)(4.18) and a priori bounds ∥ v − 1 ∥ L ∞ , ∥ u ∥ L ∞ , . . . , ∥ z y ∥ L ∞ ≤ δ √ 1+ τ from G ( τ ) < δ , w e ﬁnd |R z 1 | ≤      Z t − 1 2 0 Z R ∂ y G 42 ( x − y , t − τ ) ï a ( v − 1) 2 v + a ( θ − 1)(1 − v ) v − au 2 2 c v + µu y ( v − 1) v ò d y d τ      +      Z t − 1 2 0 Z R ∂ y G 43 ( x − y , t − τ ) ïÅ a ( θ − 1) + a (1 − v ) v ã u + ν θ y ( v − 1) v + Å ν c v − µ v ã uu y + q D ( v 2 − 1) v 2 z y ò d y d τ     +      Z t − 1 2 0 Z R ∂ y G 44 ( x − y , t − τ ) D ( v 2 − 1) v 2 z y d y d τ      69 +      Z t − 1 t − 1 2 Z R ∂ y G 42 ( x − y , t − τ ) Å a ( v − 1) 2 v + a ( θ − 1)(1 − v ) v − au 2 2 c v + µu y ( v − 1) v ã d y d τ      +      Z t − 1 t − 1 2 Z R ∂ y G 43 ( x − y , t − τ ) ïÅ a ( θ − 1) + a (1 − v ) v ã u + ν θ y ( v − 1) v + Å ν c v − µ v ã uu y + q D ( v 2 − 1) v 2 z y ò d y d τ     +      Z t − 1 t − 1 2 Z R ∂ y G 44 ( x − y , t − τ ) D ( v 2 − 1) v 2 z y d y d τ      +      Z t − 2 ν 0 t − 1 Z R ∂ y G 42 ( x − y , t − τ ) Å a ( v − 1) 2 v + a ( θ − 1)(1 − v ) v − au 2 2 c v + µu y ( v − 1) v ã d y d τ      +      Z t − 2 ν 0 t − 1 Z R ∂ y G 43 ( x − y , t − τ ) ïÅ a ( θ − 1) + a (1 − v ) v ã u + ν θ y ( v − 1) v + Å ν c v − µ v ã uu y + q D ( v 2 − 1) v 2 z y ò d y d τ     +      Z t − 2 ν 0 t − 1 Z R ∂ y G 44 ( x − y , t − τ ) D ( v 2 − 1) v 2 z y d y d τ      ≤ O (1) Z t − 1 2 0 Z R Ö 4 X j =1 O (1) e − ( x − y + β j ( t − τ ) ) 2 4( t − τ ) t − τ + C ( ν 0 ) e − σ ∗ 0 ( t − τ ) − σ 0 | x − y | è δ √ 1 + τ ( | v − 1 | + | θ − 1 | + | u | + | u y | + | θ y | + | z y | ) d y d τ + O (1) Z t − 1 t − 1 2 Z R Ö 4 X j =1 O (1) e − ( x − y + β j ( t − τ ) ) 2 4( t − τ ) t − τ + C ( ν 0 ) e − σ ∗ 0 ( t − τ ) − σ 0 | x − y | è δ √ 1 + τ ( | v − 1 | + | θ − 1 | + | u | + | u y | + | θ y | + | z y | ) d y d τ + O (1) Z t − 2 ν 0 t − 1 Z R Ä e − σ ∗ 0 ( t − τ ) − σ 0 | x − y | + ( t − τ ) e − σ 0 | x − y | ä δ √ 1 + τ ( | v − 1 | + | θ − 1 | + | u | + | u y | + | θ y | + | z y | ) d y d τ + O (1) Z t − 2 ν 0 t − 1 Z R Ñ e − ( x − y ) 2 4 D ( t − τ ) t − τ + e − σ ∗ 0 ( t − τ ) − σ 0 | x − y | + O (1)( t − τ ) e − σ 0 | x − y | é δ √ 1 + τ | z y | d y d τ ≤ O (1) Z t − 1 2 0 Å 1 t − τ + e − σ ∗ 0 ( t − τ ) ã δ 2 √ 1 + τ d τ + O (1) Z t − 1 t − 1 2 1 √ t − τ δ 2 √ 1 + τ √ τ d τ + O (1) Z t − 1 t − 1 2 e − σ ∗ 0 ( t − τ ) δ 2 √ 1 + τ d τ + O (1) Z t − 2 ν 0 t − 1 1 √ t − τ δ 2 √ 1 + τ √ τ d τ + O (1) Z t − 2 ν 0 t − 1 e − σ ∗ 0 ( t − τ ) δ 2 √ 1 + τ d τ ≤ O (1) δ 2 √ 1 + t . F or a more accurate estimate, since ∂ τ G 44 − q a c v ∂ y G 42 + Å − q ν c v + q D ã ∂ 2 y G 43 + D ∂ 2 y G 44 = 0 , with the initial data G 44 ( x, 0) = δ ( x ) , w e hav e the explicit represen tation of G 44 G 44 ( x − y , t − τ ) = H ( x, t ; y , τ ; D ) − Z t τ Z R H ( ω, s ; y , τ ; D ) (4.22) 70 Å q a c v ∂ ω G 42 ( x − ω , t − s ) + Å q ν c v − q D ã ∂ 2 ω G 43 ( x − ω , t − s ) ã d ω d s. Therefore, w e substitute this into R z 2 and then follow the same strategy as for R z 1 . A dditionally , w e use the comparison estimates for H ( x, t ; y , τ ; D ) − H ( x, t ; y , τ ; D v 2 ) in Lemma 1.5 to obtain |R z 2 ( x, t ) | ≤     Z t − ν 0 t − 2 ν 0 Z R ∂ y G 42 ( x − y , t − τ ) ï a ( v − 1) 2 v + a ( θ − 1)(1 − v ) v − au 2 2 c v + µu y ( v − 1) v −X ( t − τ ν 0 ) q a c v z ò ( y , τ ) d y d τ     +     Z t − ν 0 t − 2 ν 0 Z R ∂ y G 43 ( x − y ; t − τ ) ï a ( θ − 1) + a (1 − v ) v u + ν ( v − 1) v θ y + Å ν c v v − µ v ã uu y + q D ( v 2 − 1) v 2 z y ò ( y , τ ) d y d τ     +     Z t − ν 0 t − 2 ν 0 Z R ∂ y G 43 ( x − y ; t − τ ) X ( t − τ ν 0 ) Å q ν c v − q D ã z y d y d τ     +     Z t − ν 0 t − 2 ν 0 Z R ∂ y G 44 ( x − y ; t − τ ) D ( v 2 − 1) v 2 z y ( y , τ ) d y d τ     + Z t − ν 0 t − 2 ν 0 Z R 1 ν 0     H ( x, t ; y , τ ; D ) − H ( x, t ; y , τ ; D v 2 )     z ( y , τ ) d y d τ + Z t − ν 0 t − 2 ν 0 Z R 1 ν 0     Z t τ Z R H ( ω, s ; y , τ ; D ) q a c v ∂ ω G 42 ( x − ω , t − s ) d ω d s     z ( y , τ ) d y d τ + Z t − ν 0 t − 2 ν 0 Z R 1 ν 0     Z t τ Z R H ( ω, s ; y , τ ; D ) Å q ν c v − q D ã ∂ 2 ω G 43 ( x − ω , t − s ) d ω d s     z ( y , τ ) d y d τ ≤ O (1) Z t − ν 0 t − 2 ν 0 Z R Ä e − σ ∗ 0 ( t − τ ) − σ 0 | x − y | + ( t − τ ) e − σ 0 | x − y | ä ï δ √ 1 + τ ( | v − 1 | + | θ − 1 | + | u | + | u y | + | θ y | + | z y | ) + z ] d y d τ + O (1) Z t − ν 0 t − 2 ν 0 Z R Ñ e − ( x − y ) 2 4 D ( t − τ ) t − τ + e − σ ∗ 0 ( t − τ ) − σ 0 | x − y | + O (1)( t − τ ) e − σ 0 | x − y | é δ √ 1 + τ | z y | d y d τ + O (1) Z t − ν 0 t − 2 ν 0 Z R 1 ν 0     D − D v 2     L ∞ x e − ( x − y ) 2 t − τ √ t − τ z ( y , τ ) d y