Long-time behaviour of rouleau formation models

LONG-TIME BEHA VIOUR OF R OULEA U F ORMA TION MODELS EUGENIA FRANCO AND BERNHARD KEPKA Abstract. In this pap er w e study a tw o-component coagulation equation that models the aggregation of rouleaux in blo o d. W e consider pro duct k ernels that ha ve homogeneit y 2 and we c haracterize the initial data that lead to gelation. W e pro ve that, when gelation o ccurs, the solution to the t w o-comp onent coagulation equation localizes along a direction of the space of cluster as t approac hes the gelation time 0 < T ∗ < ∞ . The localization direction is determined b y the initial datum. W e also prov e that the solution con verges to a self-similar solution along the direction of localization. Contents 1. In tro duction 1 1.1. State of the art 1 1.2. Discrete and con tin uous coagulation equation for rouleaux 2 1.3. Main results: lo calization and asymptotic self-similarit y 6 1.4. Notation 7 2. Main results 8 3. Heuristic arguments 10 4. W ell-p osedness of coagulation models 13 5. Lo calization and self-similar asymptotics of coagulation mo dels 19 5.1. Study of momen ts 19 5.2. Self-similar v ariables and localization 29 5.3. Con v ergence to w ards self-similar solution 30 A c knowledgmen ts 41 References 41 1. Introduction 1.1. State of the art. The classical Smolucho wski’s coagulation equation ∂ t f ( t, x ) = 1 2 Z x 0 K ( x, y ) f ( t, y ) f ( t, x − y ) dy − f ( t, x ) Z ∞ 0 K ( x, y ) f ( t, y ) dy, f | t =0 = f 0 , is a mean-ﬁeld mo del that was deriv ed in order to describ e the evolution in time of the size distribution f ( t, x ) of a system of spherical particles that coalesce up on binary collision. The coagulation k ernel K ( x, y ) describes the rate at which a particle of size x > 0 merges with a particle of size y > 0 . The microscopic mec hanisms behind coagulation ev en ts are summarized by the coagulation k ernel, whic h is assumed to dep end only on the size of the coalescing particles. Smoluc ho wski’s coagulation equation (and v ariations of that) has b een used in order to mo del diﬀeren t phenomena. F or instance to mo del the coagulation of aerosols in the atmosphere [9], the grouping of animals [10], hemagglutination [17] and p olymerization pro cesses [2]. 2020 Mathematics Subje ct Classiﬁc ation. 92-10, 35R09, 82C21. K ey wor ds and phr ases. Coagulation equation, self-similar asymptotics, lo calization, rouleaux mo del. 1 2 E. FRANCO AND B. KEPKA A generalization of the classical Smolucho wski’s coagulation equation that has b een studied by several authors is the multicomponent coagulation equation (see [1, 3, 4, 5, 7, 8, 12, 16]). The solution f ( t, z ) to a n -comp onent coagulation equation represen ts the concentration of particles of type z ∈ R n + for n ≥ 1 at a certain time. The v ariable z ∈ R n + could represent diﬀeren t features. F or instance w e can assume that particles are aggregates of diﬀeren t t yp es of molecules. In this case the v ector z = ( z i ) n i =1 represen ts the c hemical comp ositions of clusters. Another p ossibility is that each particle is c haracterized by its size and its geometry . W e could then assume that eac h particle is describ ed b y the v ector z = ( z 1 , z 2 ) , where z 1 is the v olume and z 2 is a parameter that describ es the geometry of the particle. The existence of time dep endent solutions to a general class of multicomponent coagulation equations is studied in [8, 16]. The existence results prov en in [8, 16] include kernels that exhibit gelation, i.e. loss of mass conserv ation in ﬁnite time due to the formation of a particle of inﬁnite size. A relev ant feature of m ulticomp onen t coagulation equations is the so-called lo c alization phenomena. In particular, lo calization o ccurs when the solution to the multicomponent coagulation equation lo calizes asymptotically (as time tends to in ﬁnit y) along a line that dep ends on the initial condition. Lo calization results ha ve b een prov en for a class of multicomponent coagulation equations that do not exhibit gelation in [5, 7]. A lo calization result has b een pro ven in [11] for a class of kernels that giv e rise to gelation and v ery speciﬁc initial data (monodisp erse initial data). A common assumption concerning the coagulation k ernel is the homogeneity of the kenerl, i.e. K ( ax, ay ) = a γ K ( x, y ) , a, x, y ≥ 0 for some γ ∈ R . An imp ortant class of solutions to the classical coagulation equations with homogeneous k ernels are the so-called self-similar solutions of the form f ( t, x ) = s ( t ) α Φ  x s ( t )  where α ∈ R and s ( t ) is a suitable function of time. The precise form of s ( t ) and α can be obtained b y a dimensional analysis. Suc h self-similar solutions are exp ected to describ e the long-time b ehaviour of solutions to the coagulation equation. How ev er, the domain of attraction of self-similar solutions has b een c haracterized only for the solv able k ernels, i.e. for K ( x, y ) = 1 , K ( x, y ) = x + y , K ( x, y ) = xy , using Laplace transform metho ds (see [15]). In [7] the long time asymptotics of the solution to a class of multicomponent coagulation equations with a speciﬁc choice of coagulation k ernels has b een analysed. In particular it is pro ven in [7] that, for k ernels that are constant along eac h direction (in the state space) the solution lo calizes along a line as time tends to inﬁnity and it conv erges to a self-similar solution along that line. Finally , a m ulticomp onent coagulation equation describing the coagulation of a system of particles with ramiﬁcations has b een formulated and studied b y Bertoin (see [1]). In that pap er explicit solutions are obtained under sp eciﬁc assumptions on the initial condition and the o ccurrence of gelation is analysed in detail. 1.2. Discrete and con tin uous coagulation equation for rouleaux. In this pap er w e are interested in mo delling the coagulation of rouleaux in blo o d. A rouleau is a stack of red blo o d cells, also called erythro cytes. The erythro cytes hav e a sp eciﬁc shape; they are biconcav e thin discs. The aggregates that erythro cytes form app ear to ha v e v ery ramiﬁed structures due to v arious p ossible coagulation even ts. F or instance, t wo erythrocytes can adhere on their faces, forming a cylinder. Another p ossibility is that an erythro cyte adheres to the wall of a rouleau. The goal of this pap er is to formulate a m ulticomp onent coagulation equation that allo ws to study the evolution in time of the geometry and of the size of rouleaux. A model describing the coagulation of rouleaux in blo o d has b een prop osed and analysed by Perelson and Samsel in [17]. In this pap er, a system of ordinary diﬀeren tial equations describing the concentration LONG-TIME BEHA VIOUR OF R OULEA U F ORMA TION MODELS 3 + = Figure 1. Coagulation of the ﬁrst type. Here the vertices of degree one are denoted with " | ", the ones of degree t w o with an empty diamond ♢ . + = Figure 2. Coagulation of the second t yp e. Here the v ertices of degree one are denoted with " | ", the one of degree tw o with an empty diamond ♢ . The vertices with degree three with a • . of erythro cytes and the concen tration of (possibly ramiﬁed) rouleaux is formulated and studied numer- ically . In [18] the mo del studied in [17] was generalized to describ e a system of rouleaux that coalesce and fragmen t. Sp eciﬁcally the stability of steady states is analysed in [18] under the assumption that the fragmentation and coagulation rates satisfy the detailed balance assumption. The goal of this pap er is to formulate and analyse a class of multicomponent coagulation equations that describ e the dynamics of rouleaux taking in to account their shap e. More precisely , we assume that eac h particle, that is rouleau, is c haracterized by three v ariables ( c, a, s ) where a ≥ 0 , c ≥ 2 and s ≥ 1 . T o this end, we asso ciate to eac h rouleau a tree with vertices with degree d ∈ { 1 , 2 , 3 } . The v ariable s represen ts the size of the rouleau, i.e. the num b er of edges in the tree. The v ariable c is the num ber of faces of the rouleau and it is identiﬁed with the n um b er of vertices with degree 1 , i.e. the leafs of the tree. Finally , a ≥ 0 is the num b er of sites on the w all of a rouleau, to whic h another rouleau, or an erythro cyte, can adhere. Alternatively , we can identify a with the num b er of edges of degree 2 in the tree asso ciated with the rouleau. A monomer (i.e. an erythrocyte) is c haracterized by the triple (2 , 0 , 1) . Based on this, we mo del three types of coagulation ev en ts. (1) T w o rouleaux can adhere on their faces: R 1 : ( c 1 , a 1 , s 1 ) + ( c 2 , a 2 , s 2 ) 7→ ( c 1 + c 2 − 2 , a 1 + a 2 + 1 , s 1 + s 2 ) , c 1 ≥ 2 , c 2 ≥ 2 , a 1 ≥ 0 , a 2 ≥ 0 . W e assume that when t wo rouleaux adhere on a face a site for the attac hment of a rouleau is formed at the junction. If w e iden tify the rouleau with a tree, then this coagulation consists in the attac hmen t of one lea v e of one tree to a leaf of the other tree. The vertex that is formed due to the coagulation of these t wo leafs is a vertex of degree 2 (i.e. an arm). W e refer to Figure 1 for an example of coagulation of t yp e R 1 . (2) The second type of coagulation ev en t that w e model is that the face of a rouleau adheres on a free site of another rouleau: R 2 : ( c 1 , a 1 , s 1 ) + ( c 2 , a 2 , s 2 ) 7→ ( c 1 + c 2 − 1 , a 1 + a 2 − 1 , s 1 + s 2 ) , c 1 ≥ 2 , c 2 ≥ 2 , a 1 ≥ 1 , a 2 ≥ 1 . In this coagulation even t a leaf of a tree is attac hed to a vertex of degree tw o of another tree. Notice that during this t yp e of coagulation a vertex of degree three is formed due to the coalescence of a v ertex of degree one and a vertex of degree t w o. W e refer to Figure 2 for an example of coagulation of type R 2 . 4 E. FRANCO AND B. KEPKA + = Figure 3. Coagulation of the third type. Here the vertices of degree one are denoted with " | ", the one of degree tw o with an empty diamond ♢ . The vertices with degree three with a • . (3) Finally , we assume that free sites of t w o rouleaux can adhere: R 3 : ( c 1 , a 1 , s 1 ) + ( c 2 , a 2 , s 2 ) 7→ ( c 1 + c 2 , a 1 + a 2 − 2 , s 1 + s 2 + 1) , c 1 ≥ 2 , c 2 ≥ 2 , a 1 ≥ 2 , a 2 ≥ 2 . This coagulation consists in the attachmen t of a v ertex of degree t w o of a tree to a v ertex of degree t w o of another tree. During this type of coagulation we assume that t wo v ertices of degree three are formed and an edge is added to the tree. W e refer to Figure 3 for an example. As men tioned before, rouleaux are stac ks of erythro cytes that are formed by a sequence of reactions of the form R 1 , R 2 , R 3 . The triple (2 , 0 , 1) that characterize the erythro cytes satisfy the condition a + 2 c = s + 3 . (1.1) On the other hand, all the reactions in tro duced ab ov e conserv e the law ( 1.1 ). This conserv ation la w allo ws us to assume that particles are characterized only b y the v ariables c and a . W e now describ e our choice of coagulation rates, yielding the form of the coagulation kernel. W e assume that the coagulation ev ent R 1 tak es place at rate that is prop ortional to the pro duct of the n um b er of faces of the t w o rouleaux, more precisely w e assume that K 1 (( c 1 , a 1 ) , ( c 2 , a 2 )) = c 1 c 2 . Similarly , w e assume that the rate at whic h the reaction R ca tak es place is given by K 1 (( c 1 , a 1 ) , ( c 2 , a 2 )) = 1 2 ( c 1 a 2 + c 2 a 1 ) . and the rate at whic h the reaction R 3 tak es place is K 3 (( c 1 , a 1 ) , ( c 2 , a 2 )) = a 1 a 2 . W e stress that the choice of the coagulation k ernels that w e make in this pap er is in agreemen t with the c hoices made in [1, 17]. How ever, it w ould b e relev an t to deriv e the coagulation k ernels starting from a precise analysis of the mec hanisms b ehind the coagulation of rouleaux in blo o d. The multicomponent coagulation equations that w e study in this paper is the follo wing ∂ t f ( t, v ) = 1 2 3 X i =1 X v ′ ∈ B ( d ) i ( v ) α i f ( t, v − v ′ + ξ i ) f ( t, v ′ ) K i ( v − v ′ + ξ i , v ′ ) − f ( t, v ) X v ′ ∈S ( d ) f ( t, v ′ ) 3 X i =1 α i K i ( v , v ′ ) (1.2) where we are using the follo wing notations: α ∈ R 3 + is such that α  = 0 , v = ( c, a ) ⊤ ∈ N ≥ 2 × N ≥ 2 , ξ 1 = − 2 1 ! , ξ 2 = − 1 − 1 ! , ξ 3 = 0 − 2 ! , (1.3) LONG-TIME BEHA VIOUR OF R OULEA U F ORMA TION MODELS 5 and S ( d ) = N ≥ 2 × N ≥ 2 , B ( d ) i ( v ) := { v ′ ∈ S ( d ) : v − v ′ + ξ i ∈ S ( d ) } . The solution f ( t, v ) to equation ( 1.2 ) describ es the ev olution in time of a system of rouleaux that coalesce according to the coagulation mec hanisms { R i } 3 i =1 . The vector α ∈ R 3 + allo ws to give diﬀerent weigh ts to the coagulation even ts { R i } 3 i =1 . In particular, setting α j = 0 allo ws to to switch oﬀ individual coagulation mec hanisms. Notice that we assumed the state space to b e N ≥ 2 × N ≥ 2 ev en if coagulation even ts of t yp e i ∈ { 1 , 2 } can o ccur b et ween clusters ( c, a ) with a ∈ { 0 , 1 } . This is an assumption that w e make in order to simplify our analysis. Ho w ever, w e expect to hav e similar results if we assume that the state space is N ≥ 2 × N . As is standard in the theory of coagulation mo dels, w e can extend the previous mo del to a con tinuous coagulation mo del, that is, the discrete state space S ( d ) is replaced b y S = [2 , ∞ ) × [2 , ∞ ) . The contin uous version of the discrete coagulation equation ( 1.2 ) is then given b y ∂ t f = K ( f , f ) , f | t =0 = f 0 , (1.4) where for z = ( x, y ) ∈ S K ( f , f )[ z ] = 1 2 3 X i =1 α i Z B i z K i ( z ′ , z − z ′ − ξ i ) f ( t, z ′ ) f ( t, z − z ′ − ξ i ) dz ′ − f ( t, z ) 3 X i =1 α i Z S K i ( z , z ′ ) f ( t, z ′ ) dz ′ . (1.5) Here, we are using the abbreviation in ( 1.3 ) and B i z := { z ′ = ( x ′ , y ′ ) ∈ S : x ′ ≤ ξ i, 1 + x, y ′ ≤ ξ i, 2 + y } , as well as K 1 ( z , z ′ ) = z ⊤ K 1 z ′ = xx ′ , K 1 = 1 0 0 0 ! , (1.6) K 2 ( z , z ′ ) = z ⊤ K 2 z ′ = 1 2  xy ′ + x ′ y  , K 2 = 1 2 0 1 1 0 ! , (1.7) K 3 ( z , z ′ ) = z ⊤ K 3 z ′ = y y ′ , K 3 = 0 0 0 1 ! . (1.8) In this paper w e study measure-v alued solutions to ( 1.4 ). In particular, choosing measures supp orted on S ( d ) allo ws to em b ed the dynamics of the discrete mo del ( 1.2 ) into the con tinuous one. W e conclude this section discussing the conserv ation laws asso ciated with equation ( 1.4 ). Notice that if α i > 0 , for all i = 1 , 2 , 3 , then the system do es not hav e an y conserved quantit y . This is due to the fact that the coagulation even ts { R i } 3 i =1 do not hav e a common conserv ed quan tity . Note that such a conserv ed quan tity is necessarily linear in the v ariables ( c, a, s ) . It is imp ortant to mention that the coagulation kernels that w e consider hav e homogeneit y γ = 2 . Therefore, w e exp ect gelation to o ccur. F or the classical coagulation equation an indicator for this phenomenon is the violation of mass conserv ation after the so-called gelation time T ∗ ∈ [0 , ∞ ) . This is due to the formation of particles of inﬁnite size at time T ∗ and so the ﬂux of mass tow ards inﬁnit y is p ositiv e at time t = T ∗ . As w e will sho w, this is in fact the case for our rouleau formation mo dels with the exception of sp eciﬁc initial conditions. A similar t yp e of gelation phenomena, for systems that do not hav e conserved quantities, o ccurs for coagulation equation with source (see for instance [6]). 6 E. FRANCO AND B. KEPKA 1.3. Main results: localization and asymptotic self-similarity. In this pap er we study the exis- tence and the uniqueness of a time dep endent solution to equation ( 1.4 ). Notice that the coagulation k ernels ( 1.6 )-( 1.7 )-( 1.8 ) that we consider in this pap er hav e homogeneity γ = 2 and could produce gela- tion. In Prop osition 5.2 we determine conditions on α and on the initial datum that induce gelation. W e also identify a particular case in whic h gelation do es not o ccur (see Remark 2.3 ). One of the results that w e prov e in this pap er is the localization of the time dependent solution that tak es place as t approac hes the gelation time T ∗ < ∞ . W e study the lo calization phenomena only under the assumption that gelation o ccurs in ﬁnite time. In order to explain the lo calization result that we pro v e it is con v enien t to rescale the time dependent solution f ( t, z ) according to the self-similar c hange of v ariables F ( τ , η ) := ( T ∗ − t ) − 7 f ( t, z ) , t = t ( τ ) = T ∗ (1 − e − τ ) , z = ( T ∗ − t ) 2 η . (1.9) In Subsection 3 w e motiv ate the choice of the self-similar scaling using dimensional analysis arguments. W e prov e that there exists a direction ω θ = θ/ | θ | ∈ R 2 + , that dep ends only on the initial condition f 0 in ( 1.4 ) and on the parameter α , along which the function | η | 2 F ( τ , η ) lo calizes. The lo calization takes place as τ → ∞ , hence when | η | is large. More precisely w e pro ve that lim τ →∞ Z R 2 | η | 2     η | η | − ω θ     2 F ( τ , η ) dη = 0 . Moreo v er w e also prov e that under natural regularit y assumptions on the initial datum f 0 (i.e. w e assume that f 0 has ﬁnite fourth momen t), the fu nction | η | 2 F ( τ , η ) con v erges to a self-similar solution along the lo calization line ω θ . More precisely , our result can be summarized as follo ws F ( τ , η ) → 1 | η | δ ω θ  η | η |  F s ( | η | ) as τ → ∞ . The function F s is given by F s ( r ) := 1 √ 2 π K 0 r − 5 / 2 e − r/ (2 K 0 ) (1.10) where K 0 > 0 is a parameter that depends on the lo calization line ω θ and hence is determined by the initial datum and by the parameter α . The precise statements are summarized in Theorem 2.4 below. W e recall that F s is the self-similar proﬁle that characterizes the self-similar solution to the coagulation equation with product kernel and initial datum with ﬁnite momen ts (see [15]). This is exp ected as the lo calization result allows us to reduce the dynamics of the tw o-dimensional coagulation equation to a one-dimensional equation along the lo calization line. In order to prov e our result we study in detail the time evolution of the second moments associated with the solution to equation ( 1.4 ) written in self-similar v ariables, i.e. asso ciated with F . Since w e are working with a tw o-dimensional multicomponent equation the second momen t M 2 is a matrix that satisﬁes a matrix Riccati equation. Analysing the equation in detail it is p ossible to see that the second momen t tensorizes as t → T ∗ , i.e. M 2 [ F ]( t ) → θ ⊗ θ , as t → T ∗ . Here θ ∈ R 2 + dep ends on the initial datum and on α . The fact that the second moment is tensorized as t → T ∗ allo ws us to prov e our lo calization result. F or the con v ergence to war ds the self-similar solution along the lo calization line we adapt the approach used by Menon and P ego in [15] for the one-dimensional coagulation equation with pro duct kernel. In particular, we use Laplace transform methods. LONG-TIME BEHA VIOUR OF R OULEA U F ORMA TION MODELS 7 Plan of the pap er. The pap er is organized as follo ws. In Section 2 w e present the main results that w e pro v e in this pap er. In Section 3 w e presen t some heuristic arguments on the lo calization and on the c hoice of the self-similar v ariables. In Section 4 w e pro v e the existence and uniqueness of time dep endent solutions to equation ( 1.4 ). In Section 5.1 w e stu dy in detailed the ﬁrst four momen ts of the solution to equation ( 1.4 ). In Section 5.2 we infer the lo calization of the solution to equation ( 1.4 ) in self-similar v ariables. In Section 5.3 w e give a proof of the con v ergence to the self-similar proﬁle. 1.4. Notation. Here, w e summarize some notations used in this pap er. W e will write R + = [0 , ∞ ) and R ∗ = (0 , ∞ ) . F or z ∈ R 2 w e use tw o diﬀerent norms. W e write | z | = | z 1 | + | z 2 | for the 1 -norm and ∥ z ∥ = p | z 1 | 2 + | z 2 | 2 for the euclidean norm. F or a vector z ∈ R 2 w e also use the notation z ⊥ = ( − z 2 , z 1 ) ⊤ . A notational con v enience for the study of higher moments is the introduction of tensors. T o this end, let V b e a ﬁnite dimensional v ector space. W e mostly consider the case V = R 2 . F or n ∈ N denote b y V ⊗ n the vector space of n -tensors. They are deﬁned as n -linear forms V n → R . Given a basis ( b i ) d i =1 of V a basis of V ⊗ n is given by the set of pure tensors b i 1 ⊗ · · · ⊗ b i n , i 1 , . . . , i n ∈ { 1 , . . . , d } . In particular, dim V ⊗ n = d n . Giv en a basis of V ⊗ n a tensor T ∈ V ⊗ n can be identiﬁed with a family of real num b ers T i 1 ,...,i n , i 1 , . . . , i n ∈ { 1 , . . . , d } . Note that for n = 2 one can identify 2 -tensors with matrices. Via the co ordinate represen tation w e can also deﬁne a norm via ∥ T ∥ =   X i 1 ,...,i n ∈{ 1 ,...,d } T 2 i 1 ,...,i n   1 / 2 . It is con v enient to deﬁne for a v ector v ∈ V the shorthand notation v ⊗ n = v ⊗ · · · ⊗ v ( n times) . F urthermore, w e deﬁne the subspace of symmetric tensors V ⊗ n sym , that is n -linear forms symmetric in all n comp onents. On the lev el of the co ordinate representation a symmetric tensor T ∈ V ⊗ n sym satisﬁes for an y permutation σ ∈ S ( n ) T i 1 ,...,i n = T σ ( i 1 ) ,...,σ ( i n ) , ∀ i 1 , . . . , i n ∈ { 1 , . . . , d } . Note that dim V ⊗ n sym =  n + d − 1 n  . F urthermore, w e can deﬁne the pro jection P n : V ⊗ n → V ⊗ n sym via P n ( T ) i 1 ,...,i n = 1 n ! X σ ∈S ( n ) T σ ( i 1 ) ,...,σ ( i n ) , ∀ i 1 , . . . , i n ∈ { 1 , . . . , d } . In fact, this deﬁnition is indep endent of the co ordinate represen tation. Moreo v er, for S ⊂ R 2 w e denote by M ( S ) the space of ﬁnite Borel measures on S . The space of non- negativ e measures is denoted by M + ( S ) . F or p ≥ 0 we write M p, + ( S ) for measures in M + ( S ) having ﬁnite moments of order p ≥ 0 . W e equip the space M ( S ) with the weak top ology of measures, i.e. in dualit y with b ounded con tin uous functions C b ( S ) . On the other hand, for µ ∈ M ( S ) w e denote by ∥ µ ∥ the total v ariation norm and ∥ µ ∥ p = ∥ (1 + | · | p ) µ ∥ for p > 0 . F or some time interv al I ⊂ R + , op en or closed, we denote b y C ( I ; M ( S )) the space of functions con tin uous into M ( S ) on I . Similarly , also for C ( I ; M + ( S )) , C ([0 , T ]; M p, + ( S )) . In addition, w e write L ∞ ( I ; M + ( S )) for functions b ounded in to M + ( S ) . Let us men tion that for µ ∈ L ∞ ( I ; M + ( S )) this is equiv alent to sup t ∈ I ∥ µ ( t ) ∥ due to µ ( t ) ≥ 0 . Moreov er, we write L ∞ loc instead of L ∞ when b oundedness holds on compact subsets of I . In the following w e deﬁne the Laplace transform of measures. As is necessary for our study we in tro duce a vector-v alued, desingularized v arian t of it. More precisely , for ζ ∈ R 2 + and a measure µ ∈ 8 E. FRANCO AND B. KEPKA M 1 , + ( R 2 + ) we deﬁne ˆ µ : R 2 + → R 2 : ˆ µ ( ζ ) = Z R 2 + z  1 − e − z · ζ  µ ( dz ) . The desingularized v arian t allows for singularities at the origin z = 0 . Let us men tion that b y the W eierstrass appro ximation one can sho w that if ˆ µ 1 ( ζ ) = ˆ µ 2 ( ζ ) for all ζ ∈ R 2 + and µ 1 (0) = µ 2 (0) = 0 , then µ 1 = µ 2 . Hence, the desingularized Laplace transform iden tiﬁes the measure outside of the origin. W e will make also use of the desingularized Laplace transform deﬁned on measure on S . In this case, the Laplace transform identiﬁes the measure completely , since 0 ∈ S . 2. Main resul ts In this pap er w e will w ork with measure-v alued solutions to equation ( 1.4 ). T o this end, w e will mostly work with the weak form ulation of equation ( 1.4 ). The corresp onding weak form ulation of the coagulation equation ( 1.4 ) is giv en b y Z S φ ( z ) f ( t, dz ) = Z S φ ( z ) f 0 ( dz ) + 1 2 Z t 0 Z S Z S 3 X i =1 α i K i ( z , z ′ ) ∆ i φ ( z , z ′ ) f ( s, dz ) f ( s, dz ′ ) ds, t ≥ 0 , (2.1) for any φ ∈ C b ( S ) . Accordingly , we deﬁne, for all φ ∈ C b ( S ) ∆ i φ ( z , z ′ ) = φ  z + z ′ + ξ i  − φ ( z ) − φ ( z ′ ) , ( z , z ′ ) ∈ S 2 , i ∈ { 1 , 2 , 3 } . Observ e that since we assume that ( z , z ′ ) ∈ S 2 the function ∆ i φ is w ell-deﬁned. Our main results concerns the well-posedness of the rouleau mo dels as w ell as the long-time asymp- totics. As is t ypical in kinetic theory , a ﬁrst step to w ards the b ehaviour of solutions is the study of momen t equations. In the case of the rouleau mo dels ﬁrst, second and third order moments pro vide suﬃcien t information: M 1 j ( t ) = Z S z j f ( t, dz ) , M 2 j k ( t ) = Z S z j z k f ( t, dz ) , M 2 j k ℓ ( t ) = Z S z j z k z ℓ f ( t, dz ) , j, k , ℓ ∈ { 1 , 2 } . In fact, the b ehaviour is lead by the asymptotics of the second and third order moments. Due to the quadratic nature of the coagulation k ernel the matrix M 2 ( t ) of second moments satisfy a matrix Riccati diﬀeren tial equation of the form d dt M 2 = M 2 K M 2 + M 2 A + A ⊤ M 2 + B , K = 3 X i =1 α i K i , (2.2) for some matrices A = A ( t ) , B = B ( t ) ∈ R 2 × 2 . In fact, A, B dep end on M 1 ( t ) . As is well-kno wn, the equation ( 2.2 ) can lead to ﬁnite time blow-ups. This is in accordance with the occurrence of gelation for coagulation mo dels with quadratic k ernels, as the rouleau mo dels studied here. Let us therefore deﬁne the blow-up time T α, ∗ ( f 0 ) := sup ( T ≥ 0 : sup t ∈ [0 ,T ] |M 2 ( t ) | < ∞ ) ∈ [ 0 , ∞ ] . (2.3) Our ﬁrst main result concerns the w ell-p osedness of the rouleau mo dels. Theorem 2.1. L et α ∈ R 3 + , α  = 0 , p ∈ N , p ≥ 3 . Consider f 0 ∈ M p, + ( S ) and T ∗ = T α, ∗ ( f 0 ) given in ( 2.3 ) . Then, t her e exists a unique we ak solution f ∈ C 1 ([0 , T ∗ ); M + ( S )) ∩ L ∞ loc ([0 , T ∗ ); M p, + ( S ))) to ( 2.1 ) . Remark 2.2. The existence of a time dep enden t solution to equation ( 2.1 ) is prov en for t < T α, ∗ ( f 0 ) . W e do not assert that this time is in fact the maximal time of existence and we will not discuss a p ossible extension b eyond T α, ∗ . The ab ov e theorem just ensures that the solution exists as long as the second momen ts do not blo w-up. Nevertheless, as is well-kno wn in the theory of coagulation mo dels, this time LONG-TIME BEHA VIOUR OF R OULEA U F ORMA TION MODELS 9 corresp onds to the case of gelation, i.e. formation of coagulants of inﬁnity size in ﬁnite time. Let us men tion that for certain coagulation mo dels the extension b eyond the gelation time was pro ved (see [14, 16]). Our next result concerns the long-time asymptotics of solutions at the gelation time. Due to the quadratic nature of the coagulation kernel gelation o ccurs for most initial conditions. W e therefore fo cus in the situation of a ﬁnite time blo w-up, that is T ∗ ( f 0 ) < ∞ . W e then identify the asymptotic b ehaviour of the solution close to T ∗ ( f 0 ) . How ev er there are sp eciﬁc situations in which T ∗ = ∞ , as in the following remark. Remark 2.3. Assume that f 0 ( z ) = δ y =2 ( z ) µ ( dx ) . In this case blo w-up do es not o ccur for the coagulation equation ( 1.4 ) when α = (0 , 0 , α 3 ) with α 3 > 0 . Indeed, due to the form of the coagulation reaction R 3 w e will hav e that f ( t, z ) = δ y =2 ( z ) µ t ( dx ) for ev ery t ≥ 0 . Then we deduce that µ t satisﬁes the classical Smoluc ho wski’s coagulation equation with constant k ernel K = 4 , i.e. the follo wing equation ∂ t µ t ( y ) = 2 Z y 0 µ t ( t, y ′ ) µ t ( y − y ′ ) dy − 4 µ t ( y ) Z ∞ 0 µ t ( y ) dy . It is kno wn that gelation in this case do es not tak e place. In fact, as w e will see later on, see Prop osition 5.2 , the ab o ve situation is the only instance for whic h gelation do es not o ccur. T o study rigorously the asymptotic b ehaviour of the solution close to T ∗ ( f 0 ) a detailed analysis of the asymptotic b eha viour of the third momen t is necessary M 3 j k ℓ ( t ) = Z S z j z k z ℓ f ( t, dz ) , j, k , ℓ ∈ { 1 , 2 } . It is con venien t to consider the whole set of third momen ts as a 3-tensor ov er R 2 , which is symmetric in all of its comp onents, i.e. M 3 ∈ ( R 2 ) ⊗ 3 . W e refer to the notation section (Section 1.4 ) for precise deﬁnitions. The equation for the third order momen ts is in fact linear in M 3 and is lead b y the asymptotics of the second order momen ts. The study of the third momen ts allows to determine the tail for large clusters of the distribution. The main theorem that we pro ve in this pap er deals with the long-time behaviour of the solution to equation ( 1.4 ). In particular we pro ve that the solution f , written in the self-similar v ariables that we will obtain in Section 3 with a heuristic argumen t, lo calizes along the direction w θ that dep ends on the initial datum f 0 . Moreov er, w e pro v e that the solution f approaches a self-similar solution along the direction ω θ . This self-similar solution b elongs to the one-parameter family of con tinuous distributions (see [15]), of the form F s ( r ) = 1 √ 2 π K 0 r − 5 / 2 e − r/ (2 K 0 ) , (2.4) for K 0 > 0 . Theorem 2.4. L et α ∈ R 3 + b e such that α  = 0 . L et f 0 ∈ M 4 , + ( S ) and T ∗ = T ∗ ( f 0 ) b e given as in ( 2.3 ) . L et f ∈ C ([0 , T ∗ ); M + , 3 ( S )) ∩ L ∞ loc ([0 , T ∗ ); M 4 , + ( S ))) b e the solution to ( 2.1 ) as in The or em 2.1 . F urthermor e, assume that T ∗ < ∞ is ﬁnite and deﬁne F ( τ , η ) = ( T ∗ − t ( τ )) − 7 f  t ( τ ) , ( T ∗ − t ( τ )) − 2 η  , t ( τ ) = T ∗  1 − e − τ  . Then, we have the fol lowing statements. (i) (A symptotics of 2nd and 3r d moments) Ther e is θ ∈ R 2 + and a c onstant C > 0 such that the matrix M 2 ( τ ) of se c ond moments of F ( τ ) satisﬁes for al l τ ≥ 0    M 2 ( τ ) − θ ⊗ θ    ≤ C e − τ . (2.5) 10 E. FRANCO AND B. KEPKA F urthermor e, we have for the set of thir d moments M 3 ( τ ) of F ( τ )    M 3 ( τ ) − c 0 θ ⊗ θ ⊗ θ    ≤ C e − τ (2.6) wher e c 0 > 0 . Final ly, the fourth moments M 4 ( τ ) of F ( τ ) satisfy sup τ ∈ [0 , ∞ )   M 4 ( τ )   < ∞ . (ii) (L o c alization) the me asur e F lo c alizes along the line { η = λθ , λ ≥ 0 } , wher e θ is given in (i). Mor e pr e cisely, ther e is a c onstant C > 0 such that for any τ ≥ 0 Z R 2 + | η | p     η | η | − θ | θ |     2 F ( τ , dη ) ≤ C e − τ , p ∈ { 2 , 3 } . (iii) (Self-similar asymptotics) the distribution F c onver ges to a self-similar solution as τ → ∞ . Mor e pr e cisely, we have lim τ →∞ | η | 2 F ( τ , dη ) Z ( τ ) = | η | F s ( | η | ) δ  η | η | − θ | θ |  dη in the sense of me asur es. Her e, Z ( τ ) = R R 2 + | η | 2 F ( τ , dη ) and F s is given in ( 2.4 ) with the c onstant K 0 = c 0 | θ | as in (i). Remark 2.5. Let us giv e some remarks on the preceding theorem. (i) The lo calization line is deﬁned by the v ector θ . As it turns out it satisﬁes P i α i θ ⊤ K i θ = 1 , see Prop osition 5.5 . How ev er, this do es not iden tify θ and solving the second order momen t equations is necessary to do so. As a consequence θ dep ends on the second moments of the initial datum. This is in contrast with the result in [7] for non-gelling kernels where the direction of the lo calization dep ends on the initial mass distribution Z R n + z f 0 ( dz ) . The reason wh y for the equations studied in this pap er the vector θ dep ends only on the second momen t of the initial datum is that gelation o ccurs. (ii) Observe that the measure | η | F s ( | η | ) δ  η | η | − θ | θ |  dη in p oint (iii) of Theorem 2.4 is a probability measure. This is due to the c hange of v ariables η 7→ ( | η | , η / | η | ) = ( r, ω ) with Jacobian r , i.e. dη = r dr dω , and R r 2 F s ( r ) dr = 1 . (iii) Observe that the self-similar asymptotics is form ulated for the measure | η | 2 F ( τ , dη ) / Z ( τ ) . This is the correct quan tit y for the behaviour close to the gelation times, since in self-similar v ariables the second moments conv erge θ ⊗ θ , i.e. do not blo w up. On the other hand, the measures F ( τ , dη ) as well as | η | F ( τ , dη ) ha v e inﬁnite mass as τ → ∞ . This is due to the formation of a singularit y for clusters of size η = 0 . Observ e that this can also b e seen from the proﬁle F s ( r ) , which diverges for r → 0 like r − 5 / 2 . Since via the self-similar c hange of v ariables w e zo om in around the region of clusters whose mass tends to inﬁnity as t → T ∗ , the singularity in the self-similar proﬁle corresp onds to an excess mass of particles remaining of ﬁnite size at the time of gelation. 3. Heuristic arguments Before going into the details of the proofs of the results stated in Section 2 w e pro vide some heuristic argumen ts that aim at motiv ating these results. W e think that these arguments could help the reader to understand the main statemen ts prov ed in this paper, but they do not substitute the rigorous pro ofs that can b e found in the next sections. LONG-TIME BEHA VIOUR OF R OULEA U F ORMA TION MODELS 11 p 1 2 p x y p 1 2 p x y Figure 4. Left: Initial conﬁguration. the particles are lo calized in the directions p 1 = p β 1 and p 2 = p β 2 . All the clusters will remain b etw een the lines p 1 and p 2 for all times. Right: p ossible directions after three coagulation even ts. In red (solid line) after one coagulation ev en t. In green (dashed lines) after tw o coagulation even ts. In blue (dotted lines) after three coagulation ev en ts. W e start by presen ting a heuristic argument for the lo calization phenomena and later for the c hoice of the self-similar change of v ariables that w e use in this pap er. Lo calization is a phenomenon that o ccurs when inﬁnitely man y collisions take place. This could happ en in ﬁnite or inﬁnite time depending on whether or not gelation occurs. In this pap er, the latter is the case and lo calization takes place as t approac hes the gelation time T ∗ . T o illustrate the lo calization phenomenon, let us consider f 0 ( dz ) = δ (1 ,β 1 ) + δ (1 ,β 2 ) for 0 < β 1 < β 2 . Denoting p β := { z ∈ R 2 + : z 2 /z 1 = β } , this measure is supp orted on p β 1 ∪ p β 2 . W e now lo ok at the p ossible outcomes of coagulation even ts ( z , z ′ ) 7→ z + z ′ , so w e assume for the momen t ξ i = 0 . If one coagulation even t o ccurs, then the only new cluster is (2 , β 1 + β 2 ) ∈ p ( β 1 + β 2 ) / 2 . After another coagulation even t also clusters of the form (3 , 2 β 1 + β 2 ) ∈ p (2 β 1 + β 3 ) / 3 and (3 , β 1 + 2 β 2 ) ∈ p ( β 1 +2 β 3 ) / 3 can b e produced. After a third coagulation even t w e ha ve in addition also the clusters (4 , 3 β 1 + β 2 ) ∈ p 3 β 1 + β 2 4 , (4 , β 1 + 3 β 2 ) ∈ p β 1 +3 β 2 4 , (5 , 3 β 1 + 2 β 2 ) ∈ p 3 β 1 +2 β 2 5 , (5 , 2 β 1 + 3 β 2 ) ∈ p 2 β 1 +3 β 2 5 , (6 , 3 β 1 + 3 β 2 ) ∈ p β 1 + β 2 2 . See Figure 4 for an illustration of the p ossible lines to whic h a particle can b elong. W e observe that as the size (or here equiv alent z 1 ) of the cluster increases the cone of p ossible directions b ecomes narrow er. In particular, when lo oking at large clusters (as is the case in self-similar v ariables) the distribution will lo calize. The mec hanism of this phenomenon is due to the coagulation rule ( z , z ′ ) 7→ z + z ′ whic h leads to the eﬀect that the coagulant of tw o clusters b ecomes bigger and relativ e to this its direction b ecome closer. As a result particles on the boundary of the cone of all clusters alwa ys mov e closer to the cen tre of this cone by a coagulation pro cess. Thus, the bigger the particles the more their directions align. In particular, in self-similar v ariables (whic h cen tres around large clusters) the supp ort of the distribution of clusters lo calizes. Let us men tion that this heuristics also applies to the coagulation rules { R i } 3 i =1 , i.e. ( z , z ′ ) 7→ z + z ′ + ξ i , with ξ i as in ( 1.3 ). Indeed, as the size of the clusters z , z ′ increases the eﬀect of the term ξ i b ecomes smaller. F urthermore, observe that the ab ov e heuristics assumes that all coagulation processes are p ossible. If certain coagulation ev ents are not p ossible (as is the case when the coagulation k ernel v anishes), localization migh t not o ccur. W e now motiv ate the self-similar change of v ariables introduced in ( 1.9 ). T o this end we assume that lo calization occurs and we study the natural self-similar scaling that this assumption suggests. 