d τ + O (1) Z t − ν 0 t − 2 ν 0 Z R 1 ν 0 Z t τ Z R e − ( ω − y ) 2 s − τ √ s − τ Ä e − σ ∗ 0 ( t − s ) − σ 0 | x − ω | + ( t − s ) e − σ 0 | x − ω | ä z ( y , τ ) d ω d s d y d τ ≤ O (1) Z t − ν 0 t − 2 ν 0 Ä e − σ ∗ 0 ( t − τ ) + ( t − τ ) ä δ 2 √ 1 + τ d τ + O (1) Z t − ν 0 t − 2 ν 0 Ä e − σ ∗ 0 ( t − τ ) + ( t − τ ) ä δ 2 √ 1 + τ d τ + O (1) Z t − ν 0 t − 2 ν 0 Å 1 t − τ + e − σ ∗ 0 ( t − τ ) + ( t − τ ) ã δ 2 √ 1 + τ d τ + O (1) Z t − ν 0 t − 2 ν 0 1 ν 0 δ δ √ 1 + τ d τ + O (1) Z t − ν 0 t − 2 ν 0 Z t τ 1 ν 0 Ä e − σ ∗ 0 ( t − s ) + ( t − s ) ä δ √ 1 + τ d s d τ ≤ O (1) Å ν 0 δ 2 √ 1 + t + ν 2 0 δ 2 √ 1 + t + δ 2 √ 1 + t + δ √ 1 + t + ν 0 δ √ 1 + t ã . F or R z 3 , we rep eat the argumen ts for R z 1 to obtain |R z 3 ( x, t ) | 71 ≤ O (1) Z t − ν 0 t − 2 ν 0 Z R Ä e − σ ∗ 0 ( t − τ ) − σ 0 | x − y | + ( t − τ ) e − σ 0 | x − y | ä δ √ 1 + τ ( | v − 1 | + | θ − 1 | + | u | + | u y | ) d y d τ + O (1) Z t − ν 0 t − 2 ν 0 Z R Ä e − σ ∗ 0 ( t − τ ) − σ 0 | x − y | + ( t − τ ) e − σ 0 | x − y | ä z ( y , τ ) d y d τ ≤ O (1) Z t − ν 0 t − 2 ν 0 Ä e − σ ∗ 0 ( t − τ ) + ( t − τ ) ä δ 2 √ 1 + τ d τ + Z t − ν 0 t − 2 ν 0 Ä e − σ ∗ 0 ( t − τ ) + ( t − τ ) ä δ √ 1 + τ d τ ≤ O (1) Å ν 0 δ 2 √ 1 + t + ν 2 0 δ 2 √ 1 + t + ν 0 δ √ 1 + t + ν 2 0 δ √ 1 + t ã . Com bining the abov e estimates, for suﬃciently small δ and t ≥ t ♯ ≥ 4 ν 0 , we obtain that ∥ z ( · , t ) ∥ L ∞ x ≤ C ( ν 0 ) δ ∗ √ 1 + t + O (1) δ + δ 2 √ 1 + t . (4.23) Estimate of ∥ θ ( · , t ) ∥ L ∞ x : Using the represen tation of E ( x, t ) from (4.8), w e express θ as c v ( θ ( x, t ) − 1) = Z R G 31 ( x − y , t )( v ( y , 0) − 1)d y + Z R G 32 ( x − y , t ) u ( y , 0) d y + Z R G 33 ( x, t ; y , 0)( E ( y , 0) − c v ) d y + 3 X i =1 R θ i − ( u ( x, t )) 2 2 − q z ( x, t ) F ollowing the pro of of [31, Lemma 4.7], we hav e that     Z R G 31 ( x − y , t ) ( v ( y , 0) − 1) d y     ≤ C ( ν 0 ) δ ∗ √ 1 + t ,     Z R G 32 ( x − y , t ) u ( y , 0) d y     ≤ C ( ν 0 ) δ ∗ √ 1 + t ,     Z R G 33 ( x, t ; y , 0) ( E ( y, 0) − c v ) d y     ≤ C ( ν 0 ) δ ∗ √ 1 + t . F or the remainder term R θ i , w e follo w a similar pro cedure as for the estimates of z ( x, t ) and apply the bounds in [31], obtaining that |R θ 1 | ≤ O (1) δ 2 √ 1 + t , |R θ 2 | ≤ O (1)( √ ν 0 δ + δ 2 ) 1 √ 1 + t , |R θ 3 | ≤ O (1) √ ν 0 δ √ 1 + t . Therefore, we com bine the ab ov e estimates with ∥ u ( · , t ) ∥ L ∞ x in Lemma 4.5 and the b ound for ∥ z ( · , t ) ∥ L ∞ x to derive that ∥ θ − 1 ∥ L ∞ x ≤ C ( ν 0 ) δ ∗ √ 1 + t + O (1) √ ν 0 δ + δ 2 √ 1 + t + ∥ u ( · , t ) ∥ 2 L ∞ x 2 + q ∥ z ( · , t ) ∥ L ∞ x ≤ C ( ν 0 ) δ ∗ √ 1 + t + O (1) √ ν 0 δ + δ 2 √ 1 + t + Ç C ( ν 0 ) δ ∗ √ 1 + t + O (1) √ ν 0 δ + δ 2 √ 1 + t å 2 + C ( ν 0 ) δ ∗ √ 1 + t + O (1) δ + δ 2 √ 1 + t ≤ C ( ν 0 ) δ ∗ √ 1 + t + O (1) √ ν 0 δ + δ 2 √ 1 + t + Ç C ( ν 0 ) δ ∗ √ 1 + t + O (1) √ ν 0 δ + δ 2 √ 1 + t å 2 . (4.24) □ 72 Lemma 4.7. L et ( v , u, θ , z ) b e the lo c al solution c onstructe d in The or em 2.1. A lso assume the c ondition (4.16) . Then, for t > t ♯ , z ( x, t ) and θ ( x, t ) satisfy the fol lowing ﬁrst-or der estimates:                ∥ z x ( · , t ) ∥ L 1 x ≤ C ( ν 0 ) δ ∗ + O (1)  | log( ν 0 ) | δ 2 √ ν 0 + √ ν 0 δ  ,   √ tz x ( · , t )   L ∞ x ≤ C ( ν 0 ) δ ∗ + O (1)  | log( ν 0 ) | δ 2 √ ν 0 + √ ν 0 δ  , ∥ θ x ( · , t ) ∥ L 1 x ≤ C ( ν 0 ) δ ∗ + O (1)  | log( ν 0 ) | δ 2 √ ν 0 + √ ν 0 δ  , ∥ √ t θ x ( · , t ) ∥ L ∞ x ≤ C ( ν 0 ) δ ∗ + O (1)  | log( ν 0 ) | δ 2 √ ν 0 + √ ν 0 δ  . Pr o of. F rom the represen tation formula (4.6) for z ( x, t ) , w e deduce that z x ( x, t ) = Z R ∂ x G 41 ( x − y , t )( v ( y , 0) − 1)d y + Z R ∂ x G 42 ( x − y , t ) u ( y , 0) d y + Z R ∂ x G 43 ( x − y , t )( E ( y, 0) − c v ) d y + Z R ∂ x G 44 ( x, t ; y , 0) z ( y , 0) d y − Z t 0 Z R ∂ x G 44 ( x, t ; y , τ ) K ϕ ( θ ) z ( y , τ ) d y d τ + 3 X i =1 ∂ x R z i , (4.25) where R z i deﬁned as in Lemma 4.1. Recall the second-order deriv ative estimates of the Green’s function from Lemma 4.4. F or large time t ≥ 1 , the second deriv ativ e ∂ xx G 4 k satisﬁes | ∂ xx G 4 k ( x, t ) | ≤ 4 X j =1 O (1) e − ( x + β j t ) 2 4 C t t 3 2 + 3 X j =1 O (1) e − ( x + β j t ) 2 4 C t t 2 + 3 X j =1 O (1) e − ( x + β j t ) 2 4 C t t 5 2 + O (1) e − σ ∗ 0 t − σ 0 | x | ≤ 4 X j =1 O (1) e − ( x + β j t ) 2 4 C t t 3 2 + O (1) e − σ ∗ 0 t − σ 0 | x | for k = 1 , 2 , 3 , 4 , (4.26) while for short time t ≤ 1 , | ∂ xx G 4 k ( x, t ) | ≤ O (1) e − σ ∗ 0 t − σ 0 | x | + O (1) te − σ 0 | x | , k = 1 , 2 , 3 , | ∂ xx G 44 ( x, t ) | ≤ O (1)       ∂ xx Ñ e − x 2 4 Dt √ 4 π D t é       + O (1) e − σ ∗ 0 t − σ 0 | x | + O (1) te − σ 0 | x | . (4.27) Estimate of ∥ z x ( · , t ) ∥ L ∞ x : F or t ≥ t ♯ ≥ 4 ν 0 , w e start with the integral term inv olving the initial p erturbation v 0 − 1 . Using the large-time estimate of ∂ x G 41 from (4.17) and the initial smallness condition (4.16), w e obtain that     Z R ∂ x G 41 ( x − y , t ) ( v ( y , 0) − 1) d y     ≤ C ( ν 0 ) Z R 4 X j =1 e − ( x − y + β j t ) 2 4 C t t | v ( y , 0) − 1 | d y + C ( ν 0 ) Z R e − σ ∗ 0 t − σ 0 | x − y | | v ( y , 0) − 1 | d y ≤ C ( ν 0 ) δ ∗ √ t . 