12 E. FRANCO AND B. KEPKA Let us assume that x ∼ s ( t ) as t → T ∗ and that s ( t ) → ∞ as t → T ∗ . A c hange of v ariable that keeps the coagulation equation inv arian t m ust satisfy ∂ t [ f ] = [ K ( f , f )] (3.1) where we are denoting with [ f ] the dimension of f and with [ K ( f , f )] the dimension of K ( f , f ) . Now notice that for large times, i.e. as t → T ∗ , it holds that [ K ( f , f )] ∼ [ f ] 2 [ x ] 4 . This follo ws by the homogeneity of the k ernel and b y the fact that s ( t ) → ∞ as t → T ∗ . Using ( 3.1 ) deduce that [ f ] ∼ 1 ( t − T ∗ ) s ( t ) 4 as t → T ∗ . (3.2) The fact that gelation o ccurs means that we ha ve an escap e of mass to w ards inﬁnit y at time T ∗ . As for the classical coagulation equation, this allows us to inf er the precise form of the scaling s ( t ) . The follo wing argumen t is an adaptation of the dimensional analysis p erformed in [4] in order to ﬁnd the scaling for the self-similar solution to the coagulation equation with product k ernel. W e can deﬁne the ﬂux of mass J R [ f ] at size R in an analogous wa y as for the classical coagulation equa- tion. The ﬂux can b e obtained multiplying equation ( 1.4 ) against the test function φ ( z ) = 1 {| z | >R } ( z ) | z | and integrating. W e deduce that d dt Z {| z | >R } | z | f ( t, dz ) = J 1 R [ f ]( t ) + J 2 R [ f ]( t ) where with ζ i = − ξ i, 1 − ξ i, 2 ∈ { 1 , 2 } J 1 R [ f ]( t ) = 3 X i =1 α i Z {| z | + | z ′ |≥ R + ζ i , | z |≤ R } z ⊤ K i z ′ | z | f ( t, dz ) f ( t, dz ′ ) , J 2 R [ f ]( t ) = − 3 X i =1 ζ i α i Z {| z | + | z ′ |≥ R + ζ i , | z |≤ R } z ⊤ K i z ′ f ( t, dz ) f ( t, dz ′ ) . Assuming that gelation o ccurs at time t = T ∗ w e necessarily ha ve 0 < lim R →∞  J 1 R [ f ]( T ∗ ) + J 2 R [ f ]( T ∗ )  < ∞ . (3.3) F urthermore, since lo calization o ccurs for t → T ∗ w e ha v e f ( t, z ) ∼ F ( t, | z | ) as t → T ∗ . Hence, in order to ensure ( 3.3 ), F ( t, | z | ) has to decay slow enough, in particular p olynomially , i.e. F ( t, | z | ) ∼ | z | − κ when | z | → ∞ for some κ > 0 . W e no w identify the v alue of κ . T o this end, w e notice that J 1 R [ f ]( T ∗ ) ∼ 3 X i =1 α i Z {| z | + | z ′ |≥ R + ζ i , | z |≤ R } z ⊤ K i z ′ | z | 1 − κ | z ′ | − κ dz dz ′ ∼ Z {| z | + | z ′ |≥ R, | z |≤ R } | z | 2 − κ | z ′ | 1 − κ dz dz ′ ∼ Z R 0 dr r 3 − κ Z ∞ R − r dr ′ ( r ′ ) 2 − κ ∼ R 7 − 2 κ as R → ∞ . Similarly , we obtain J 2 R [ f ]( T ∗ ) ∼ R 6 − 2 κ as R → ∞ . As a consequence J 1 R [ f ]( t ) + J 2 R [ f ]( t ) ∼ R 7 − 2 κ as R → ∞ . But then ( 3.3 ) requires κ = − 7 / 2 . Therefore using ( 3.2 ) we deduce that [ f ] ∼ 1 ( t − T ∗ ) s ( t ) 4 ∼ 1 s ( t ) 7 / 2 This implies s ( t ) ∼ ( t − T ∗ ) − 2 and [ f ] ∼ ( t − T ∗ ) − 7 . which coincides with ( 1.9 ). Notice that this scaling suggests that the second moment of f blows up like ( T ∗ − t ) − 1 and the third moment lik e ( T ∗ − t ) − 3 as consisten t with Theorem 2.4 (i). LONG-TIME BEHA VIOUR OF R OULEA U F ORMA TION MODELS 13 Notice that applying the c hange of v ariables ( 1.9 ) in equation ( 1.4 ) yields ∂ τ F ( τ , η ) = 7 F ( τ , η ) + 2 η · ∇ F ( τ , η ) + 1 2 3 X i =1 α i Z B i z K i ( η ′ , η − η ′ − δ i ( τ )) F ( τ , η ′ ) F ( τ , η − η ′ − δ i ( τ )) dη ′ − F ( τ , η ) Z S 3 X i =1 α i K i ( η , η ′ ) F ( τ , η ′ ) dη ′ (3.4) where δ i ( τ ) → 0 as τ → ∞ . Let us now set δ i ( τ ) = 0 and assume that F is lo calized along a line. W e then exp ect F to by a self-similar rescaling of a one-dimensional distribution f which satisﬁes a one-dimensional coagulation equation with pro duct k ernel of the form ∂ t f ( t, z ) = 1 2 Z z 0 z z ′ f ( t, y ) f ( t, z − z ′ ) dz ′ − f ( t, z ) Z ∞ 0 z z ′ f ( t, z ′ ) dz ′ . As is kno w from [15] solutions to this equation hav e a self-similar b eha viour close to the gelation time. More precisely , in self-similar v ariables solutions approac h the one-parameter family of distributions F s giv en b y ( 1.10 ). This argumen t suggests the long-time asymptotics formulated in Theorem 2.4 (iii). 4. Well-posedness of coa gula tion models In this section we pro ve w ell-p osedness of the coagulation mo dels. T o this end, w e ﬁrst give a deﬁnition of weak solutions to the coagulation equations. Deﬁnition 4.1 (W eak solutions) . Let α ∈ R 3 + and f 0 ∈ M 1 , + ( S ) . W e sa y that f ∈ C ([0 , T ); M + ( S ))) ∩ L ∞ loc ([0 , T ); M 1 , + ( S ))) is a w eak solution to ( 2.1 ) if we hav e for any φ ∈ C b ( S ) and all t ∈ [0 , T ) Z S φ ( z ) f ( t, dz ) = Z S φ ( z ) f 0 ( dz ) + 1 2 3 X i =1 Z t 0 Z S Z S α i K i ( z , z ′ ) ∆ i φ ( z , z ′ ) f ( s, dz ) f ( s, dz ′ ) ds. Remark 4.2. Let us men tion that the assumption f ∈ L ∞ loc ([0 , T ); M 1 , + ( S ))) ensures that the integral on the righ t hand side is w ell-deﬁned. Prop osition 4.3. L et α ∈ R 3 + and f ∈ C 1 ([0 , T ]; M + ) ∩ L ∞ ([0 , T ]; M 3 , + ) b e a we ak solution to ( 2.1 ) for some T > 0 . Then, the fol lowing holds: (i) the function t 7→ M 1 ( f ( t )) b elongs to C 1 ([0 , T ]; R 2 + ) , it satisﬁes the fol lowing system of ODEs d dt M 1 = 3 X i =1 α i 2 ( M 1 ) ⊤ K i M 1 ξ i and   M 1 ( f ( t ))   ≤ C   M 1 ( f (0))   for some C > 0 . (ii) the function t 7→ M 2 ( f ( t )) b elongs to C 1 ([0 , T ]; R 2 × 2 + ) and satisﬁes the Ric c ati e quation d dt M 2 = 3 X i =1 α i ( M 2 + ξ i ⊗ M 1 ) K i ( M 2 + M 1 ⊗ ξ i ) − 3 X i =1 α i 2 ( ξ i ⊗ M 1 ) K i ( ξ i ⊗ M 1 ) . (4.1) Pr o of. W e pro ve this result using a cutoﬀ of the test functions z and z ⊗ z . T o this end, deﬁne for some R > 0 the smo oth function Ψ R suc h that Ψ R ( v ) = 1 for v ≤ R and Ψ R ( v ) = 0 for v > R + 1 . 14 E. FRANCO AND B. KEPKA W e no w prov e (i) . W e tak e φ R ( z ) = Ψ R ( | z | ) z as a test function in ( 2.1 ). Deﬁne also φ ( z ) = z and notice that for i ∈ { 1 , 2 , 3 } | ∆ i φ R ( z , z ′ ) − ∆ i φ ( z , z ′ ) | ≤ | (Ψ R ( | z + z ′ + ξ i | ) − 1)( z + z ′ + ξ i ) − (Ψ R ( | z | ) − 1) z −  Ψ R ( | z ′ | ) − 1  z ′ | ≤  1 | z |≥ R −| ξ i | 2 ( z , z ′ ) + 1 | z ′ |≥ R −| ξ i | 2 ( z , z ′ )  | z + z ′ + ξ i | . Due to f ∈ L ∞ loc ([0 , T ); M 2 , + ) and the pro duct structure of the k ernels, we obtain Z S Z S K i ( z , z ′ )   ∆ i φ R ( z , z ′ ) − ∆ i φ ( z , z ′ )   f ( t, dz ) f ( t, dz ′ ) → 0 as R → ∞ . As a consequence the ﬁrst moments M 1 satisfy the follo wing system of equations M 1 ( t ) = M 1 (0) + 3 X i =1 α i 2 Z t 0 ξ i  M 1 ( s )  ⊤ K i M 1 ( s ) ds. In particular, we deduce that M 1 ∈ C 1 ([0 , T ]; R 2 + ) and it satisﬁes the asserted ODE. Summing this v ectorial equation yields with ξ i, 1 + ξ i, 2 ≤ 0 for all i = 1 , 2 , 3 0 ≤ M 1 1 ( t ) + M 1 2 ( t ) ≤ M 1 1 (0) + M 1 2 (0) . W e no w prov e (ii) . W e no w choose φ ( z ) = z ⊗ z and φ R ( z ) = Ψ R ( | z | )( z ⊗ z ) . W e again ha v e | ∆ i φ R ( z , z ′ ) − ∆ i φ ( z , z ′ ) | ≤ | (Ψ R ( | z + z ′ + ξ i | ) − 1)(( z + z ′ + ξ i ) ⊗ ( z + z ′ + ξ i )) + (Ψ R ( | z | ) − 1)( z ⊗ z ) + (Ψ R ( | z ′ | ) − 1)( z ′ ⊗ z ′ ) | ≤ 1 | z + z ′ + ξ i |∈ ( R,R +1) ( z , z ′ )(( z + z ′ + ξ i ) ⊗ ( z + z ′ + ξ i )) + 1 | z |∈ ( R,R +1) ( z , z ′ )( z ⊗ z ) + 1 | z ′ |∈ ( R,R +1) ( z , z ′ )( z ′ ⊗ z ′ ) . Since f ∈ L ∞ loc ([0 , T ); M 3 , + ) we can pass to the limit R → ∞ in the w eak form ulation. Observe that ∆ i φ ( z , z ′ ) = ( z + z ′ + ξ i ) ⊗ ( z + z ′ + ξ i ) − z ⊗ z − z ′ ⊗ z ′ = ( z + ξ i ) ⊗ ( z ′ + ξ i ) + ( z ′ + ξ i ) ⊗ ( z + ξ i ) − ξ i ⊗ ξ i . This yields for i = 1 , 2 , 3 Z S Z S K i ( z , z ′ )∆ i φ ( z , z ′ ) f ( t, dz ) f ( t, dz ′ ) = Z S Z S ( z ⊤ K i z ′ )[( z + ξ i ) ⊗ ( z ′ + ξ i )] f ( t, dz ) f ( t, dz ′ ) − 1 2 Z S Z S ( z ⊤ K i z ′ )( ξ i ⊗ ξ i ) f ( t, dz ) f ( t, dz ′ ) =  M 2 ( t ) + ξ i ⊗ M 1 ( t )  K i  M 2 ( t ) + ξ i ⊗ M 1 ( t )  − 1 2  ξ i ⊗ M 1 ( t )  K i  ξ i ⊗ M 1 ( t )  . In particular, w e obtain the equation M 2 ( t ) = M 2 (0) + 3 X i =1 α i Z t 0 h M 2 ( s ) + ξ i ⊗ M 1 ( s )  K i  M 2 ( s ) + ξ i ⊗ M 1 ( s )  − 1 2  ξ i ⊗ M 1 ( s )  K i  ξ i ⊗ M 1 ( s )   ds. As a consequence M 2 ∈ C 1 ((0 , T ]; R 2 × 2 + ) and M 2 satisﬁes the ODE ( 4.1 ). □ Deﬁnition 4.4. Let α ∈ R 3 + . Let M 1 , M 2 b e the solution to the ﬁrst and second order moment equations in Prop osition 4.3 . W e deﬁne the blo w-up time T α, ∗ ( f 0 ) := sup ( T ≥ 0 : sup t ∈ [0 ,T ] |M 2 ( t ) | < ∞ ) ∈ (0 , ∞ ] . LONG-TIME BEHA VIOUR OF R OULEA U F ORMA TION MODELS 15 Theorem 4.5. Consider α ∈ R 3 + and p ∈ N , p ≥ 3 . L et f 0 ∈ M p, + ( S ) and T ∗ = T α, ∗ ( f 0 ) b e given as in Deﬁnition 4.4 . Then, ther e is a unique we ak solution f ∈ C 1 ([0 , T ∗ ); M + ) ∩ L ∞ loc ([0 , T ∗ ); M p, + ) to ( 2.1 ) . F urthermor e, the solution satisﬁes the fol lowing pr op erties: (i) for every t ∈ [0 , T ∗ ) it holds that   M 1 ( f ( t ))   ≤ C   M 1 ( f (0))   for some c onstant C > 0 . (ii) the function t 7→ M 2 ( f ( t )) b elongs to C 1 ([0 , T ]; R 2 × 2 sym ) and satisﬁes the Ric c ati e quation ( 4.1 ) . (iii) we have for al l t < T ∗ ∥ f ( t ) ∥ p ≤ C p exp  C p Z t 0  1 +    M 2 ( s )     ds  , for a time-indep endent c onstant C p > 0 . W e will prov e this result in several steps: (i) At ﬁrst w e show that weak solutions in C ([ 0 , T ); M + ) ∩ L ∞ loc ([0 , T ); M 3 , + ) are unique. The following steps then concerns the construction of the weak solution. (ii) W e prov e well-posedness of an appropriately truncated equation for whic h the coagulation op erator is b ounded. (iii) W e then derive uniform b ounds on the p -th momen ts indep endent of the truncation parameter. These b ounds are a priori only av ailable on some small, non-empt y time in terv al [0 , T ] . (iv) Based on (ii) w e extract a w eakly con v erging subsequence, yielding a weak solution on the time in terv al [ 0 , T ] by passing to the limit in the weak form ulation. F urthermore, we sho w that the so constructed solution satisﬁes (i)-(iv) in Theorem 4.5 . (v) Finally , we show that the solution can b e extended to the interv al [0 , T ∗ ) . Here, we mak e use of the fact that the p -th momen t remain b ounded as long as the second momen t does not blow-up. As a preparation for the pro of of Theorem 4.5 we provide the follo wing lemma. Lemma 4.6. L et α ∈ R 3 + , T > 0 , f 0 ∈ M 3 , + ( S ) . A ny two we ak solutions in the class C ([ 0 , T ); M + ) ∩ L ∞ loc ([0 , T ); M 3 , + ) to ( 2.1 ) with initial c ondition f 0 c oincide. Pr o of. W e mak e use of the desingularized Laplace transform. In fact, w e show that the Laplace transform ˆ f satisﬁes a quasilinear ﬁrst order PDE, whic h has a unique classical solution. T o this end, w e use in ( 2.1 ) the test function φ ( z ) = z (1 − e − z · ζ ) for ζ ∈ R 2 + . Be the regularit y assumption and a similar argumentation as in Prop osition 4.3 we obtain ˆ f ∈ C 1 t C 0 ζ ∩ C 0 t C 1 ζ . F urthermore, w e obtain for all t ∈ [0 , T ) , ζ ∈ R 2 + ∂ t ˆ f ( t, ζ ) = 3 X i =1 α i 2 Z S Z S K i ( z , z ′ ) h  z + z ′ + ξ i   1 − e − ( z + z ′ + ξ i ) · ζ  − z (1 − e − z · ζ ) − z ′ (1 − e − z ′ · ζ ) i f f ′ = 3 X i =1 α i Z S Z S K i ( z , z ′ ) z e − z · ζ  1 − e − ( z ′ + ξ i ) · ζ  f f ′ + 3 X i =1 α i 1 2 Z S Z S K i ( z , z ′ ) ξ i  1 − e − ( z + z ′ + ξ i ) · ζ  f f ′ . Using the form of the kernel K i w e then obtain ∂ t ˆ f ( t, ζ ) = D ζ ˆ f " 3 X i =1 α i K i  ˆ f + (1 − e − ξ i · ζ )( M 1 − ˆ f )  # + 3 X i =1 α i 1 2  ( M 1 ) ⊤ K i M 1 − ( M 1 − ˆ f ) ⊤ K i ( M 1 − ˆ f ) e − ξ i · ζ  ξ i . 