73 W e also ha ve     Z R ∂ x G 42 ( x − y , t ) u ( y , 0) d y     ≤ C ( ν 0 ) δ ∗ √ t ,     Z R ∂ x G 43 ( x − y , t ) ( E ( y , 0) − c v ) d y     ≤ C ( ν 0 ) δ ∗ √ t ,     Z R ∂ x G 44 ( x, t ; y , 0) z ( y , 0) d y     =     Z R ∂ x G 44 ( x − y , t ) z ( y , 0) d y     ≤ C ( ν 0 ) δ ∗ √ t , follo wing the similar argument as in Lemma 4.6. W e split the interv al ( 0 , t − 2 ν 0 ) into ( 0 , t − 1+2 ν 0 2 ), ( t − 1+2 ν 0 2 , t − 1 ) and ( t − 1 , t − 2 ν 0 ) and use the Lipschitz con tin uit y of ϕ , the Green’s function b ounds in (4.17) and (4.18), and a priori b ound G ( τ ) < δ from (4.16). In this w a y , we obtain that     Z t 0 Z R ∂ x G 44 ( x, t ; y , τ ) K ϕ ( θ ) z ( y , τ ) d y d τ     ≤ O (1) Z t − ν 0 t − 2 ν 0 Z R X Å t − τ ν 0 ã     H x ( x, t ; y , τ ; D v 2 )     z ( y , τ ) d y d τ + O (1) Z t t − ν 0 Z R     H x ( x, t ; y , τ ; D v 2 )     z ( y , τ ) d y d τ + O (1) Z t − 1+2 ν 0 2 0 Z R | ∂ x G 44 ( x − y , t − τ ) | K ϕ ( θ ) z ( y , τ ) d y d τ + O (1) Z t − 1 t − 1+2 ν 0 2 Z R | ∂ x G 44 ( x − y , t − τ ) | K ϕ ( θ ) z ( y , τ ) d y d τ + O (1) Z t − 2 ν 0 t − 1 Z R | ∂ x G 44 ( x − y , t − τ ) | z ( y , τ ) d y d τ + O (1) Z t − ν 0 t − 2 ν 0 Z R Å 1 − X Å t − τ ν 0 ãã | ∂ x G 44 ( x − y , t − τ ) | z ( y , τ ) d y d τ ≤ O (1) Z t − ν 0 t − 2 ν 0 Z R e − ( x − y ) 2 C ∗ ( t − τ ) t − τ δ √ 1 + τ d y d τ + O (1) Z t t − ν 0 Z R e − ( x − y ) 2 C ∗ ( t − τ ) t − τ δ √ 1 + τ d y d τ + O (1) Z t − 1+2 ν 0 2 0 Z R Ö 4 X j =1 O (1) e − ( x − y + β j ( t − τ ) ) 2 4 C ( t − τ ) t − τ + e − σ ∗ 0 ( t − τ ) − σ 0 | x − y | è K ϕ ( θ ) z ( y , τ ) d y d τ + O (1) Z t − 1 t − 1+2 ν 0 2 Z R Ö 4 X j =1 O (1) e − ( x − y + β j ( t − τ ) ) 2 4 C ( t − τ ) t − τ + e − σ ∗ 0 ( t − τ ) − σ 0 | x − y | è K ϕ ( θ ) z ( y , τ ) d y d τ + O (1) Z t − ν 0 t − 1 Z R Ñ e − ( x − y ) 2 4 D ( t − τ ) t − τ + e − σ ∗ 0 ( t − τ ) − σ 0 | x − y | + ( t − τ ) e − σ 0 | x − y | é z ( y , τ ) d y d τ ≤ O (1) Z t t − 2 ν 0 1 √ t − τ δ √ 1 + τ d τ + O (1) Z t − 1+2 ν 0 2 0 Å 1 t − τ + e − σ ∗ 0 ( t − τ ) ã M ( τ ) d τ + O (1) Z t − 1 t − 1+2 ν 0 2 Å 1 √ t − τ + e − σ ∗ 0 ( t − τ ) ã M ( τ ) d τ + O (1) Z t − ν 0 t − 1 Å 1 √ t − τ + e − σ ∗ 0 ( t − τ ) + ( t − τ ) ã δ √ 1 + τ d τ 74 ≤ O (1) √ ν 0 δ √ 1 + t + O (1) 1 + t − 2 ν 0 Z t − 1+2 ν 0 2 0 M ( τ ) d τ + O (1) Z t − 1 t − 1+2 ν 0 2 Å 1 √ t − τ + e − σ ∗ 0 ( t − τ ) ã δ ∗ 1 + τ d τ ≤ O (1) Å √ ν 0 δ √ t + δ √ t + δ ∗ √ t ã . F or the remainder terms ∂ x R z 1 , w e split the time integral at τ = t − 1 2 , τ = t − 1 and τ = t − 2 ν 0 , which allows us to apply short-time and long-time deriv ative estimates from (4.26) and (4.27) respectively . Using the a priori estimates G ( τ ) < δ in (4.16), w e ﬁnd that | ∂ x R z 1 | ≤      Z t − 1 2 0 Z R ∂ xy G 42 ( x − y , t − τ ) ï a ( v − 1) 2 v + a ( θ − 1)(1 − v ) v − au 2 2 c v + µu y ( v − 1) v ò d y d τ      +      Z t − 1 2 0 Z R ∂ xy G 43 ( x − y , t − τ ) ïÅ a ( θ − 1) + a (1 − v ) v ã u + ν θ y ( v − 1) v + Å ν c v − µ v ã uu y + q D ( v 2 − 1) v 2 z y ò d y d τ     +      Z t − 1 2 0 Z R ∂ xy G 44 ( x − y , t − τ ) D ( v 2 − 1) v 2 z y d y d τ      +      Z t − 1 t − 1 2 Z R ∂ xy G 42 ( x − y , t − τ ) Å a ( v − 1) 2 v + a ( θ − 1)(1 − v ) v − au 2 2 c v + µu y ( v − 1) v ã d y d τ      +      Z t − 1 t − 1 2 Z R ∂ xy G 43 ( x − y , t − τ ) ïÅ a ( θ − 1) + a (1 − v ) v ã u + ν θ y ( v − 1) v + Å ν c v − µ v ã uu y + q D ( v 2 − 1) v 2 z y ò d y d τ     +      Z t − 1 t − 1 2 Z R ∂ xy G 44 ( x − y , t − τ ) D ( v 2 − 1) v 2 z y d y d τ      +      Z t − 2 ν 0 t − 1 Z R ∂ xy G 42 ( x − y , t − τ ) Å a ( v − 1) 2 v + a ( θ − 1)(1 − v ) v − au 2 2 c v + µu y ( v − 1) v ã d y d τ      +      Z t − 2 ν 0 t − 1 Z R ∂ xy G 43 ( x − y , t − τ ) ïÅ a ( θ − 1) + a (1 − v ) v ã u + ν θ y ( v − 1) v + Å ν c v − µ v ã uu y + q D ( v 2 − 1) v 2 z y ò d y d τ     +      Z t − 2 ν 0 t − 1 Z R ∂ xy G 44 ( x − y , t − τ ) D ( v 2 − 1) v 2 z y d y d τ      ≤ O (1) Z t − 1 2 0 Z R Ö 4 X j =1 O (1) e − ( x − y + β j ( t − τ ) ) 2 4( t − τ ) ( t − τ ) 3 2 + C ( ν 0 ) e − σ ∗ 0 ( t − τ ) − σ 0 | x − y | è δ √ 1 + τ ( | v − 1 | + | θ − 1 | + | u | + | u y | + | θ y | + | z y | ) d y d τ + O (1) Z t − 1 t − 1 2 Z R Ö 4 X j =1 O (1) e − ( x − y + β j ( t − τ ) ) 2 4( t − τ ) ( t − τ ) 3 2 + C ( ν 0 ) e − σ ∗ 0 ( t − τ ) − σ 0 | x − y | è δ √ 1 + τ ( | v − 1 | + | θ − 1 | + | u | + | u y | + | θ y | + | z y | ) d y d τ + O (1) Z t − 2 ν 0 t − 1 Z R Ä e − σ ∗ 0 ( t − τ ) − σ 0 | x − y | + ( t − τ ) e − σ 0 | x − y | ä δ √ 1 + τ ( | v − 1 | + | θ − 1 | + | u | + | u y | + | θ y | + | z y | ) d y d τ + O (1) Z t − 2 ν 0 t − 1 Z R Ñ e − ( x − y ) 2 4 D ( t − τ ) ( t − τ ) 3 2 + e − σ ∗ 0 ( t − τ ) − σ 0 | x − y | + O (1)( t − τ ) e − σ 0 | x − y | é δ √ 1 + τ | z y | d y d τ 75 ≤ O (1) Z t − 1 2 0 Ç 1 ( t − τ ) 3 2 + e − σ ∗ 0 ( t − τ ) å δ 2 √ 1 + τ d τ + O (1) Z t − 1 t − 1 2 Å 1 t − τ + e − σ ∗ 0 ( t − τ ) ã δ 2 √ 1 + τ √ τ d τ + O (1) Z t − 2 ν 0 t − 1 Å 1 t − τ + e − σ ∗ 0 ( t − τ ) + ( t − τ ) ã δ 2 √ 1 + τ √ τ d τ ≤ O (1) Å δ 2 1 + t + δ 2 √ 1 + t ã + O (1) δ 2 √ 1 + t Z t − 1 t − 1 2 1 √ t − τ √ τ d τ + O (1) δ 2 √ t Z t − 2 ν 0 t − 1 1 t − τ 1 √ τ d τ + O (1) δ 2 √ t Z t − 2 ν 0 t − 1 1 √ τ d τ ≤ O (1) (1 + | log ( ν 0 ) | ) δ 2 √ t . F or ∂ x R z 2 , w e utilize the explicit represen tation for G 44 in Eq. (4.22), together with com- parison estimates for H x ( x, t ; y , τ ; D ) − H x ( x, t ; y , τ ; D v 2 ) from Lemma 1.6, the estimates (4.26) and (4.27), as well as the a priori b ounds in Eq. (4.16). One th us deduces that | ∂ x R z 2 ( x, t ) | ≤     Z t − ν 0 t − 2 ν 0 Z R ∂ xy G 42 ( x − y , t − τ ) ï a ( v − 1) 2 v + a ( θ − 1)(1 − v ) v − au 2 2 c v + µu y ( v − 1) v −X ( t − τ ν 0 ) q a c v z ò d y d τ     +     Z t − ν 0 t − 2 ν 0 Z R ∂ xy G 43 ( x − y ; t − τ ) ï a ( θ − 1) + a (1 − v ) v u + ν ( v − 1) v θ y + Å ν c v v − µ v ã uu y + q D ( v 2 − 1) v 2 z y ò d y d τ     +     Z t − ν 0 t − 2 ν 0 Z R ∂ xy G 43 ( x − y ; t − τ ) X ( t − τ ν 0 ) Å q ν c v − q D ã z y d y d τ     +     Z t − ν 0 t − 2 ν 0 Z R ∂ xy G 44 ( x − y ; t − τ ) D ( v 2 − 1) v 2 z y d y d τ     + Z t − ν 0 t − 2 ν 0 Z R 1 ν 0     H x ( x, t ; y , τ ; D ) − H x ( x, t ; y , τ ; D v 2 )     z ( y , τ ) d y d τ + Z t − ν 0 t − 2 ν 0 Z R 1 ν 0     Z t τ Z R H ω ( ω , s ; y , τ ; D ) q a c v ∂ ω G 42 ( x − ω , t − s ) d ω d s     z ( y , τ ) d y d τ + Z t − ν 0 t − 2 ν 0 Z R 1 ν 0     Z t τ Z R H ω ( ω , s ; y , τ ; D ) Å q ν c v − q D ã ∂ 2 ω G 43 ( x − ω , t − s ) d ω d s     z ( y , τ ) d y d τ ≤ O (1) Z t − ν 0 t − 2 ν 0 Z R Ä e − σ ∗ 0 ( t − τ ) − σ 0 | x − y | + ( t − τ ) e − σ 0 | x − y | ä ï δ √ 1 + τ ( | v − 1 | + | θ − 1 | + | u | + | u y | + | θ y | + | z y | ) + z ] d y d τ + O (1) Z t − ν 0 t − 2 ν 0 Z R Ñ e − ( x − y ) 2 4 D ( t − τ ) ( t − τ ) 3 2 + e − σ ∗ 0 ( t − τ ) − σ 0 | x − y | + O (1)( t − τ ) e − σ 0 | x − y | é δ √ 1 + τ | z y | d y d τ + O (1) Z t − ν 0 t − 2 ν 0 Z R 1 ν 0 e − ( x − y ) 2 C ∗ ( t − τ ) t − τ ñ | log ( t − τ ) | sup t − 2 ν 0 0 and t ≥ t ♯ ≥ 4 ν 0 w e hav e that ∥ z x ( · , t ) ∥ L ∞ x ≤ C ( ν 0 ) δ ∗ √ t + O (1) Ç | log ( ν 0 ) | δ 2 √ ν 0 √ t + √ ν 0 δ √ t å . (4.28) Estimate of ∥ z x ( · , t ) ∥ L 1 x : F or t ≥ t ♯ ≥ 4 ν 0 , using the estimates in Eq. (4.17) and the smallness condition ∥ v 0 − 1 ∥ L 1 x < δ ∗ in Eq. (4.16), we get Z R     Z R ∂ x G 41 ( x − y , t ) ( v ( y , 0) − 1) d y     d x ≤ C ( ν 0 ) Z R Z R 4 X j =1 e − ( x − y + β j t ) 2 4 C t t | v ( y , 0) − 1 | d y d x + C ( ν 0 ) Z R Z R e − σ ∗ 0 t − σ 0 | x − y | | v ( y , 0) − 1 | d y d x ≤ C ( ν 0 ) δ ∗ . Similarly , for the other G 4 j terms we hav e that Z R     Z R ∂ x G 42 ( x − y , t ) u ( y , 0) d y     d x ≤ C ( ν 0 ) δ ∗ , Z R     Z R ∂ x G 43 ( x − y , t ) ( E ( y , 0) − c v ) d y     d x ≤ C ( ν 0 ) δ ∗ , Z R     Z R ∂ x G 44 ( x, t ; y , 0) z ( y , 0) d y     d x = Z R     Z R G 44 ( x − y , t ) z ( y , 0) d y     d x ≤ C ( ν 0 ) δ ∗ . F or the integral remainder term, w e split the time integral using the cutoﬀ function X ( t − τ ν 0 ) . Applying the bounds for H x in Lemma 1.3, estimates for ∂ x G 44 in (4.17)(4.18), and a priori b ound (4.16), w e ﬁnd that Z R     Z t 0 Z R ∂ x G 44 ( x, t ; y , τ ) K ϕ ( θ ) z ( y , τ ) d y d τ     d x = Z R     Z t 0 Z R Å t − τ ν 0 ã H x ( x, t ; y , τ ; D v 2 ) K ϕ ( θ ) z ( y , τ ) d y d τ     d x + Z R     Z t 0 Z R Å 1 − X Å t − τ ν 0 ãã ∂ x G 44 ( x − y , t − τ ) K ϕ ( θ ) z ( y , τ ) d y d τ     d x ≤ O (1) Z t − ν 0 t − 2 ν 0 Z R Z R     H x ( x, t ; y , τ ; D v 2 )     z ( y , τ ) d y d x d τ + O (1) Z t t − ν 0 Z R Z R     H x ( x, t ; y , τ ; D v 2 )     K ϕ ( θ ) z ( y , τ ) d y d x d τ + O (1) Z t − 2 ν 0 0 Z R Z R | ∂ x G 44 ( x − y , t − τ ) | K ϕ ( θ ) z ( y , τ ) d y d x d τ + O (1) Z t − ν 0 t − 2 ν 0 Z R Z R | ∂ x G 44 ( x − y , t − τ ) | K ϕ ( θ ) z ( y , τ ) d y d x d τ ≤ O (1) Z t − ν 0 t − 2 ν 0 Z R Z R e − ( x − y ) 2 t − τ t − τ z ( y , τ ) d y d x d τ 78 + O (1) Z t t − ν 0 Z R Z R e − ( x − y ) 2 t − τ t − τ z ( y , τ ) d y d x d τ + O (1) Z t − 2 ν 0 0 Z R Z R Ö 4 X j =1 e − ( x − y + β j ( t − τ ) ) 2 4 C ( t − τ ) t − τ + e − σ ∗ 0 ( t − τ ) − σ 0 | x − y | è K ϕ ( θ ) z ( y , τ ) d y d x d τ + O (1) Z t − ν 0 t − 2 ν 0 Z R Z R Ñ       ∂ x Ñ e − ( x − y ) 2 4 D ( t − τ ) p 4 π D ( t − τ ) é       + e − σ ∗ 0 ( t − τ ) − σ 0 | x − y | + ( t − τ ) e − σ 0 | x − y | é · z ( y , τ ) d y d x d τ ≤ O (1) Z t t − 2 ν 0 1 √ t − τ δ d τ + O (1) Z t − 2 ν 0 0 Å 1 √ t − τ + e − σ ∗ 0 ( t − τ ) ã M ( τ ) d τ + O (1) Z t − ν 0 t − 2 ν 0 Å 1 √ t − τ + e − σ ∗ 0 ( t − τ ) + ( t − τ ) ã δ d τ ≤ O (1) ν 0 δ + O (1) Z t − 2 ν 0 0 1 √ t − τ δ ∗ 1 + τ d τ + O (1)( √ ν 0 + ν 0 + ν 2 0 ) δ ≤ O (1)( √ ν 0 δ + ν 0 δ + ν 2 0 δ ) + C ( ν 0 ) δ ∗ . W e next b ound the L 1 -norms of ∂ x R z 1 , ∂ x R z 2 and ∂ x R z 3 . F or ∂ x R z 1 , w e split the time in tegral as b efore, and note the b ounds for ∂ xy G 4 k from (4.26) (4.27) and the a priori condition G ( τ ) < δ (4.16). This gives us Z R | ∂ x R z 1 | d x ≤ Z R      Z t − 1 0 Z R ∂ xy G 42 ( x − y , t − τ ) ï a ( v − 1) 2 v + a ( θ − 1)(1 − v ) v − au 2 2 c v + µu y ( v − 1) v ò d y d τ      d x + Z R      Z t − 1 0 Z R ∂ xy G 43 ( x − y , t − τ ) ïÅ a ( θ − 1) + a (1 − v ) v ã u + ν θ y ( v − 1) v + Å ν c v − µ v ã uu y + q D ( v 2 − 1) v 2 z y ò d y d τ     d x + Z R      Z t − 1 0 Z R ∂ xy G 44 ( x − y , t − τ ) D ( v 2 − 1) v 2 z y d y d τ      d x + Z R      Z t − 2 ν 0 t − 1 Z R ∂ xy G 42 ( x − y , t − τ ) Å a ( v − 1) 2 v + a ( θ − 1)(1 − v ) v − au 2 2 c v + µu y ( v − 1) v ã d y d τ      d x + Z R      Z t − 2 ν 0 t − 1 Z R ∂ xy G 43 ( x − y , t − τ ) ïÅ a ( θ − 1) + a (1 − v ) v ã u + ν θ y ( v − 1) v + Å ν c v − µ v ã uu y + q D ( v 2 − 1) v 2 z y ò d y d τ     d x + Z R      Z t − 2 ν 0 t − 1 Z R ∂ xy G 44 ( x − y , t − τ ) D ( v 2 − 1) v 2 z y d y d τ      d x ≤ O (1) Z t − 1 0 Z R Z R Ö 4 X j =1 e − ( x − y + β j ( t − τ ) ) 2 4( t − τ ) ( t − τ ) 3 2 + C ( ν 0 ) e − σ ∗ 0 ( t − τ ) − σ 0 | x − y | è δ √ 1 + τ ( | v − 1 | + | θ − 1 | + | u | + | u y | + | θ y | + | z y | ) d y d x d τ + O (1) Z t − 2 ν 0 t − 1 Z R Z R Ä e − σ ∗ 0 ( t − τ ) − σ 0 | x − y | + ( t − τ ) e − σ 0 | x − y | ä δ √ 1 + τ ( | v − 1 | + | θ − 1 | + | u | + | u y | + | θ y | + | z y | ) d y d x d τ 79 + O (1) Z t − 2 ν 0 t − 1 Z R Z R Ñ e − ( x − y ) 2 4 D ( t − τ ) ( t − τ ) 3 2 + e − σ ∗ 0 ( t − τ ) − σ 0 | x − y | + ( t − τ ) e − σ 0 | x − y | é δ √ 1 + τ | z y | d y d x d τ ≤ O (1) Z t − 1 0 Å 1 t − τ + e − σ ∗ 0 ( t − τ ) ã δ 2 √ 1 + τ d τ + O (1) Z t − 2 ν 0 t − 1 Å 1 t − τ + e − σ ∗ 0 ( t − τ ) + ( t − τ ) ã δ 2 √ 1 + τ d τ ≤ O (1) (1 + | log ( ν 0 ) | ) δ 2 . F or ∂ x R z 2 , substituting the representation for G 44 in Eq. (4.22), and combining the same estimates for ∂ x R z 1 with the comparison estimates for H x in Lemma 1.6, we deduce that Z R | ∂ x R z 2 ( x, t ) | d x ≤ Z t − ν 0 t − 2 ν 0 Z R Z R | ∂ xy G 42 ( x − y , t − τ ) |     a ( v − 1) 2 v + a ( θ − 1)(1 − v ) v − au 2 2 c v + µu y ( v − 1) v − X ( t − τ ν 0 ) q a c v z     d y d x d τ + Z t − ν 0 t − 2 ν 0 Z R Z R | ∂ xy G 43 ( x − y ; t − τ ) |     a ( θ − 1) + a (1 − v ) v u + ν ( v − 1) v θ y + Å ν c v v − µ v ã uu y + q D ( v 2 − 1) v 2 z y     d y d x d τ + Z t − ν 0 t − 2 ν 0 Z R Z R | ∂ xy G 43 ( x − y ; t − τ ) | X ( t − τ ν 0 )     Å q ν c v − q D ã z y     d y d τ + Z t − ν 0 t − 2 ν 0 Z R Z R | ∂ xy G 44 ( x − y ; t − τ ) |     D ( v 2 − 1) v 2 z y     d y d x d τ + Z t − ν 0 t − 2 ν 0 Z R Z R 1 ν 0     H x ( x, t ; y , τ ; D ) − H x ( x, t ; y , τ ; D v 2 )     z ( y , τ ) d y d x d τ + Z t − ν 0 t − 2 ν 0 Z R Z R 1 ν 0     Z t τ Z R H ω ( ω , s ; y , τ ; D ) q a c v ∂ ω G 42 ( x − ω , t − s ) d ω d s     z ( y , τ ) d y d x d τ + Z t − ν 0 t − 2 ν 0 Z R Z R 1 ν 0     Z t τ Z R H ω ( ω , s ; y , τ ; D ) Å q ν c v − q D ã ∂ 2 ω G 43 ( x − ω , t − s ) d ω d s     z ( y , τ ) d y d x d τ ≤ O (1) Z t − ν 0 t − 2 ν 0 Z R Z R Ä e − σ ∗ 0 ( t − τ ) − σ 0 | x − y | + ( t − τ ) e − σ 0 | x − y | ä ï δ √ 1 + τ ( | v − 1 | + | θ − 1 | + | u | + | u