16 E. FRANCO AND B. KEPKA This equation is of the from ∂ t ˆ f =  b ( t, ζ , ˆ f ) · ∇ ζ  ˆ f + c ( t, ζ , ˆ f ) for vector-v alued functions b, c ∈ C 1 . This equation can b e solved using the metho d of characteristics. In particular, an y classical solution provides a solution to the characteristic system: for all ζ ∈ R 2 +    Z ′ ( t, ζ ) = b ( t, Z ( t, ζ ) , F ( t, ζ )) , F ′ ( t, ζ ) = c ( t, Z ( t, ζ ) , F ( t, ζ )) , Z (0 , ζ ) = ζ , F (0 , ζ ) = ˆ f 0 ( ζ ) . Since solutions to this system are unique, we conclude that the Laplace transforms to an y tw o w eak solutions to ( 2.1 ) coincide for all t ∈ [0 , T ) , ζ ∈ R 2 + . Hence, the w eak solutions coincide as well. □ As mentioned ab o v e, in order to construct a weak solution we introduce an appropriate truncated mo del. T o this end, let i ∈ { 1 , 2 , 3 } and R > 0 . The trunc ate d kernels K ( R ) i : S × S → R + are symmetric con tin uously diﬀeren tiable b ounded functions such that        K ( R ) i ( z , z ′ ) = K i ( z , z ′ ) , ( z , z ′ ) ∈ ([0 , R ] 2 × [0 , R ] 2 ) ∩ S 2 K ( R ) i ( z , z ′ ) = 0 ( z , z ′ ) ∈ S 2 \ ([0 , 2 R ] 2 × [0 , 2 R ] 2 ) 0 ≤  K i ( z , z ′ ) − K ( R ) i ( z , z ′ )  ≤ e − R ( z , z ′ ) ∈ ( S 2 ∩ ([0 , 2 R ] 2 × [0 , 2 R ] 2 )) \ ([0 , R ] 2 × [0 , R ] 2 ) . (4.2) The weak form to the truncated equation with these k ernels is given by Z S φ ( z ) f R ( t, dz ) = Z S φ ( z ) f 0 ( dz ) + 3 X i =1 α i 2 Z t 0 Z S Z S K ( R ) i ( z , z ′ ) ∆ i φ ( z , z ′ ) f R ( s, dz ) f R ( s, dz ′ ) ds, (4.3) whic h needs to b e satisﬁed for every φ ∈ C c ( S ) . Deﬁnition 4.7 (T runcated coagulation equation) . Let α ∈ R 3 + and let R > 0 . Assume that K ( R ) i for i = 1 , 2 , 3 are truncated kernels satisfying ( 4.2 ). A function f R ∈ C 1 ([0 , ∞ ); M + ,b ( S )) is a weak solution to the truncated mo del if ( 4.3 ) is satisﬁed for all φ ∈ C c ( S ) and t ≥ 0 . Remark 4.8. Note that b y deﬁnition K ( R ) i is con tin uous and compactly supp orted. This has the adv antage that also un b ounded test functions φ ∈ C ( S ) can be used in the weak form, since ( z , z ′ ) 7→ K ( R ) i ( z , z ′ ) ∆ i φ ( z , z ′ ) is contin uous with compact support. In particular, this will b e useful in order to study the momen ts of f R and no truncation argument as in the proof Proposition 4.3 is needed. Lemma 4.9. L et α ∈ R 3 + , p ≥ 1 and R > 0 . A ssume that f 0 ∈ M p, + ( S ) . Then ther e is a unique solution f R ∈ C ([ 0 , ∞ ); M + ) ∩ L ∞ loc ([0 , ∞ ); M p, + ) to ( 4.3 ) in the sense of Deﬁnition 4.7 with initial c ondition f 0 . Pr o of. W e rewrite equation ( 4.3 ) in ﬁxed p oin t form. T o this end we in tro duce the following notation. W e deﬁne X T := ( g ∈ C ([0 , ∞ ); M 1 , + ( S )) : sup t ∈ [0 ,T ] ∥ g ( t ) ∥ p ≤ 1 + ∥ f 0 ∥ p ) . W e equip this set with the metric d T ( f , g ) := sup t ∈ [0 ,T ] sup ∥ φ ∥ Lip ≤ 1 |⟨ f ( t ) − g ( t ) , φ ⟩| , Then, the metric space ( X T , d T ) is complete. Note that this metric corresp onds to the Kantoro vic h- R ubinstein distance on M 1 , + ( S ) . LONG-TIME BEHA VIOUR OF R OULEA U F ORMA TION MODELS 17 Giv en g ∈ X T w e deﬁne the b ounded and con tinuous function µ [ g ]( t, z ) := 3 X i =1 α i Z S K ( R ) i ( z , z ′ ) g ( t, dz ′ ) , t ≥ 0 , z ∈ S i . F urthermore, w e denote b y T 0 ( t )[ g ] the measure on S deﬁned b y duality for all φ ∈ C b ( S ) , t ≥ 0 ⟨ φ, T 0 ( t )[ g ] ⟩ := Z S φ ( z ) e − R t 0 µ [ g ]( s,z ) ds f 0 ( dz ) , φ ∈ C b ( S ) , t ≥ 0 . In addition, w e write T 1 ( t )[ g ] for the measure on S deﬁned b y dualit y ⟨ φ, T 1 ( t )[ g ] ⟩ := 3 X i =1 α i 2 Z t 0 Z S ×S K ( R ) i ( z , z ′ ) φ ( z + z ′ + ξ i ) e − R t s µ [ g ]( r,z ) dr g ( s, dz ′ ) g ( s, dz ) ds. Let T := T 0 + T 1 . W e no w prov e that the ﬁxed point equation g ( t ) = T ( t )[ g ] , t ∈ [0 , T ) (4.4) has a unique solution f R ∈ X T for a small enough T > 0 . Note that this yields a weak solution to ( 4.3 ). Step 1. W e ﬁrst show that T : X T → X T is a self-mapping for small T > 0 . F rom the deﬁnition one can see that for an y g ∈ X T the function T [ g ] is in C ([0 , ∞ ); M + ( S )) . F urthermore, ha v e the estimates    T 0 [ g ]( t )    p ≤ ∥ f 0 ∥ p ,    T 1 [ g ]( t )    p ≤ T C R sup t ∈ [0 ,T ] (1 + ∥ g ( t ) ∥ ) 2 ≤ T C R  1 + ∥ f 0 ∥ p  2 . In particular, T [ g ] ∈ X T for suﬃciently small T > 0 . Step 2. W e now show that T is a con traction. T o this end, let g 1 , g 2 ∈ X i . W e observe that for every φ ∈ Lip( S ) with ∥ φ ∥ Lip ≤ 1 ∥ µ [ g 1 ]( t ) − µ [ g 2 ]( t ) ∥ ∞ ≤ C R d T ( g 1 ( t ) , g 2 ( t )) ,    D T 0 [ g 1 ]( t ) − T 0 [ g 2 ]( t ) , φ E    ≤ T C R (1 + ∥ f 0 ∥ 1 ) d T ( g 1 ( t ) , g 2 ( t )) ,    D T 0 [ g 1 ]( t ) − T 0 [ g 2 ]( t ) , φ E    ≤ T C R (1 + ∥ f 0 ∥ ) 2 d T ( g 1 ( t ) , g 2 ( t )) . Th us, T is a con traction for suﬃcien tly small T > 0 . Using Banach ﬁxed p oint theorem we deduce that there is a unique ﬁxed p oint. Step 3. Let f b e the unique weak solution on some time interv al [0 , T ] . W e now show that there is no ﬁnite time blow-up. Observe that from the w eak form ( 4.3 ) we obtain ∥ f ( t ) ∥ ≤ ∥ f 0 ∥ . Thus, Step 1 yields the estimate sup t ∈ [0 ,T ] ∥ f ( t ) ∥ p ≤ ∥ f 0 ∥ p + T C R sup t ∈ [0 ,T ] (1 + ∥ f ( t ) ∥ ) 2 ≤ ∥ f 0 ∥ p + T C R (1 + ∥ f 0 ∥ ) 2 . Consequen tly , the solution can b e extended to all of [0 , ∞ ) . □ In the follo wing we prov e uniform bounds on the solution to the truncated model. Lemma 4.10. L et α ∈ R 3 + , p ∈ N and p ≥ 3 . Consider f 0 ∈ M p, + ( S ) and f R ∈ C ([0 , ∞ ); M + ) ∩ L ∞ loc ([0 , ∞ ); M p, + ) b e the unique solution to ( 4.3 ) with initial c ondition f 0 . Then, ther e is T 0 > 0 and C > 0 dep ending only on ∥ f 0 ∥ 2 such that sup t ∈ [0 ,T 0 ]    M 2 ( f R ( t ))    ≤ C . F urthermor e, ther e is a c onstant C > 0 dep ending only on p and ∥ f 0 ∥ p such that ∥ f R ( t ) ∥ p ≤ C exp  C Z t 0  1 +    M 2 ( f R ( s ))     ds  . 18 E. FRANCO AND B. KEPKA Remark 4.11. Notice that the abov e inequality implies that the p -th moment remains b ounded as long as the second moment is ﬁnite. In particular, in the limit R → ∞ ﬁnite time blo w-up is dictated by the blo w-up of the second momen ts. Pr o of of L emma 4.10 . Let us recall that we can use un b ounded test functions due to the compact supp ort of the k ernel in the truncated mo del. Using this we ﬁrst show that ∥ f R ( t ) ∥ 1 ≤ ∥ f 0 ∥ 1 for all t ≥ 0 . Indeed, w e ha ve with ( 4.3 ) Z S (1 + | z | ) f R ( t, dz ) = Z S (1 + | z | ) f 0 ( dz ) + 3 X i =1 α i 2 Z t 0 Z S Z S K ( R ) i ( z , z ′ )( ξ i, 1 + ξ i, 2 − 1) f R ( s, dz ) f R ( s, dz ′ ) ds ≤ ∥ f 0 ∥ 1 b y the deﬁnition of ξ i in ( 1.3 ). F or the second moments we hav e with K ( R ) i ( z , z ′ ) ≤ C | z || z ′ |    M 2 ( f R ( t ))    ≤    M 2 (0)    + C X i Z t 0 K ( R ) i ( z , z ′ )( | z | + 1)( | z ′ | + 1) f R ( s, dz ) f R ( s, dz ′ ) ds ≤    M 2 (0)    + C Z t 0     M 2 ( f R ( s ))    2 + 1  ds Hence, by a Gron w all argument there is T 0 > 0 and C > 0 dep ending on ∥ f 0 ∥ 2 suc h that sup t ∈ [0 ,T 0 ]    M 2 ( f R ( t ))    ≤ C . Concerning the p -th moment we ha ve Z S | z | p f R ( t, dz ) = Z S | z | p f 0 ( dz ) + X i α i 2 Z t 0 Z S i Z S i K ( R ) i ( z , z ′ ) ∆ i [ | · | p ] f R ( s, dz ) f R ( s, dz ′ ) ds. Using p ∈ N one can show the inequalit y for i = 1 , 2 , 3 ∆ i [ | · | p ] ( z , z ′ ) ≤ C p | z | p − 1  | z ′ | + 1  + C p | z ′ | p − 1 ( | z | + 1) + C p . F or mixed terms | z ′ | k | z | j with 1 ≤ j, k ≤ p − 1 one can split into the cases | z | ≤ | z ′ | and | z ′ | ≤ | z | . This yields with the b ound K ( R ) i ( z , z ′ ) ≤ C | z || z ′ | ∥ f R ( t ) ∥ p ≤ ∥ f 0 ∥ p + C p Z t 0 h ∥ f R ( s ) ∥ p     M 2 ( f R ( s ))    + ∥ f R ( s ) ∥ 1  + ∥ f R ( s ) ∥ 2 1 i ds ≤ ∥ f 0 ∥ p + C Z t 0 h ∥ f R ( s ) ∥ p     M 2 ( f R ( s ))    + 1  + 1 i ds. Gron w all’s lemma then concludes the proof. □ W e can now giv e the pro of to Theorem 4.5 . Pr o of of The or em 4.5 . W e divide the pro of in to tw o steps. Step 1. Limit as R → ∞ . Let ( f R ) R b e the unique weak solution to the truncated mo del for R ≥ 0 . W e ﬁrst sho w that there is a con v erging subsequence R n → ∞ . T o this end, we ﬁrst observe that by Lemma 4.10 the set ( f R ( t )) R is uniformly b ounded in M p, + for all t ∈ [0 , T 0 ] . Here, T 0 is giv en in Lemma 4.10 . In addition, ( f R ) R satisﬁes the following equicon tin uit y prop erty with resp ect to time: for an y LONG-TIME BEHA VIOUR OF R OULEA U F ORMA TION MODELS 19 φ ∈ C b ( S ) and t, s ∈ [0 , T 0 ] |⟨ f R ( t ) − f R ( s ) , φ ⟩| ≤ X i α i Z t s K ( R ) i ( z , z ′ ) | ∆ i φ ( z , z ′ ) | f R ( r , dz ) f R ( r , dz ′ ) dr ≤ C ∥ φ ∥ ∞ sup t ∈ [0 ,T 0 ] ∥ f ( t ) ∥ 2 1 | t − s | ≤ C ∥ φ ∥ ∞ sup t ∈ [0 ,T 0 ] ∥ f 0 ∥ 2 1 | t − s | . All in all, this implies that there is a subsequence R 0 → ∞ suc h that f R n con v erges w eakly to f ∈ C ([0 , T 0 ]; M + ( S )) ∩ L ∞ loc ([0 , T 0 ]; M p, + ( S )) . F urthermore, b y the uniform momen t b ounds on the time in terv al [0 , T 0 ] we can pass to the limit R n → ∞ in the w eak form ( 4.3 ). Consequently , f provides a weak solution to ( 2.1 ). In addition, b y passing into the limit in the estimate in Lemma 4.10 w e obtain ∥ f ( t ) ∥ p ≤ C exp  C Z t 0  1 +    M 2 ( s )     ds  . (4.5) Step 2. Extension. Recall that by Lemma 4.6 the solution constructed in Step 1 is unique. F urther- more, due to p ≥ 3 and Prop osition 4.3 , the equations for the ﬁrst and second moments are satisﬁed on [0 , T 0 ] . In particular, this implies the following a priori b ounds: the ﬁrst moment M 1 is b ounded for all times, while the second momen t is ﬁnite on the time interv al [0 , T ∗ ) by deﬁnition of T ∗ in ( 4.4 ). F rom ( 4.5 ) w e infer that also the p -th moment is ﬁnite on the time in terv al [0 , T ∗ ) . This allo ws to extend the solution to the whole time interv al [ 0 , T ∗ ) . Statemen ts (i)-(iii) in Theorem 4.5 follo w from the preceding arguments. This concludes the pro of. □ 5. Localiza tion and self-similar asymptotics of coa gula tion models In this section w e study the long-time behaviour of solutions to the rouleau coagulation equation. A ﬁrst step is the study of the momen ts. More precisely , we study the asymptotic b ehaviour of the second and third moments close to the gelation time. F urthermore, w e give a b ound on the fourth momen t needed for the subsequent argumen ts. This asymptotics in fact allo ws to identify the appropriate self- similar scaling to justify rigorously the scaling identiﬁed in Section 3 via heuristic arguments. With resp ect to these self-similar v ariables we then sho w that the distribution function lo calizes along a line and approaches a self-similar proﬁle. 5.1. Study of moments. In this section w e study the ﬁrst, second, third and fourth order momen ts for given α ∈ R 3 + . 5.1.1. First or der moments. By Proposition 4.3 the ﬁrst moments satisfy the system d dt M 1 = 3 X i =1 α i 2 ( M 1 ) ⊤ K i M 1 ξ i . (5.1) Since ξ i, 1 + ξ i, 2 ≤ 0 for all i ∈ { 1 , 2 , 3 } w e obtain the following uniform bound. Prop osition 5.1. L et α ∈ R 3 + , f 0 ∈ M 3 , + ( S ) and f ∈ C 1 ([0 , T ∗ ); M + ) ∩ L ∞ loc ([0 , T ∗ ); M 3 , + ) b e the unique we ak solution to ( 2.1 ) on the maximal time interval [0 , T ∗ ) . The solution M 1 to ( 5.1 ) satisﬁes    M 1 ( t )    ≤ C    M 1 (0)    , for al l t ∈ [ 0 , T ∗ ) and some c onstant C > 0 . 20 E. FRANCO AND B. KEPKA 5.1.2. Se c ond or der moments. W e no w turn to the study of the second order momen t equations, which are given by (see Proposition 4.3 ) d dt M 2 = 3 X i =1 α i ( M 2 + ξ i ⊗ M 1 ) K i ( M 2 + M 1 ⊗ ξ i ) − 3 X i =1 α i 2 ( ξ i ⊗ M 1 ) K i ( ξ i ⊗ M 1 ) . (5.2) This system is a matrix-v alued Riccati equation. As is w ell-kno wn this system can pro duce blo w-ups in ﬁnite time. W e no w characterize the initial conditions f 0 that pro duce blow-ups. Prop osition 5.2. L et α ∈ R 3 + , f 0 ∈ M 3 , + ( S ) , f 0  = 0 , and f ∈ C 1 ([0 , T ∗ ); M + ) ∩ L ∞ loc ([0 , T ∗ ); M 3 , + ) b e the unique we ak solution to ( 2.1 ) on the maximal time interval [0 , T ∗ ) . Then, T ∗ < ∞ if and only if one of the fol lowing holds (i) α 1 > 0 or α 2 > 0 , (ii) α 1 = α 2 = 0 , α 3 > 0 and Z S y ( y − 2) f 0 ( dz ) > 0 . (5.3) Remark 5.3. Observ e that ( 5.3 ) fails if and only if f 0 ( dz ) = µ ( dx ) ⊗ δ 2 ( dy ) for some measure µ ∈ M + . Pr o of of Pr op osition 5.2 . W e consider three cases. Case 1. α 1 > 0 . W e assume that α 1 > 0 . Consider the test function φ ( z ) = x 2 . Let us ﬁrst observe that ∆ i φ ( z , z ) = 2 xx ′ + ( x + x ′ ) ξ i, 1 + ξ 2 i, 1 , i = 1 , 2 , 3 . Considering all three cases for ξ i, 1 , see ( 1.3 ), one can see that the minim um of ∆ i φ ( z , z ) is attained for x = x ′ = 2 and is p ositiv e. In particular, there is c 0 > 0 suc h that for all z , z ′ ∈ S ∆ i φ ( z , z ′ ) ≥ c 0 (1 + xx ′ ) . Th us w e obtain d dt M 2 11 ( t ) = X i α i 2 Z S Z S K i ( z , z ′ ) ∆ i φ ( z , z ) f ( t, dz ) f ( t, dz ′ ) ≥ c 0 α 1 2 Z S Z S K 1 ( z , z ′ ) (1 + xx ′ ) f ( t, dz ) f ( t, dz ′ ) ≥ c 0 α 1 2 M 2 11 ( t ) 2 . Since M 2 11 (0) > 0 a comparison argumen t shows that M 2 11 ( t ) blows up in ﬁnite time. Case 2. α 2 > 0 . W e no w assume α 2 > 0 . In this case we lo ok at φ ( z ) = xy . Here, w e get ∆ i φ ( z , z ′ ) = xy ′ + x ′ y + ( x + x ′ ) ξ i, 2 + ( y + y ′ ) ξ i, 1 + ξ i, 1 ξ i, 2 , i = 1 , 2 , 3 . The minimum of this is attained at ( z , z ′ ) = (2 , 2 , 2 , 2) . W e obtain ∆ 1 φ ( z , z ′ ) ≥ 2 , ∆ 2 φ ( z , z ′ ) ≥ 1 , ∆ 3 φ ( z , z ′ ) ≥ 0 . Hence, there is c 0 > 0 suc h that for all z , z ′ ∈ S ∆ 2 φ ( z , z ′ ) ≥ c 0 (1 + xy ′ + x ′ y ) . This implies d dt M 2 12 ( t ) ≥ α 2 2 Z S Z S K 2 ( z , z ′ ) ∆ 2 φ ( z , z ) f ( t, dz ) f ( t, dz ′ ) ≥ c 0 α 2 2 Z S Z S ( xy ′ + x ′ y )(1 + xy ′ + x ′ y ) f ( t, dz ) f ( t, dz ′ ) ≥ c 0 α 2 M 2 12 ( t ) 2 . LONG-TIME BEHA VIOUR OF R OULEA U F ORMA TION MODELS 21 Th us, M 2 12 ( t ) blows up in ﬁnite time. Case 3. W e now assume α 1 = α 2 = 0 but α 3 > 0 . In this case, we compute explicitly the second momen ts. Using a time-c hange w e can assume α 3 = 1 . F or th e second momen ts w e obtain, see Prop osition 4.3 (ii), d dt M 2 = ( M 2 + ξ 3 ⊗ M 1 ) K 3 ( M 2 + M 1 ⊗ ξ 3 ) − 1 2 ( ξ 3 ⊗ M 1 ) K 3 ( ξ 3 ⊗ M 3 ) . (5.4) and deﬁning N = M 2 + M 1 ⊗ ξ 3 yields with the equations for the ﬁrst moments d dt N = N ⊤ K 3 N . W riting N = a ( t ) b ( t ) c ( t ) d ( t ) ! yields the system ˙ a = c 2 , ˙ b = cd, ˙ c = cd, ˙ d = d 2 . The solution to this system is a ( t ) = a 0 + c 2 0 t 1 − d 0 t , b ( t ) = b 0 + c 0 d 0 t 1 − d 0 t , c ( t ) = c 0 1 − d 0 t , d ( t ) = d 0 1 − d 0 t . Th us, N ( t ) and hence M 2 ( t ) blows up in ﬁnite time if and only if d 0 > 0 , i.e. 0 < M 2 22 (0) − 2 M 1 2 (0) = Z S y ( y − 2) f 0 ( dz ) . This concludes the pro of. □ The following lemma will b e useful to c haracterize the b eha viour at a blow-up. Lemma 5.4. L et α ∈ R 3 + , f 0 ∈ M 3 , + ( S ) and f ∈ C 1 ([0 , T ∗ ); M + ) ∩ L ∞ loc ([0 , T ∗ ); M 3 , + ) b e the unique we ak solution to ( 2.1 ) on the maximal time interval [0 , T ∗ ) . A ssume that α 1 , α 2 > 0 . Then, one of the fol lowing holds: (i) either M 2 11 ( f ( t )) , M 2 22 ( f ( t )) → ∞ as t → T ∗ , (ii) or sup t ∈ [0 ,T ∗ )  M 2 11 ( f ( t )) , M 2 22 ( f ( t ))  < ∞ . Pr o of. W e will sho w the follo wing diﬀerential inequalities d dt M 2 11 ( t ) ≥ c  M 2 11 ( t ) 2 + M 2 12 ( t ) 2  , d dt M 2 12 ( t ) ≥ c M 2 11 ( t ) M 2 12 ( t ) , for some constant c > 0 . This implies the claim. Indeed, this system shows that whenev er M 2 11 blo ws up then also M 2 12 and vice v erse. Note that alwa ys M 2 11 , M 2 12 > 0 since f is supp orted in S . Thus, at the blow-up time T ∗ < ∞ either b oth M 2 11 , M 2 12 tend to inﬁnity or both remain b ounded. The latter o ccurs when M 2 22 blo ws up b efore M 2 11 . T o pro ve the ab o ve inequalities, we use a similar argument as in the pro of of Prop osition 5.2 . F or φ ( z ) = x 2 w e obtain ∆ i φ ( z , z ′ ) = 2 xx ′ + ( x + x ′ ) ξ i, 1 + ξ 2 i, 1 , i = 1 , 2 , 3 . ∆ 1 φ ( z , z ′ ) ≥ 1 , ∆ 2 φ ( z , z ′ ) ≥ 5 ∆ 3 φ ( z , z ′ ) = 2 xx ′ . In particular, w e get for some δ > 0 ∆ 1 φ ( z , z ′ ) ≥ δ xx ′ . 22 E. FRANCO AND B. KEPKA This yields d dt M 2 11 ≥ 1 2 δ α 1 ( M 2 11 ) 2 + α 3 ( M 2 12 ) 2 . On the other hand, for φ ( z ) = xy we ha ve ∆ i φ ( z , z ′ ) = xy ′ + x ′ y + ( x + x ′ ) ξ i, 2 + ( y + y ′ ) ξ i, 1 + ξ i, 1 ξ i, 2 , i = 1 , 2 , 3 . ∆ 1 φ ( z , z ′ ) ≥ 2 , ∆ 2 φ ( z , z ′ ) ≥ 1 , ∆ 3 φ ( z , z ′ ) ≥ 0 . In particular, w e hav e for some δ > 0 ∆ 1 φ ( z , z ′ ) ≥ δ ( xy ′ + x ′ y ) . This implies d dt M 2 12 ( t ) ≥ δ α 1 2 Z S Z S xx ′ ( xy ′ + x ′ y ) f ( t, dz ) f ( t, dz ′ ) ≥ δ α 1 M 2 11 ( t ) M 2 12 ( t ) . This concludes the pro of. □ In the follo wing we describ e the structure of the second moments when a blow-up o ccurs. Prop osition 5.5. L et α ∈ R 3 + , f 0 ∈ M 3 , + ( S ) and f ∈ C 1 ([0 , T ∗ ); M + ) ∩ L ∞ loc ([0 , T ∗ ); M 3 , + ) b e the unique we ak solution to ( 2.1 ) on the maximal time interval [0 , T ∗ ) . L et M 2 ( t ) b e the matrix of se c ond moments of f ( t ) . A ssume that T ∗ < ∞ , i.e. M 2 blows-up in ﬁnite time. Then, ther e is a non-zer o θ ∈ R 2 + and some c onstant C > 0 such that     d dt h ( T ∗ − t ) M 2 ( t ) i     ≤ C ,    ( T ∗ − t ) M 2 ( t ) − θ ⊗ θ    ≤ C ( T ∗ − t ) for al l t ∈ [ 0 , T ∗ ) . F urthermor e, θ satisﬁes P 3 i =1 α i θ ⊤ K i θ = 1 . Pr o of. W e divide the pro of in to several steps. Step 1. Let us observe that M 2 is symmetric and non-negative deﬁnite. W e now show that t 7→ M 2 ( t ) is meromorphic on a neighbourho o d of [0 , T ∗ ] in C . T o this end, recall from Proposition 4.3 the second momen ts M 2 satisfy ( 5.2 ). W e can reform ulate this in the form d dt M 2 = M 2 K M 2 + M 2 A + A ⊤ M 2 + B , (5.5) where A = A ( t ) ∈ R 2 × 2 and B ( t ) ∈ R 2 × 2 are dep ending on M 1 and K = P i α i K i . F urthermore, B is symmetric. Since M 1 solv es ( 5.1 ) it can be extended to a analytic function on a neigh b ourho o d of [0 , T ∗ ] in C , due to the b ound in Prop osition 5.1 . In particular, A, B are analytic on a neighbourho o d of [0 , T ∗ ] in C . W e now sho w that M 2 is meromorphic on a neighbourho o d of [0 , T ∗ ] in C . T o this end, we use the standard ansatz for Riccati type equations, i.e. we write M 2 ( t ) = U ( t ) V ( t ) − 1 . The matrices U, V satisfy the system d dt U V ! = A ⊤ B − K − A ! U V ! , U (0) V (0) ! = M 2 (0) I ! . Consequen tly , U, V are analytic on a complex neigh b ourho o d of [0 , T ∗ ] . Moreov er, U ( t ) V ( t ) − 1 solv es ( 5.5 ), hence b y uniqueness M 2 ( t ) = U ( t ) V ( t ) − 1 as long as V ( t ) is inv ertible. On the other hand, this form ula sho ws that M 2 is meromorphic on a complex neighbourho o d of [0 , T ∗ ] . Step 2. W e no w sho w that the blow-up of M 2 at T ∗ is a simple p ole. W e do this by splitting into tw o cases. LONG-TIME BEHA VIOUR OF R OULEA U F ORMA TION MODELS 23 (i) W e ﬁrst assume that K is in v ertible. Note that p oles appear exactly when the matrix V ( t ) is not in v ertible and hence at the blow-up time T ∗ w e hav e det V ( T ∗ ) = 0 . Let v ∈ R 2 b e non-zero and suc h that V ( T ∗ ) v = 0 . Deﬁne now ( x ( t ) , y ( t )) = ( U ( t ) v , V ( t ) v ) . W e obtain the system d dt x y ! = A ⊤ B − K − A ! x y ! , x (0) y (0) ! = M 2 (0) v v ! . First of all notice that x ( T ∗ )  = 0 , otherwise ( x ( t ) , y ( t )) v anishes at t = T ∗ . Since it solve the ab o v e homogeneous system we w ould hav e ( x ( t ) , y ( t )) ≡ 0 . This is not consistent with the initial condition ( M 2 (0) v , v )  = 0 . W e th us obtain at t = T ∗ y ′ ( T ∗ ) = − K x ( T ∗ )  = 0 . (5.6) The latter follows from the fact that K has trivial kernel. Thus, y ( t ) = O ( t − T ∗ ) v anishes linearly in T ∗ . In particular, w e see that V ( t ) b ecomes singular only linearly . Consequently , the p ole of M 2 ( t ) = U ( t ) V − 1 ( t ) is simple. (ii) W e no w lo ok at the case of non-in v ertible K . Since K is symmetric, w e can write K = λk ⊗ k for some k ∈ R 2 and λ ∈ R . Since K has only non-negative en tries w e hav e λ > 0 and k ∈ R 2 + . Assume that M 2 has a p ole of order m ≥ 2 at t = T ∗ . Then, the Lauren t series expansion has the form M 2 ( t ) = M m ( T ∗ − t ) m + R ( t ) . Here, R ( t ) has a Lauren t series with a pole of order at most m − 1 at t = T ∗ . Since M 2 is a sym- metric matrix with non-negative entries, this also holds form M m . W e then obtain by comparing co eﬃcien ts in ( 5.5 ), since m ≥ 2 , M m K M m = 0 . This requires M m to b e non-inv ertible. Thus, M m = v ⊗ v for some vector v ∈ R 2 + , since M m has only non-negative entries. Hence, w e obtain with the form of K 0 = v ⊤ K v = λ ( v · k ) 2 . Ho w ever, w e hav e v , k ∈ R 2 + and v · k = 0 . Thus, v = 0 and hence also M m = 0 . This is contradicting that M 2 has a p ole of order m ≥ 2 at t = T ∗ . Thus, the p ole is of order m = 1 . Step 3. W e no w pro ve the last assertions. Using again the Lauren t series expansion around the blo w-up time T ∗ w e obtain M 2 ( t ) = M ∗ T ∗ − t + R ( t ) . Here, R is analytic around T ∗ . Note that M − 1 is necessarily symmetric and non-negativ e deﬁnite, since M 2 is. W e use again the equation ( 5.5 ) and matc h terms in the Lauren t series yielding M ∗ = M ∗ K M ∗ . If M ∗ is not in v ertible then necessarily M ∗ = θ ⊗ θ for some θ ∈ R 2 + , since M ∗ is symmetric with non-negativ e en tries. F urthermore, w e obtain from the preceding equation θ ⊤ K θ = 1 . W e no w show that M ∗ cannot b e in v ertible. If this would b e the case then K = M − 1 ∗ . In particular, K has to be in v ertible too. Observe that then M ∗ = K − 1 = 1 α 1 α 3 − α 2 2 α 3 − α 2 − α 2 α 1 ! . Since M ∗ has non-negative entries we necessarily hav e α 2 = 0 and α 1 , α 2 > 0 . How ev er, this asymptotics then implies that M 2 11 ( t ) → ∞ , M 2 22 ( t ) → ∞ 24 E. FRANCO AND B. KEPKA as t → T ∗ , but sup t ∈ [0 ,T ∗ ) M 2 12 ( t ) < ∞ . This is a contradiction to Lemma 5.4 . Thus, M ∗ is not in vertible. The asymptotics M 2 ( t ) = θ ⊗ θ T ∗ − t + R ( t ) . then implies also that for some constan t C > 0 it holds     d dt h ( T ∗ − t ) M 2 ( t ) i     ≤ C ,    ( T ∗ − t ) M 2 ( t ) − θ ⊗ θ    ≤ C ( T ∗ − t ) for all t < T ∗ . This concludes the proof. □ 5.1.3. Thir d or der moments. In the following result we also give the asymptotics of the third moments close to the gelation time. Prop osition 5.6. L et α ∈ R 3 + , f 0 ∈ M 4 , + ( S ) and f ∈ C 1 ([0 , T ∗ ); M + ) ∩ L ∞ loc ([0 , T ∗ ); M 4 , + ) b e the unique we ak solution to ( 2.1 ) on the maximal time interval [0 , T ∗ ) . L et M 3 ( t ) b e the 3 -tensor of thir d moments of f ( t ) . A ssume that T ∗ < ∞ and let θ ∈ R 2 + as in Pr op osition 5.5 . Then, ther e is ar e c onstants C > 0 and c 0 > 0 such that for al l t ∈ [0 , T ∗ )    ( T ∗ − t ) 3 M 3 ( t ) − c 0 θ ⊗ θ ⊗ θ    ≤ C ( T ∗ − t ) . Pr o of. W e split the pro of in to tw o steps. Step 1. ODE system. W e ﬁrst derive the ODE satisﬁed by the third moments. T o this end, we can use as a test function φ R ( z ) = ψ R ( | z | )( z ⊗ z ⊗ z ) and let R → ∞ . Here, w e make use of the ﬁnite fourth momen ts, similarly as in the pro of of Prop osition 4.3 . This allo ws to sho w that t 7→ M 3 ( f ( t )) b elongs to C 1 ([0 , T ]; ( R 2 ) ⊗ 3 ) . In order to identify the equation we mak e use of the tensor formulation. W e observ e that for φ ( z ) = z ⊗ z ⊗ z w e ha ve ∆ i φ ( z , z ′ ) = 3 P 3 ( z ′ ⊗ z ⊗ z ) + 3 P 3 ( z ⊗ z ′ ⊗ z ′ ) + 3 P 3 ( z ⊗ z ⊗ ξ i ) + 3 P 3 ( z ′ ⊗ z ′ ⊗ ξ i ) + 6 P 3 ( z ⊗ z ′ ⊗ ξ i ) + 3 P 3 ( z ⊗ ξ i ⊗ ξ i ) + 3 P 3 ( z ′ ⊗ ξ i ⊗ ξ i ) + ξ i ⊗ ξ ⊗ ξ i Making use of the quadratic form of the k ernel we obtain an equation of the form d dt M 3 = B ( M 3 , M 2 ) + R 1 ( M 3 , M 1 ) + R 2 ( M 2 , M 2 ) + R 3 ( M 2 , M 1 ) + R 4 ( M 1 , M 1 ) (5.7) Here, B as well as R j for j = 1 , 2 , 3 , 4 are bilinear operators with v alues in ( R 2 ) ⊗ 3 sym . In the following only the precise form of the ﬁrst op erator is relev ant. It has the form B ( M 3 , M 2 ) kℓm = 2 X a,b =1 K ab  M 2 ka M 3 bℓm + M 2 ℓa M 3 bmk + M 2 ma M 3 bℓk  , k , ℓ, m ∈ { 1 , 2 } , where K = P 3 i =1 α i K i . W e can rewrite this as B ( M 3 , M 2 ) = 3 P 3 A ( M 3 , M 2 ) , where A : ( R 2 ) ⊗ 3 × ( R 2 ) ⊗ 2 → ( R 2 ) ⊗ 3 is given by A ( J, T ) kℓm = 2 X a,b =1 K ab J ka T bℓm . The most relev ant contribution on the righ t hand side in ( 5.7 ) is the ﬁrst term when M 2 is replaced b y ( T ∗ − t ) − 1 θ ⊗ θ . Recall that N 2 ( t ) := M 2 ( t ) − ( T ∗ − t ) − 1 θ ⊗ θ is uniformly b ounded on [0 , T ∗ ) b y Prop osition 5.5 . W e then deﬁne LM 3 = B ( M 3 , θ ⊗ θ ) . LONG-TIME BEHA VIOUR OF R OULEA U F ORMA TION MODELS 25 F or the remaining terms in 5.5 we obtain the b ounds    B  M 3 , M 2 − ( T ∗ − t ) − 1 θ ⊗ θ     ≤ C    M 3    ,    R 1 ( M 3 , M 1 )    ≤ C    M 3    ,    R 2 ( M 2 , M 2 )    ≤ C ( T ∗ − t ) 2 ,    R 3 ( M 2 , M 1 )    ≤ C T ∗ − t ,    R 4 ( M 1 , M 1 )    ≤ C for all t ∈ [0 , T ∗ ) . W e no w compute a co ordinate representation of the op erator L . T o this end, w e use the following basis for ( R 2 ) ⊗ 3 B 1 := θ ⊗ 3 , B 2 := P 3  θ ⊗ θ ⊗ θ ⊥  , B 3 := P 3  θ ⊗ θ ⊥ ⊗ θ ⊥  , B 4 := ( θ ⊥ ) ⊗ 3 . Let us also deﬁne β := θ ⊤ K θ ⊥ . Recall that θ ⊤ K θ = 1 b y Prop osition 5.5 . With this one can see that L B 1 = 3 B 1 , L B 2 = β B 1 + 2 B 2 , L B 3 = 2 β B 2 + B 3 , L B 4 = 3 β B 3 . Th us, with resp ect to this basis w e can identify L with the matrix ˆ L :=      3 β 0 0 0 2 2 β 0 0 0 1 3 β 0 0 0 0      Note that this matrix has the eigen v alues 0 , 1 , 2 , 3 . Let us denote by V ( t ) ∈ R 4 the co ordinate representation of M 3 ( t ) with resp ect to the ab o ve basis. The ab o ve argumen ts yields the following equation d dt V = 1 T ∗ − t ˆ L V + ˆ B V + ˆ R, (5.8) where ˆ B ( t ) , ˆ R ( t ) ∈ R 4 × 4 satisfy    ˆ B ( t )    ≤ C ,    ˆ R ( t )    ≤ C ( T ∗ − t ) 2 . Using no w the time-c hange t ( τ ) = T ∗ (1 − e − τ ) and writing V ( t ( τ )) e − 3 τ in the basis formed b y the eigen v ectors of ˆ L yields the equation d dτ W = ¯ L W + ¯ B ( τ ) W + ¯ R ( τ ) . (5.9) Here, W ( τ ) are the co ordinates of V ( t ( τ )) e − 3 τ in this basis, ¯ L = diag { 0 , − 1 , − 2 , − 3 } and the matrices ¯ B , ¯ R satisfy the bounds    ¯ B ( τ )    ≤ C e − τ ,    ¯ R ( τ )    ≤ C e − 2 τ . Step 2. Study of A symptotics. W e no w sho w that M 3 ( t ) blo ws up like c 0 ( T ∗ − t ) − 3 B 1 at t = T ∗ . Note that B 1 corresp onds to the eigen vector to the eigenv alue 0 of ¯ L . Here, c 0 > 0 is some constan t. T o this end, we ﬁrst sho w that M 3 ( t ) necessarily blo ws up at t = T ∗ . This follows from the follo wing inequalit y relating momen ts of diﬀerent order    M 2 ( t )    ≤ C    M 1 ( t )    1 / 2    M 3 ( t )    1 / 2 . Since   M 1 ( t )   > 0 for t ∈ [0 , T ∗ ] we conclude from the asymptotics of the second momen ts that    M 3 ( t )    ≥ c ( T ∗ − t ) 2 . W riting this in terms of W ( τ ) yields ∥ W ( τ ) ∥ ≥ ce − τ for some constan t c > 0 . 