y | + | θ y | + | z y | ) + z + | z y | ] ( y , τ ) d y d x d τ + O (1) Z t − ν 0 t − 2 ν 0 Z R Z R Ñ e − ( x − y ) 2 4 D ( t − τ ) ( t − τ ) 3 2 + e − σ ∗ 0 ( t − τ ) − σ 0 | x − y | + ( t − τ ) e − σ 0 | x − y | é δ √ 1 + τ | z y | ( y , τ ) d y d x d τ + O (1) Z t − ν 0 t − 2 ν 0 Z R Z R 1 ν 0 e − ( x − y ) 2 C ∗ ( t − τ ) t − τ ñ | log ( t − τ ) | sup t − 2 ν 0 t ♯ , v ( x, t ) satisﬁes the fol lowing estimates      ∥ v ( · , t ) − 1 ∥ L 1 x ≤ C ( ν 0 ) δ ∗ + O (1) δ 2 , ∥ √ 1 + t ( v ( · , t ) − 1) ∥ L ∞ x ≤ C ( ν 0 ) δ ∗ + O (1) δ 2 , ∥ v ( · , t ) ∥ B V ≤ C ( ν 0 ) δ ∗ + O (1) δ 2 √ ν 0 . With the abov e preparations, no w we ma y deduce the main theorem of this section. Theorem 4.3 (Global existence) . Ther e exists a universal c onstant δ ∗ > 0 such that the fol lowing holds. Supp ose that the initial data ( v 0 , u 0 , θ 0 , z 0 ) satisﬁes ∥ v 0 − 1 ∥ L 1 x + ∥ v 0 ∥ B V + ∥ u 0 ∥ L 1 x + ∥ u 0 ∥ B V + ∥ θ 0 − 1 ∥ L 1 x + ∥ θ 0 ∥ B V + ∥ z 0 ∥ L 1 x + ∥ z 0 ∥ B V ≤ δ ∗ . Then the lo c al solution to Eq. (0.4) c onstructe d in The or ems 2.1 and 2.5 extends glob al ly in time. Mor e over, ther e exists a p ositive c onstant C 3 such that the glob al solution satisﬁes the fol lowing lar ge-time b ehaviour: ∥ √ t + 1( v ( · , t ) − 1) ∥ L ∞ x + ∥ √ t + 1 u ( · , t ) ∥ L ∞ x + ∥ √ t + 1( θ ( · , t ) − 1) ∥ L ∞ x + ∥ √ t + 1 z ( · , t ) ∥ L ∞ x +    √ tu x ( · , t )    L ∞ x +    √ tθ x ( · , t )    L ∞ x +    √ tz x ( · , t )    L ∞ x ≤ C 3 δ ∗ for t ∈ (0 , + ∞ ) . (4.30) Pr o of. The assertion follo ws from a standard con tin uit y argument. Let C ♯ , t ♯ and δ b e the parameters as in Theorem 2.1. In view of Theorems 2.1 and 2.5, the unique w eak solution ( v, u, θ , z ) exists on [0 , t ♯ ) . Under the smallness assumption ∥ v 0 − 1 ∥ L 1 x + ∥ v 0 ∥ B V + ∥ u 0 ∥ L 1 x + ∥ u 0 ∥ B V + ∥ θ 0 − 1 ∥ L 1 x + ∥ θ 0 ∥ B V + ∥ z 0 ∥ L 1 x + ∥ z 0 ∥ B V ≤ δ ∗ , (4.31) w e may apply Lemma 4.3 to deﬁne a stopping time T as in Eq. (4.10), suc h that T > t ♯ , G ( T ) ≥ δ, G ( τ ) < δ for all τ < T , pro vided that δ ∗ is suﬃcien tly small. By the deﬁnition of G ( τ ) and Lemma 4.3, the lifespan of the solution ( v , u, θ, z ) is larger than T . Thus, to establish the global existence of the solution, it suﬃces to show T = + ∞ for suﬃcien t small δ ∗ . Supp ose for con tradiction that T < + ∞ and G ( T ) ≥ δ, for arbitrary positive δ ∗ . (4.32) Applying Lemmas 4.5 – 4.8, we obtain the follo wing estimate at time T : G ( T ) ≤ C ( ν 0 ) δ ∗ + O (1) | log ( ν 0 ) | √ ν 0 δ 2 + O (1) √ ν 0 δ + Å C ( ν 0 ) δ ∗ + O (1) | log ( ν 0 ) | √ ν 0 δ 2 + O (1) √ ν 0 δ ã 2 , where ν 0 is a small p ositiv e constan t with ν 0 ≤ t ♯ 4 . Note that C ♯ and t ♯ remain uniformly b ounded as δ ∗ and δ b ecome small, and the O(1) coeﬃcients are indep endent of ν 0 , δ , δ ∗ . Therefore, we can ﬁrst c hoose ν 0 to b e suﬃcien tly small suc h that O (1) √ ν 0 δ ≤ δ 6 . Fix this ν 0 and then c hoose δ so small that O (1) | log ( ν 0 ) | √ ν 0 δ 2 ≤ δ 6 , 82 and then c hoose δ ∗ so small that C ( ν 0 ) δ ∗ ≤ δ 6 . (4.33) Then we hav e G ( T ) < δ, whic h contradicts the assumption (4.32). Hence, T = + ∞ . Therefore, for suﬃcien tly small δ ∗ > 0 that veriﬁes Eq. (4.31), there exists δ suc h that G ( τ ) < δ for all τ > 0 . This implies the global existence, uniqueness and the large time b ehaviour of the w eak solution. The positive constan t C 3 is directly determined by Eq. (4.33). The pro of is now complete. □ 83 Appendix A. In the appendix, we collect several length y yet direct computations omitted from the main text. W e ﬁrst tabulate the co eﬃcients in the expansion of λ j at inﬁnity as in §3.1, Eq. (3.16). See Y u–W ang–Zhang [31]. β ∗ 1 = v p v µ , A 1 , 1 = − v 3  ν θ e p 2 v + µpp e p v  ν µ 3 θ e , A 1 , 2 = v 3  µ 2 p 2 v 2 p 2 e p v + 2 ν 2 v 2 θ 2 e p 3 v + 3 ν µpv 2 θ e p e p 2 v + µ 2 pv 2 p e p 2 v − ν 2 µ 2 θ 2 e p 2 v + ν µ 3 ( − p ) θ e p e p v  ν 2 µ 5 θ 2 e , A 1 , 3 = − v 3 p v  µ 3 p 3 v 4 p 3 e + µ 2 p 2 v 2 p 2 e  3 v 2 p v (2 ν θ e + µ ) − 2 ν µ 2 θ e  + ν 3 θ 3 e p v  µ 4 + 5 v 4 p 2 v − 4 µ 2 v 2 p v  ν 3 µ 7 θ 3 e − v 3 p v  µpp e  ν 2 µ 4 θ 2 e + v 4 p 2 v  10 ν 2 θ 2 e + 4 ν µθ e + µ 2  − 2 ν µ 2 v 2 θ e p v (3 ν θ e + µ )  ν 3 µ 7 θ 3 e , α ∗ 2 = µ v , β ∗ 2 = v ( µpp e + ν θ e p v − µp v ) µ ( µ − ν θ e ) , A 2 , 1 = v 3 Ä µ 3 p 2 p 2 e − µpp e p v  ν 2 θ 2 e − 3 ν µθ e + 2 µ 2  + p 2 v ( µ − ν θ e ) 3 ä µ 3 ( µ − ν θ e ) 3 , A 2 , 2 = v 3 ( µpp e + p v ( ν θ e − µ )) Ä 2 µ 4 p 2 v 2 p 2 e + µpp e ( µ − ν θ e ) Ä