26 E. FRANCO AND B. KEPKA W e no w study the solution W to the ODE system ( 5.9 ). Due to the bounds on ¯ B , ¯ R w e obtain sup τ ∈ [0 , ∞ ) ∥ W ( τ ) ∥ < ∞ . Using Duhamel’s form ula we obtain W ( τ ) = W ∞ + ¯ W ( τ ) ,    ¯ W ( τ )    ≤ C e − τ . Using this expression in ( 5.9 ) implies that ¯ L W ∞ = 0 , i.e. W ∞ = c 0 (1 , 0 , 0 , 0) ⊤ . If c 0  = 0 , w e conclude that    ( T ∗ − t ) 3 M 3 ( t ) − c 0 B 1    ≤ C ( T ∗ − t ) . This in particular sho ws that c 0 > 0 since M 3 has only non-negative comp onents M 3 kℓm ≥ 0 for all k , ℓ, m ∈ { 1 , 2 } . W e now assume for contradiction that c 0 = 0 . In particular, we obtain W ( τ ) = ¯ W ( τ ) . W e no w consider only the ﬁrst comp onen t in ( 5.9 ) yielding up on in tegration W 1 ( τ ) = W 1 (0) + Z τ 0  ( ¯ B ( σ ) W ( σ )) 1 + ¯ R 1 ( σ )  dσ = − Z ∞ τ  ( ¯ B ( σ ) ¯ W ( σ )) 1 + ¯ R 1 ( σ )  dσ. Here, we used that W ∞ 1 = 0 . Due to the estimate a v ailable for ¯ W we conclude that | W 1 ( τ ) | ≤ C e − 2 τ . W riting U ( τ ) = e τ ( W 2 ( τ ) , W 3 ( τ ) , W 4 ( τ )) yields then the equation d dτ U = ˜ L U + ˜ B ( τ ) U + ˜ R ( τ ) . Here, ˜ L = diag { 0 , − 1 , − 2 } , ˜ B is restricted to the last three comp onents and    ˜ R ( τ )    ≤ C e − τ . Note that the terms con taining W 1 are absorb ed in ˜ R . As before this yields U ( τ ) = U ∞ + ¯ U ( τ ) ,    ¯ U ( τ )    ≤ C e − τ with U ∞ = c 1 (1 , 0 , 0) . In particular, we obtain W ( τ ) = c 1 e − τ (0 , 1 , 0 , 0) ⊤ + ˜ W ( τ ) ,    ˜ W ( τ )    ≤ C e − 2 τ . Since we know a priori that ∥ W ( τ ) ∥ ≥ ce − τ , we concluded that c 1  = 0 . F orm ulating this in terms of the second moments yields ( T ∗ − t ) 2 M 3 ( t ) → c 1 Ξ , Ξ := β B 1 − 3 B 2 . (5.10) Note that Ξ is the eigenv ector with eigen v alue 2 to ˆ L whic h corresp onds to the eigen vector with eigenv alue − 1 to ¯ L . W e now show that ( 5.10 ) cannot hold, since M 3 satisﬁes some p ositive deﬁniteness prop erties. Indeed, the matrix ( k , ℓ ) 7→ X a,b =1 M 3 ( t ) kℓa K ab θ b = Z S i z k z ℓ  z ⊤ K θ  f ( t, dz ) is non-negative deﬁnite for all t ∈ [0 , T ∗ ) , since z ⊤ K θ ≥ 0 . Recall that θ ∈ R 2 + and K has only non-negative en tries. But then ( 5.10 ) w ould imply the same for ( k , ℓ ) 7→ c X a,b =1 Ξ kℓa K ab θ b = c  β ( θ ⊗ θ ) kℓ − 1 2  β ( θ ⊗ θ ) kℓ + ( θ ⊗ θ ⊥ ) kℓ + ( θ ⊥ ⊗ θ ) kℓ   = c 2 h β ( θ ⊗ θ ) kℓ − ( θ ⊗ θ ⊥ ) kℓ − ( θ ⊥ ⊗ θ ) kℓ i . LONG-TIME BEHA VIOUR OF R OULEA U F ORMA TION MODELS 27 This matrix has the follo wing form with resp ect to the basis { θ , θ ⊥ } c 2 β − 1 − 1 0 ! . This matrix has determinant − 1 and hence is not p ositiv e deﬁnite for any choice c  = 0 . This yields a con tradiction. W e hence infer our assumption c 0 = 0 cannot hold so that the conclusion of c 0  = 0 holds. This concludes the pro of. □ 5.1.4. F ourth or der moments. In the follo wing we study the fourth order momen ts. In this case only an upp er b ound is prov en rather than an asymptotics (as was done for the second and third order moments). Prop osition 5.7. L et α ∈ R 3 + , f 0 ∈ M 4 , + ( S ) and f ∈ C 1 ([0 , T ∗ ); M + ) ∩ L ∞ loc ([0 , T ∗ ); M 4 , + ) b e the unique we ak solution to ( 2.1 ) on the maximal time interval [0 , T ∗ ) . L et M 4 ( t ) b e the 4 -tensor of fourth moments of f ( t ) . A ssume that T ∗ < ∞ . Then, ther e is a c onstant C > 0 such that for al l t ∈ [0 , T ∗ )    ( T ∗ − t ) 5 M 4 ( t )    ≤ C . Pr o of. W e split the proof into t w o steps. In the ﬁrst tw o steps we consider a solution to with initial condition f 0 ∈ M 5 , + ( S ) . By Theorem 4.5 we kno w that there is a unique solution f ∈ C ([ 0 , T ∗ ); M + ) ∩ L ∞ loc ([0 , T ∗ ); M 5 , + ) . In Step 3 w e sho w how to reduce the general case to this situation. Step 1. W e ﬁrst deriv e the ODEs satisﬁed b y the 4 -tensor of fourth momen ts. Since the solution has ﬁnite 5 -th moments, we can use the test functions φ R ( z ) = ψ R ( | z | ) z ⊗ 4 and let R → ∞ . W e observe that for the test function φ ( z ) = z ⊗ 4 w e ha ve ∆ i φ ( z , z ′ ) = 4 P 4  z ⊗ 3 ⊗ z ′  + 4 P 4  z ⊗ ( z ′ ) ⊗ 3  + 6 P 4  z ⊗ z ⊗ z ′ ⊗ z ′  + 4 P 4  z ⊗ 3 ⊗ ξ i  + 4 P 4  ( z ′ ) ⊗ 3 ⊗ ξ i  + 12 P 4  z ⊗ 2 ⊗ z ′ ⊗ ξ i  + 12 P 4  ( z ′ ) ⊗ 2 ⊗ z ⊗ ξ i  + 6 P 4  z ⊗ 2 ⊗ ξ ⊗ 2 i  + 6  ( z ′ ) ⊗ 2 ⊗ ξ ⊗ 2 i  + 12 P 4  z ⊗ z ′ ⊗ ξ ⊗ 2 i  + 4 P 4  z ⊗ ξ ⊗ 3 i  + 4 P 4  z ′ ⊗ ξ ⊗ 3 i  + ξ ⊗ 4 i . With the quadratic form of the k ernel this yields an ODE system for M 4 of the form d dt M 4 = B 1 ( M 4 , M 2 ) + B 2 ( M 3 , M 3 ) + R 1 ( M 4 , M 1 ) + R 2 ( M 3 , M 2 ) + R 3 ( M 3 , M 1 ) + R 4 ( M 2 , M 2 ) + R 5 ( M 2 , M 1 ) + R 6 ( M 1 , M 1 ) . (5.11) Here, the op erators B 1 , B 2 and R j , j = 1 , . . . , 6 , are bilinear with v alues in ( R 2 ) ⊗ 4 sym . One can see that B 1 ( M 3 , M 2 ) = 4 P 4 A 2 ( M 4 , M 2 ) , where A 2 : ( R 2 ) ⊗ 4 × ( R 2 ) ⊗ 2 → ( R 2 ) ⊗ 4 is given by A 2 ( J, T ) kℓmn = 2 X a,b =1 K ab J ka T bℓmn , K = 3 X i =1 α i K i . Step 2. W e no w study the ODE ( 5.11 ). In the ﬁrst term in ( 5.11 ) w e use similarly as in the pro of or Prop osition 5.6 the asymptotics of M 2 according to Prop osition 5.5 . W e henceforth deﬁne LM 4 = 3 P 4 A 2 ( M 4 , θ ⊗ θ ) . W e then obtain an equation of the form d dt M 4 = 1 T ∗ − t LM 3 + B ( t ) M 4 + R ( t ) , (5.12) 28 E. FRANCO AND B. KEPKA where B ( t ) is a linear op erator ( R 2 ) ⊗ 4 sym → ( R 2 ) ⊗ 4 sym and R ( t ) ∈ ( R 2 ) ⊗ 4 sym . Using Prop osition 5.5 and 5.6 w e obtain the b ounds    B M 4    ≤ C    M 4    , ∥R ( t ) ∥ ≤ C ( T ∗ − t ) 6 . In order to analyse L we use the following basis on ( R 2 ) ⊗ 4 sym B 1 := θ ⊗ 4 , B 2 := P 4  θ ⊗ 3 ⊗ θ ⊥  , B 3 := P 4  θ ⊗ 2 ⊗ ( θ ⊥ ) ⊗ 2  , B 4 := P 4  θ ⊗ ( θ ⊥ ) ⊗ 3  , B 5 := ( θ ⊥ ) ⊗ 4 . Recall that θ ⊤ K θ = 1 and let us write again β = θ ⊤ K θ ⊥ . The operator L can then b e written in matrix form as ˆ L =         4 β 0 0 0 0 3 2 β 0 0 0 0 2 3 β 0 0 0 0 1 4 β 0 0 0 0 0         . In particular, we obtain    e ˆ L t    ≤ C e 4 t for all t ≥ 0 . Let us write V ( t ) for the co ordinate representation of M 4 ( t ) in the ab o v e basis. W e then obtain from ( 5.12 ) the equation d dt V = 1 T ∗ − t ˆ L V + ˆ B ( t ) V + ˆ R ( t ) . W e no w introduce the time-change t ( τ ) = T ∗ (1 − e − τ ) yielding for W ( τ ) = V ( t ( τ )) d dτ W = ˆ L W + T ∗ e − τ ˆ B ( t ( τ )) W + T ∗ e − τ ˆ R ( t ( τ )) . Observ e that    ˆ B ( t ( τ )) W    ≤ C ∥ W ∥ ,    ˆ R ( t ( τ ))    ≤ C e 6 τ . By Gronw all’s lemma w e obtain ∥ W ( τ ) ∥ ≤ C e 5 τ = ⇒ ∥ V ( t ) ∥ ≤ C ( T ∗ − t ) 5 . This yields the asserted b ound. Let us mention that C > 0 only dep ends on M j ( f 0 ) for j = 1 , 2 , 3 , 4 . Step 3. W e now prov e the b ound when f 0 ∈ M 4 , + ( S ) and hence the solution satisﬁes merely f ∈ L ∞ loc ([0 , T ∗ ); M 4 , + ) . W e no w choose a sequence ( f n 0 ) n in M 5 , + ( S ) such that for all n ∈ N M j ( f n 0 ) = M j ( f 0 ) , j ∈ { 1 , 2 , 3 , 4 } , (5.13) and f n 0 → f 0 w eakly as n → ∞ . A w a y to construct these measures is to take the restriction f 0 1 [0 ,n ] 2 and add a ﬁnite n umber of Diracs at sp eciﬁc points with sp eciﬁc w eigh ts to ensure ( 5.13 ). This yields a set of equations for the w eights whic h can b e solved. In fact, this is related to the Caratheo dory theorem in conv ex geometry (see Theorem B.12 in [13]). F rom Step 2 w e then obtain for the solutions f n to f n 0    M 4 ( f n ( t ))    ≤ C ( T ∗ − t ) 5 . Note that T ∗ as w ell as C is indep enden t of n ∈ N by ( 5.13 ). By a compactness argument, similar to the one in the pro of of Theorem 4.5 , we obtain f n ( t ) → f ( t ) w eakly for all t ∈ [0 , T ∗ ) . Thus, w e obtain the asserted b ound also for M 4 ( f ( t )) . This concludes the proof. □ LONG-TIME BEHA VIOUR OF R OULEA U F ORMA TION MODELS 29 5.2. Self-similar v ariables and lo calization. In this subsection we deﬁne the self-similar change of v ariables, whic h was already introduced in the in tro duction. W e then reform ulate the results concerning the momen ts prov ed in the preceding subsection. F urthermore, w e then sho w that the distribution lo calizes along a line. These results corresp ond to statements (i) and (ii) in Theorem 2.4 . Giv en the solution f as in Theorem 4.5 with initial condition f 0 ∈ M 4 , + w e introduce the self-similar c hange of v ariables via F ( τ , η ) = ( T ∗ − t ( τ )) − 7 f  t ( τ ) , ( T ∗ − t ( τ )) − 2 η  , t ( τ ) = T ∗  1 − e − τ  . (5.14) Recall that T ∗ is the blo w-up time of the second momen ts, see Deﬁnition 4.4 . W e assume here that T ∗ < ∞ . Let us note that ( 5.14 ) has to b e deﬁned by duality , since f is merely a measure. Deﬁnition 5.8. Consider α ∈ R 3 + , f 0 ∈ M 4 , + ( S ) and assume T α, ∗ < ∞ . Let f be the solution to ( 2.1 ) with initial condition f 0 according to Theorem 4.5 . W e deﬁne the measure F ∈ C 1 ([0 , ∞ ); M + ( R 2 + )) ∩ L ∞ loc ([0 , ∞ ); M 4 , + ( R 2 + )) by duality via Z R 2 + φ ( η ) F ( τ , dη ) = ( T ∗ − t ( τ )) − 3 Z S φ ( z ( T ∗ − t ( τ )) 2 ) f ( t ( τ ) , dz ) t ( τ ) = T ∗  1 − e − τ  for every φ ∈ C b ( R 2 + ) . Remark 5.9. Note that the supp ort of the measure on the right hand side in ( 5.14 ) is given b y e − 2 τ S . In Deﬁnition 5.8 we implicitly extend the measure F ( τ ) b y zero outside of e − 2 τ S to all of R 2 + . W e no w derive the equation satisﬁed b y F as in Deﬁnition 5.8 . Lemma 5.10. Consider α ∈ R 3 + , f 0 ∈ M 4 , + ( S ) and assume T ∗ = T α, ∗ < ∞ . L et F b e deﬁne d as in Deﬁnition 5.8 . Then, F satisﬁes for al l φ ∈ C b ( R 2 + ) and τ ≥ 0 d dτ Z R 2 + φ ( η ) F ( τ , dη ) = 3 Z R 2 + φ ( η ) F ( τ , dη ) − 2 Z R 2 + ( η · ∇ φ ( η )) F ( τ , dz ) + 3 X i =1 α i 2 Z R 2 + Z R 2 + K i ( η , η ′ ) ∆ ∗ i φ ( τ ; η , η ′ ) F ( τ , dη ) F ( τ , dη ′ ) . (5.15) wher e ∆ ∗ i φ ( τ ; η , η ′ ) = φ z + z ′ + ξ i e − 2 τ T 2 ∗ ! − φ ( z ′ ) − φ ( z ) , i ∈ { 1 , 2 , 3 } . (5.16) Pr o of. The assertion follows from ( 2.1 ) and the deﬁnition of F . □ The study of the moments of the solution f no w yields the following corollary for the momen ts of F . Corollary 5.11. Consider α ∈ R 3 + , f 0 ∈ M 4 , + ( S ) and assume T ∗ = T α, ∗ < ∞ . L et F b e deﬁne d as in Deﬁnition 5.8 . Then, we have for some c onstant C > 0 and al l τ ≥ 0    M 1 ( F ( τ ))    ≤ C e τ ,     d dτ M 2 ( F ( τ ))     ≤ C e − τ ,    M 2 ( F ( τ )) − θ ⊗ θ    ≤ C e − τ ,    M 3 ( F ( τ )) − c 0 θ ⊗ θ ⊗ θ    ≤ C e − τ ,    M 4 ( F ( τ ))    ≤ C . Her e, θ ∈ R 2 + is given as in Pr op osition 5.5 and c 0 > 0 is deﬁne d in Pr op osition 5.6 . 