µ 3 ( µ − ν θ e ) − v 2 p v ( ν θ e − 2 µ ) 2 ää µ 5 ( µ − ν θ e ) 5 + v 3 ( µpp e + p v ( ν θ e − µ )) Ä p v  2 v 2 p v − µ 2  ( µ − ν θ e ) 4 ä µ 5 ( µ − ν θ e ) 5 , A 2 , 3 = v 3  5 µ 7 p 4 v 4 p 4 e + µ 3 p 3 v 2 p 3 e ( µ − ν θ e )  4 µ 4 ( µ − ν θ e ) + v 2 p v  ν 3 θ 2 e − 6 ν 2 µθ 2 e + 15 ν µ 2 θ e − 20 µ 3  µ 7 ( µ − ν θ e ) 7 + v 3 µ 2 p 2 p 2 e ( µ − ν θ e ) 2 Ä µ 5 ( µ − ν θ e ) 2 + 3 v 4 p 2 v  − 2 ν 3 θ 3 e + 9 ν 2 µθ 2 e − 15 ν µ 2 θ e + 10 µ 3  ä µ 7 ( µ − ν θ e ) 7 − v 3 µ 2 p 2 p 2 e ( µ − ν θ e ) 2 2 µ 2 v 2 p v  − ν 3 θ 3 e + 5 ν 2 µθ 2 e − 10 ν µ 2 θ e + 6 µ 3  µ 7 ( µ − ν θ e ) 7 − v 3 µpp e p v ( µ − ν θ e ) 3 Ä µ 4 ( µ − ν θ e ) 2 (2 µ − ν θ e ) + v 4 p 2 v  − 10 ν 3 θ 3 e + 36 ν 2 µθ 2 e − 45 ν µ 2 θ e + 20 µ 3  ä µ 7 ( µ − ν θ e ) 7 − v 3 µpp e p v ( µ − ν θ e ) 3  2 µ 2 v 2 p v  3 ν 3 θ 2 e − 11 ν 2 µθ 2 e + 14 ν µ 2 θ e − 6 µ 3  µ 7 ( µ − ν θ e ) 7 + v 3 p 2 v  µ 4 + 5 v 4 p 2 v − 4 µ 2 v 2 p v  ( µ − ν θ e ) 7 µ 7 ( µ − ν θ e ) 7 . α ∗ 3 = ν θ e v , β ∗ 3 = pv p e ν θ e − µ , A 3 , 1 = pv 3 p e ( ν pθ e p e + p v ( ν θ e − µ )) ν θ e ( ν θ e − µ ) 3 , 84 A 3 , 2 = pv 3 p e  2 ν 2 p 2 v 2 θ 2 e p 2 e + pp e ( µ − ν θ e )  ν 2 θ 2 e ( µ − ν θ e ) + v 2 p v ( µ − 4 ν θ e )  ν 2 θ 2 e ( ν θ e − µ ) 5 + pv 3 p e Ä p v ( µ − ν θ e ) 2  ν θ e ( ν θ e − µ ) + v 2 p v  ä ν 2 θ 2 e ( ν θ e − µ ) 5 , A 3 , 3 = pv 3 p e  5 ν 3 p 3 v 4 θ 3 e p 2 e − p 2 v 2 p 2 e ( µ − ν θ e )  4 ν 3 θ 3 e ( ν θ e − µ ) + v 2 p v  15 ν 2 θ 2 e − 6 ν µθ e + µ 2  ν 3 θ 3 e ( ν θ e − µ ) 7 + pv 3 p e pp e ( µ − ν θ e ) 2 Ä ν 3 θ 3 e ( µ − ν θ e ) 2 − 3 v 4 p 2 v ( µ − 3 ν θ e ) + 2 ν v 2 θ e p v  4 ν 2 θ 2 e − 5 ν µθ e + µ 2  ä ν 3 θ 3 e ( ν θ e − µ ) 7 + pv 3 p e Ä − p v ( µ − ν θ e ) 3  ν θ e ( ν θ e − µ ) + v 2 p v  2 ä ν 3 θ 3 e ( ν θ e − µ ) 7 . Next, the matrices M ∗ ,k j in Lemma 3.4 are as follo ws: M ∗ , 0 1 = á 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ë , M ∗ , 1 1 = à 0 v µ 0 0 − v p v µ 0 0 0 − uv p v µ 0 0 0 0 0 0 0 í , M ∗ , 2 1 = à − v 2 p v µ 2 − uv 2 p e ν µθ e v 2 p e ν µθ e − q v 2 p e ν µθ e 0 v 2 p v µ 2 0 0 − pv 2 p v ν µθ e uv 2 p v µ 2 0 0 0 0 0 0 í , M ∗ , 3 1 = â 0 − v 3 ( pµp e +2 ν p v θ e ) ν µ 3 θ e 0 0 v 3 p v ( pµp e +2 ν p v θ e ) ν µ 3 θ e uv 3 p e p v ν µ 2 θ e − v 3 p e p v ν µ 2 θ e q v 3 p e p v ν µ 2 θ e uv 3 p v ( pµp e +2 ν p v θ e ) ν µ 3 θ e − v 3 ( pp v − u 2 p e p v ) ν µ 2 θ e − uv 3 p e p v ν µ 2 θ e q uv 3 p e p v ν µ 2 θ e 0 0 0 0 ì , M ∗ , 4 1 = ( ξ 1 , ξ 2 ) , ξ 1 = â v 4 ( pp e p v µ 2 +2 pν p e p v θ e µ +3 ν 2 p 2 v θ 2 e ) ν 2 µ 4 θ 2 e uv 4 p e ( pµp e + µp v +2 ν p v θ e ) ν 2 µ 3 θ 2 0 − 2 pµp e p v v 3 − 3 ν p 2 v θ e v 4 ν µ 4 θ e pv 4 p v ( pµp e + µp v +2 ν p v θ e ) ν 2 µ 3 θ 2 e v 4 ( pup e p v µ 2 − 2 puν p e p v θ e µ − 3 uν 2 p 2 v θ 2 e ) ν 2 µ 4 θ 2 e 0 0 ì , ξ 2 = à − v 4 p e ( pµp e + µp v +2 ν p v θ e ) ν 2 µ 3 θ 2 e q v 4 p e ( pµp e + µp v +2 ν p v θ e ) ν 2 µ 3 θ 2 e 0 0 − pv 4 p e p v ν 2 µ 2 θ 2 e q pv 4 p e p v ν 2 µ 2 θ 2 e 0 0 í M ∗ , 0 2 = á 0 0 0 0 0 1 0 0 0 u 0 0 0 0 0 0 ë , M ∗ , 1 2 = à 0 − v µ 0 0 v p v µ − uv p e µ − ν θ e v p e µ − ν θ e − q v p e µ − ν θ e uv p v µ v ( p − u 2 p e ) µ − ν θ e uv p e µ − ν θ e − q uv p e µ − ν θ e 0 0 0 0 í , 85 M ∗ , 2 2 = â v 2 p v µ 2 − uv 2 p e µ 2 − ν ν θ e v 2 p e µ 2 − ν µθ e − q v 2 p e µ 2 − ν µθ e 0 v 2 ( pµ 2 p e − p v ( µ − ν θ e ) 2 ) µ 2 ( µ − ν θ e ) 2 0 0 − pv 2 p v µ 2 − ν µθ e uv 2 ( 2 pµ 2 p e − p v ( µ − ν θ e ) 2 ) µ 2 ( µ − ν θ e ) 2 − pv 2 p e ( µ − ν θ e ) 2 q pv 2 p e ( µ − ν θ e ) 2 0 0 0 0 ì , M ∗ , 3 2 = ( ζ 1 , ζ 2 ) , ζ 1 =          0 v 3  2 p v + pµp e ( ν θ e − 2 µ ) ( µ − ν θ e ) 2  µ 3 − v 3 p v  2 p v + pµp e ( ν θ e − 2 µ ) ( µ − ν θ e ) 2  µ 3 − uv 3 p e ( 2 pµ 2 p e − p v ( 2 µ 2 − 3 ν θ e µ + ν 2 θ 2 e )) µ 2 ( µ − ν θ e ) 3 − uv 3 p v ( 2 p v ( µ − ν θ e ) 2 + pµp e ( ν θ e − 2 µ ) ) µ 3 ( µ − ν θ e ) 2 v 3 ( p − u 2 p e )( 2 pµ 2 p e − p v ( 2 µ 2 − 3 ν θ e µ + ν 2 θ 2 e )) µ 2 ( µ − ν θ e ) 3 0 0          , ζ 2 = â 0 0 v 3 p e ( 2 pµ 2 p e − p v ( 2 µ 2 − 3 ν θ e µ + ν 2 θ 2 e )) µ 2 ( µ − ν θ e ) 3 − q v 3 p e ( 2 pµ 2 p e − p v ( 2 µ 2 − 3 ν θ e µ + ν 2 θ 2 e )) µ 2 ( µ − ν θ e ) 3 uv 3 p e ( 2 pµ 2 p e − p v ( 2 µ 2 − 3 ν θ e µ + ν 2 θ 2 e )) µ 2 ( µ − ν θ e ) 3 − q uv 3 p e ( 2 pµ 2 p e − p v ( 2 µ 2 − 3 ν θ e µ + ν 2 θ 2 e )) µ 2 ( µ − ν θ e ) 3 0 0 ì , M ∗ , 4 2 = ( η 1 , η 2 , η 3 ) , η 1 = â v 4 p v  pµp e (3 µ − 2 ν θ e ) ( µ − ν θ e ) 2 − 3 p v  µ 4 0 − pv 4 p v ( pµp e (3 µ − ν θ e )+ p v ( − 3 µ 2 +5 ν θ e µ − 2 ν 2 θ 2 e )) µ 3 ( µ − ν θ e ) 3 0 ì , η 2 = à − uv 4 p e ( pµp e (3 µ − ν θ e )+ p v ( − 3 µ 2 +5 ν θ e µ − 2 ν 2 θ 2 e )) µ 3 ( µ − ν θ e ) 3 v 4 ( 3 p 2 p 2 e µ 4 +2 pp e p v ( − 3 µ 3 +6 ν e µ 2 − 4 ν 2 θ 2 e µ + ν 3 θ 3 e ) µ +3 p 2 v ( µ − ν θ e ) 4 ) µ 4 ( µ − ν θ e ) 4 uv 4 ( 6 p 2 p 2 e µ 4 + pp e p v ( − 9 µ 3 +16 ν θ e µ 2 − 9 ν 2 θ 2 e µ +2 ν 3 θ 3 e ) µ +3 p 2 v ( µ − ν θ e ) 4 ) µ 4 ( µ − ν θ e ) 4 í , η 3 = â v 4 p e ( pµp e (3 µ − ν θ e )+ p v ( − 3 µ 2 +5 ν θ e µ − 2 ν 2 θ 2 e )) µ 3 ( µ − ν θ e ) 3 − q v 4 p e ( pµp e (3 µ − ν θ e )+ p v ( − 3 µ 2 +5 ν θ e µ − 2 ν 2 θ 2 e )) µ 3 ( µ − ν θ e ) 3 0 0 pv 4 p e ( p v ( 3 µ 2 − 4 ν θ e µ + ν 2 θ 2 e ) − 3 pµ 2 p e ) µ 2 ( µ − ν θ e ) 4 − q pv 4 p e ( p v ( 3 µ 2 − 4 ν θ e µ + ν 2 θ 2 e ) − 3 pµ 2 p e ) µ 2 ( µ − ν θ e ) 4 0 0 ì The following are the matrices M 0 j and M 1 j in Lemma 3.7 and Theorem 3.4: M 0 1 = á pp e pp e − p v − up e pp e − p v p e pp e − p v 0 0 0 0 0 − pp v pp e − p v up v pp e − p v − p v pp e − p v 0 0 0 0 0 ë , M 0 2 = à − p v 2 pp e − 2 p v up e − √ pp e − p v 2 pp e − 2 p v − p e 2 pp e − 2 p v 0 p v 2 √ pp e − p v 1 2 − up e 2 √ pp e − p v p e 2 √ pp e − p v 0 ( p + u √ pp e − p v ) p v 2 pp e − 2 p v − p e √ pp e − p v u 2 − p v u + p √ pp e − p v 2 pp e − 2 p v p e ( p + u √ pp e − p v ) 2 pp e − 2 p v 0 0 0 0 0 í , 86 M 0 3 = à − p v 2 pp e − 2 p v up e + √ pp e − p v 2 pp e − 2 p v − p e 2 pp e − 2 p v 0 − p v 2 √ pp e − p v 1 2 Ä up e √ pp e − p v + 1 ä − p e 2 √ pp e − p v 0 ( p − u √ pp e − p v ) p v 2 pp e − 2 p v − − p e √ pp e − p v u 2 + p v u + p √ pp e − p v 2 pp e − 2 p v p e ( p − u √ pp e − p v ) 2 pp e − 2 p v 0 0 0 0 0 í . M 1 1 = â 0 pν p e θ e v ( p v − pp e ) 2 0 0 − pν p e p v θ e v ( p v − pp e ) 2 uν p e p v θ e v ( p v − pp e ) 2 − ν p e p v θ e v ( p v − pp e ) 2 0 − puν p e p v θ e v ( p v − pp e ) 2 − ν ( p − u 2 p e ) p v θ e v ( p v − pp e ) 2 − uν p e p v θ e v ( p v − pp e ) 2 0 0 0 0 0 ì , M 1 2 = ( ξ 1 , ξ 2 , ξ 3 ) , ξ 1 = â p v ( µp v − pp e ( µ +3 ν θ e )) 4 v ( pp e − p v ) 5 / 2 pν p e p v θ e 2 v ( p v − pp e ) 2 pp v ( p e ( pµ + ν ( p +2 u √ pp e − p v ) θ e ) − p v ( µ − 2 ν θ e )) 4 v ( pp e − p v ) 5 / 2 0 ì , ξ 2 = â p e ( − 2 pν √ pp e − p v θ e − up v ( µ − 2 ν θ e )+ pup e ( µ + ν θ e )) 4 v ( pp e − p v ) 5 / 2 − ( p e ( p √ pp e − p v µ +(2 uν p v − pν √ pp e − p v ) θ e ) − µ √ pp e − p v p v ) 4 v ( p v − pp e ) 2 − ( 2 p 2 u ( µ − ν θ e ) p 2 e + up v ( ν (5 p +2 u √ pp e − p v ) θ e − 3 pµ ) p e + p v ( uµp v − 2 pν √ pp e − p v θ e ) ) 4 v ( pp e − p v ) 5 / 2 0 ì , ξ 3 = â − p e ( pp e ( µ + ν θ e ) − p v ( µ − 2 ν θ e )) 4 v ( pp e − p v ) 5 / 2 0 ν p e p v θ e 2 v ( p v − pp e ) 2 0 p e ( p e ( µ − ν θ e ) p 2 + p v (2 ν (2 p + u √ pp e − p v ) θ e − pµ ) ) 4 v ( pp e − p v ) 5 / 2 0 0 0 ì , M 1 3 = ( ζ 1 , ζ 2 , ζ 3 ) , ζ 1 = â p v ( pp e ( µ +3 ν θ e ) − µp v ) 4 v ( pp e − p v ) 5 / 2 pν p e p v θ e 2 v ( p v − pp e ) 2 p v ( pp v ( µ − 2 ν θ e ) − pp e ( pµ + ν ( p − 2 u √ pp e − p v ) θ e )) 4 v ( pp e − p v ) 5 / 2 0 ì , ζ 2 = â − p e (2 pν √ pp e − p v θ e − up v ( µ − 2 ν θ e )+ pup e ( µ + ν θ e )) 4 v ( pp e − p v ) 5 / 2 ( p e ( pµ √ pp e − p v − ν ( √ pp e − p v p +2 up v ) θ e ) − µ √ pp e − p v p v ) 4 v ( p v − pp e ) 2 ( 2 p 2 u ( µ − ν θ e ) p 2 e + up v ( ν (5 p − 2 u √ pp e − p v ) θ e − 3 pµ ) p e + p v ( uµp v +2 pν √ pp e − p v θ e ) ) 4 v ( pp e − p v ) 5 / 2 0 ì , ζ 3 = â p e ( pp e ( µ + ν θ e ) − p v ( µ − 2 ν θ e )) 4 v ( pp e − p v ) 5 / 2 0 ν p e p v θ e 2 v ( p v − pp e ) 2 0 p e ( p e ( ν θ e − µ ) p 2 + p v ( pµ +(2 uν √ pp e − p v − 4 pν ) θ e ) ) 4 v ( pp e − p v ) 5 / 2 0 0 0 ì . 87 A c kno wledgemen t . 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Siran Li: School of Ma thema tical Sciences & CMA-Shanghai, Shanghai Jiao Tong Univer- sity, No. 6 Science Buildings, 800 Dongchuan R oad, Minhang District, Shanghai, China (200240) Email addr ess : siran.li@sjtu.edu.cn Hait ao W ang: School of Ma thema tical Sciences, Institute of Na tural Sciences, MOE-LSC, CMA-Shanghai, Shanghai Jiao Tong University, Shanghai, China (200240) Email addr ess : haitallica@sjtu.edu.cn Jianing Y ang: School of Ma thema tical Sciences, Shanghai Jiao Tong University, No. 6 Science Buildings, 800 Dongchuan Ro ad, Minhang District, Shanghai, China (200240) Email addr ess : jnyang22@sjtu.edu.cn 89

Global well-posedness for one-dimensional compressible Navier--Stokes system in dynamic combustion with small $BV\cap L^1$ initial data

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