30 E. FRANCO AND B. KEPKA Pr o of. The proof follo ws from the following scaling principle relating the moments of f ( t ) and F ( τ ) M k ( F ( τ )) = ( T ∗ − t ( τ )) 2 k − 3 M k ( f ( t ( τ ))) , k ∈ N 0 . W e can then emplo y the results of Prop ositions 5.1 , 5.5 , 5.6 and 5.7 . T o this end, observe that T ∗ − t ( τ ) = T ∗ e − τ . F or the estimate of the deriv ativ e of the second momen ts w e use Prop osition 5.5 yielding     d dτ M 2 ( F ( τ ))     =     d dt h ( T ∗ − t ) M 2 ( f ( t )) i | t = t ( τ )         dt ( τ ) dτ     ≤ C e − τ . This concludes the pro of. □ The study of the momen ts yield in fact the lo calization of the whole distribution function. Prop osition 5.12. L et α ∈ R 3 + , f 0 ∈ M 4 , + ( S ) and assume T ∗ = T α, ∗ < ∞ . Consider F as in Deﬁnition 5.8 and θ as in Pr op osition 5.11 . Then, we have the fol lowing estimate Z R 2 + | η | p     η | η | − θ | θ |     2 F ( τ , dη ) ≤ C e − τ , p ∈ { 2 , 3 } for al l τ ≥ 0 and some c onstant C > 0 . Pr o of. Observe that for p ∈ { 2 , 3 } | η | p     η | η | − θ | θ |     2 = | η | p − 2 ∥ η ∥ 2 + | η | p | θ | 2 ∥ θ ∥ 2 − 2 | η | p − 1 | θ | η · θ . F or p = 2 we then ha v e Z R 2 + | η | 2     η | η | − θ | θ |     2 F ( τ , dη ) = Z R 2 + ∥ η ∥ 2 + | η | 2 | θ | 2 ∥ θ ∥ 2 − 2 | η | | θ | η · θ ! F ( τ , dη ) = 2 X k =1 M 2 kk ( τ ) + ∥ θ ∥ 2 | θ | 2 2 X j,k =1 M 2 kj ( τ ) − 2 | θ | 2 X j,k =1 M 2 kj ( τ ) θ j . W e now use Corollary 5.11 . Observe that replacing M kj ( τ ) by θ k θ j in the preceding formula yields zero. In particular, w e hav e Z R 2 + | η | 2     η | η | − θ | θ |     2 F ( τ , dη ) = 2 X k =1  M 2 kk ( τ ) − θ 2 k  + ∥ θ ∥ 2 | θ | 2 2 X j,k =1  M 2 kj ( τ ) − θ k θ j  − 2 | θ | 2 X j,k =1  M 2 kj ( τ ) − θ k θ j  θ j . ≤ C e − τ . In the last inequality w e used the estimate in Corollary 5.11 . The pro of for p = 3 is analogous using the estimate for the third order moments in Corollary 5.11 . □ 5.3. Con v ergence to w ards self-similar solution. In the previous subsection we prov e that the distri- bution in self-similar v ariables F ( τ ) concen trates on the line { λθ, λ ≥ 0 } . In this subsection w e study the long-time asymptotics of the proﬁle on the lo calization line itself, see (iii) in Theorem 2.4 . T o this end, it is conv enien t to introduce a v arian t of p olar co ordinates as follows. Let us denote ∆ = { η ∈ R 2 : | η | = 1 } . W e deﬁne the mapping T : (0 , ∞ ) 2 → (0 , ∞ ) × ∆ : η 7→ ( r, ω ) =  | η | , η | η |  . LONG-TIME BEHA VIOUR OF R OULEA U F ORMA TION MODELS 31 Note that T is a diﬀeomorphism with Jacobian | det D T − 1 | = r . F urthermore, let us also deﬁne the pro jection P : (0 , ∞ ) × ∆ → (0 , ∞ ) : ( r, ω ) → r. W e then deﬁne the following distributions. Deﬁnition 5.13. Let α ∈ R 3 + , f 0 ∈ M 4 , + ( S ) and assume T α, ∗ < ∞ . Consider F as in Deﬁnition 5.8 . W e then deﬁne G ( τ ) = T # F ( τ ) ∈ M + ( R + × ∆) , g ( τ ) = 1 Z ( τ ) P # G ( τ ) ∈ M + ( R + ) , where Z ( τ ) = P 2 j,k =1 M 2 j k ( F ( τ )) . The measure g ( τ ) can b e regarded as a measure on the line, indeed the ω -comp onent has b een in tegrated. An adv antage of choosing the p olar co ordinates r = | η | and ω = η | η | as the radial v ariable, is the fact that g will satisfy a one-dimensional coagulation equation. Indeed, the coagulation rule is ( η , η ′ ) 7→ η + η ′ + e − 2 τ ξ i /T 2 ∗ , see ( 5.16 ), so that the coagulation rule for the radial v ariable is ( r , r ′ ) 7→ r + r ′ + e − 2 τ T 2 ∗ ( ξ i, 1 + ξ i, 2 ) . When τ → ∞ this is exactly the standard coagulation rule in one dimension. As men tioned in the in tro duction the self-similar asymptotics is exp ected to be v alid for the w eigh ted measure | η | 2 F ( τ ) . Thus, on the lo calization line w e are interested in the measure r 2 g ( τ ) , where the normalization factor Z ( τ ) has b een in tro duced in order to ensure that r 2 g ( τ ) is a probabilit y measure: Z R + r 2 g ( τ , dr ) = 1 Z ( τ ) Z R + × ∆ r 2 G ( τ , dr, dω ) = 1 Z ( τ ) Z R 2 + | η | 2 F ( τ , dη ) = 1 Z ( τ ) 2 X j,k =1 M 2 j k ( F ( τ )) = 1 . Also observ e that by Corollary 5.11 w e ha ve Z ( τ ) → | θ | 2 . Let us mention that Z ( τ ) > 0 for all times. Otherwise F ≡ 0 by the uniqueness of the solution to the coagulation equation. The main result in this subsection is then the following theorem. Theorem 5.14. L et α ∈ R 3 + , f 0 ∈ M 4 , + ( S ) and assume T ∗ = T α, ∗ < ∞ . Consider g as in Deﬁnition 5.13 and the pr ob ability me asur e on R + g ∞ ( dr ) = 1 √ 2 π K 0 r e − r/ 2 K 0 dr , wher e K 0 = c 0 | θ | . Her e, θ and c 0 is given in Cor ol lary 5.11 . Then, we have lim τ →∞ r 2 g ( τ ) = g ∞ , with r esp e ct to the we ak c onver genc e of me asur es. The pro of of this result will b e achiev ed via the following steps. (i) W e ﬁrst derive a coagulation equation satisﬁed by g . This equation will be one-dimensional. How- ev er, the defect of G b eing not lo calized for tr < T ∗ will introduce forcing terms, that can b e shown to decay as τ → ∞ . (ii) W e then study the desingularized Laplace transform of g , giv en b y ˆ g ( τ , ρ ) = Z R + r (1 − e − rρ ) g ( τ , dr ) . (5.17) Using (i) w e can derive an equation satisﬁed by ˆ g . This equation has the structure of a Burger’s equation with forcing. Again the forcing term dep ends on G and can b e shown to decay as τ → ∞ . 32 E. FRANCO AND B. KEPKA (iii) Finally , using the momen t b ounds in Corollary 5.11 w e obtain a priori knowledge on the regularit y of ˆ g . In fact, w e hav e ˆ g ∈ C 1 b ( R + ; C b ( R + )) ∩ C b ( R + ; C 3 b ( R + )) . This and the study of the characteristics to the Burger’s equation allo ws to pro v e for all ρ ≥ 0 lim τ →∞ ∂ ρ ˆ g ( τ , ρ ) = 1 √ 1 + 2 K 0 ρ . Observ e that ∂ ρ ˆ g ( τ , ρ ) is the standard Laplace transform of r 2 g ( τ , dr ) , see [15]. Recall that the Laplace transform to g ∞ in Theorem 5.14 is giv en by 1 / √ 1 + 2 K 0 ρ . F urthermore, recall that weak con v ergence in the sense of measures is equiv alen t to the p oin twise conv ergence of the Laplace transform. F ollowing the preceding plan we hav e the follo wing lemma. Lemma 5.15. L et α ∈ R 3 + , f 0 ∈ M 4 , + ( S ) and assume T ∗ = T α, ∗ < ∞ . Consider g and G as in Deﬁnition 5.8 . L et θ ∈ R 2 + b e given as in Cor ol lary 5.11 and deﬁne ω θ = θ/ | θ | . Then, we have the fol lowing e quation for al l φ ∈ C 1 b ( R + ) d dτ Z R + φ ( r ) g ( τ , dr ) = 3 Z R + φ ( r ) g ( τ , dr ) − 2 Z R 2 + r ∂ r φ ( r ) g ( τ , dr ) − ˙ Z ( τ ) Z ( τ ) Z R + φ ( r ) g ( τ , dr ) + 3 X i =1 α i Z ( τ ) 2 θ ⊤ K i θ | θ | 2 Z R + Z R + r r ′ ¯ ∆ i φ ( τ ; r, r ′ ) g ( τ , dr ) g ( τ , dr ′ ) + 3 X i =1 α i 2 Z ( τ ) Z R + × ∆ Z R + × ∆ r r ′  K i ( ω , ω ′ ) − K i ( ω θ , ω θ )  ¯ ∆ i φ ( τ ; r, r ′ ) G ( τ , dr, dω ) G ( τ , dr ′ , dω ) . Her e, we deﬁne d ¯ ∆ i φ ( τ , r, r ′ ) = 1 { r + r ′ − δ i ( τ ) ≥ 0 }  φ  r + r ′ − δ i ( τ )  − φ ( r ) − φ ( r ′ )  , δ i ( τ ) = − ( ξ i, 1 + ξ i, 2 ) e − 2 τ T 2 ∗ ≥ 0 . Pr o of. The equation follows using Lemma 5.10 and the deﬁnition of g and G . Recall that F ( τ ) deﬁned in Deﬁnition 5.8 is supp orted on the set e − 2 τ S i . As a consequence G ( τ ) is supp orted on the set { r + r ′ − δ i ( τ ) ≥ 0 } . The coagulation op erator has then the form 3 X i =1 α i 2 Z ( τ ) Z R + × ∆ Z R + × ∆ r r ′ K i ( ω , ω ′ ) ¯ ∆ i φ ( τ ; r, r ′ ) G ( τ , dr, dω ) G ( τ , dr ′ , dω ) . W e then add and subtract the k ernel K i ( ω θ , ω θ ) = θ ⊤ K i θ / | θ | 2 . This yields then the last t wo terms in the asserted equation. □ In order to estimate the con tribution of the coagulation kernel containing G we make use of the follo wing lemma, whic h immediately follows from Prop osition 5.12 . Lemma 5.16. L et G b e given as in Deﬁnition 5.13 . Then, we have the estimates for al l τ ≥ 0 Z R + × ∆ r p ∥ ω − ω θ ∥ 2 G ( τ , dr, dω ) ≤ C e − τ , p ∈ { 2 , 3 } . Using Lemma 5.15 we now derive the equation satisﬁed b y the desingularized Laplace transform of g , see ( 5.17 ). Lemma 5.17. L et α ∈ R 3 + , f 0 ∈ M 4 , + ( S ) and assume T ∗ = T α, ∗ < ∞ . Consider g and G as in Deﬁni- tion 5.8 . Then, the desingularize d L aplac e tr ansform ˆ g ∈ C 1 ( R + ; C b ( R + )) ∩ C b ( R + ; C 3 b ( R + )) satisﬁes the e quation ∂ τ ˆ g = ˆ g + ( ˆ g − 2 ρ ) ∂ ρ ˆ g + R . (5.18) LONG-TIME BEHA VIOUR OF R OULEA U F ORMA TION MODELS 33 Her e, we deﬁne d R ( τ , ρ ) = − ˙ Z ( τ ) Z ( τ ) ˆ g +  Z ( τ ) | θ | 2 − 1  ˆ g ∂ ρ ˆ g + 3 X i =1 α i Z ( τ ) θ ⊤ K i θ | θ | 2 Z R + Z R + h h i 1 + h i 2 i g ( τ , dr ) g ( τ , dr ′ ) + 3 X i =1 α i Z ( τ ) Z R + × ∆ Z R + × ∆  K i ( ω , ω ′ ) − K i ( ω θ , ω θ )  h h i 1 + h i 2 + h 3 i G ( τ , dr, dω ) G ( τ , dr ′ , dω ′ ) , as wel l as h i 1 ( τ , ρ, r, r ′ ) = 1 { r + r ′ − δ i ( τ ) ≥ 0 } r ′ r 2 e − ρ ( r + r ′ )  1 − e δ i ( τ ) ρ  , i ∈ { 1 , 2 , 3 } , h i 2 ( τ , ρ, r, r ′ ) = − 1 { r + r ′ − δ i ( τ ) ≥ 0 } 1 2 r r ′ δ i ( τ )  1 − e − ( r + r ′ − δ i ( τ )) ρ  , i ∈ { 1 , 2 , 3 } , h 3 ( τ , ρ, r, r ′ ) = r ′ r 2 e − rρ  1 − e − r ′ ρ  . Pr o of. First of all the regularity ˆ g ∈ C 1 ( R + ; C b ( R + )) ∩ C b ( R + ; C 3 b ( R + )) follows from the moment b ounds giv en in Corollary 5.11 together with the deﬁnition of g and its desingularized Laplace transform. The asserted equation can b e derived using Lemma 5.15 and the test function φ ( r ) = r (1 − e − rρ ) for given ρ ≥ 0 . T o this end, we observe that ¯ ∆ i φ ( τ , r, r ′ ) = 1 { r + r ′ − δ i ( τ ) ≥ 0 } h r ′ r 2 e − ρr (1 − e − r ′ ρ ) + r ′ r 2 e − ρ ( r + r ′ ) (1 − e δ i ( τ ) ρ ) i + + 1 { r + r ′ − δ i ( τ ) ≥ 0 } h r ( r ′ ) 2 e − ρr ′ (1 − e − rρ ) + r ( r ′ ) 2 e − ρ ( r + r ′ ) (1 − e δ i ( τ ) ρ ) i − 1 { r + r ′ − δ i ( τ ) ≥ 0 } h r r ′ δ i ( τ )(1 − e − ( r + r ′ − δ i ( τ )) ρ ) i = 1 { r + r ′ − δ i ( τ ) ≥ 0 } 2 X j =1 h h i j ( τ , r, r ′ ) + h i j ( τ , r ′ , r ) i + 1 { r + r ′ − δ i ( τ ) ≥ 0 }  h 3 ( τ , r, r ′ ) + h 3 ( τ , r ′ , r )  . W e can reduce this in the integral b y making use of the symmetry r ↔ r ′ . This yields 3 X i =1 α i Z ( τ ) 2 θ ⊤ K i θ | θ | 2 Z R + Z R + r r ′ ¯ ∆ i φ ( τ ; r, r ′ ) g ( τ , dr ) g ( τ , dr ′ ) = Z ( τ ) | θ | 2 ˆ g ∂ ρ ˆ g + 3 X i =1 α i Z ( τ ) θ ⊤ K i θ | θ | 2 Z R + Z R + h h i 1 + h i 2 i g ( τ , dr ) g ( τ , dr ′ ) . Here, w e used for the ﬁrst term con taining h 3 that P 3 i =1 α i θ ⊤ K i θ = 1 , see Prop osition 5.5 . F urthermore, w e ha ve 3 Z R + φ ( r ) g ( τ , dr ) − 2 Z R 2 + r ∂ r φ ( r ) g ( τ , dr ) − ˙ Z ( τ ) Z ( τ ) Z R + φ ( r ) g ( τ , dr ) = ˆ g − 2 ρ∂ ρ ˆ g − ˙ Z ( τ ) Z ( τ ) ˆ g . Putting all terms together yields the equation ( 5.18 ) and the form of R . □ Next we provide estimates on the reminder term R and its deriv ative ∂ ρ R . Lemma 5.18. Under the assumptions of The or em 5.14 we have | R ( τ , ρ ) | ≤ C ρe − τ / 2 , for al l τ ≥ 0 , ρ ≥ 0 and some c onstant C > 0 . 34 E. FRANCO AND B. KEPKA Pr o of. W e ﬁrst consider the follo wing splitting of the reminder R = P 6 j =1 R k , where w e hav e R 1 ( τ , ρ ) = − ˙ Z ( τ ) Z ( τ ) ˆ g +  Z ( τ ) | θ | 2 − 1  ˆ g ∂ ρ ˆ g , R 2 ( τ , ρ ) = 3 X i =1 α i Z ( τ ) θ ⊤ K i θ | θ | 2 Z R + Z R + h i 1 ( τ , ρ, r, r ′ ) g ( τ , dr ) g ( τ , dr ′ ) , R 3 ( τ , ρ ) = 3 X i =1 α i Z ( τ ) θ ⊤ K i θ | θ | 2 Z R + Z R + h i 2 ( τ , ρ, r, r ′ ) g ( τ , dr ) g ( τ , dr ′ ) , R 4 ( τ , ρ ) = 3 X i =1 α i Z ( τ ) Z R + × ∆ Z R + × ∆  K i ( ω , ω ′ ) − K i ( ω θ , ω θ )  h i 1 ( τ , ρ, r, r ′ ) G ( τ , dr, dω ) G ( τ , dr ′ , dω ′ ) , R 5 ( τ , ρ ) = 3 X i =1 α i Z ( τ ) Z R + × ∆ Z R + × ∆  K i ( ω , ω ′ ) − K i ( ω θ , ω θ )  h i 2 ( τ , ρ, r, r ′ ) G ( τ , dr, dω ) G ( τ , dr ′ , dω ′ ) , R 6 ( τ , ρ ) = 3 X i =1 α i Z ( τ ) Z R + × ∆ Z R + × ∆  K i ( ω , ω ′ ) − K i ( ω θ , ω θ )  h 3 ( τ , ρ, r, r ′ ) G ( τ , dr, dω ) G ( τ , dr ′ , dω ′ ) . W e no w estimate eac h term separately . First of all, we observe for R 1 that ˆ g ( τ , ρ ) ≤ C ρ , ∥ ∂ ρ ˆ g ∥ ≤ C and hence | R 1 | ≤ C ρ      ˙ Z ( τ ) Z ( τ )      + ρ     Z ( τ ) | θ | 2 − 1     ≤ C ρe − τ . Here, we used the estimates in Corollary ( 5.11 ) for Z ( τ ) = P 2 j,k =1 M 2 j k ( F ( τ )) . F or R 2 w e mak e use of | h i 1 ( τ , ρ, r, r ′ ) | = 1 { r + r ′ − δ i ( τ ) ≥ 0 } r ′ r 2 e − ρ ( r + r ′ )    e δ i ( τ ) ρ − 1    ≤ 1 { r + r ′ − δ i ( τ ) ≥ 0 } r ′ r 2 e − ρ ( r + r ′ ) ρδ i ( τ ) e δ i ( τ ) ρ ≤ r ′ r 2 δ i ( τ ) ρ. This yields then | R 2 ( τ , ρ ) | ≤ C ρ M 1 ( g ( τ )) M 2 ( g ( τ )) X i δ i ( τ ) ≤ C ρe − τ . Here, we used the fact that M 1 ( g ( τ )) ≤ C e τ and M 2 ( g ( τ )) = 1 as follo ws from Corollary 5.11 and the deﬁnition of g ( τ ) . Considering R 3 w e ﬁrst observ e that | h i 2 ( τ , ρ, r, r ′ ) | ≤ 1 { r + r ′ − δ i ( τ ) ≥ 0 } 1 2 r r ′ δ i ( τ )  r + r ′ − δ i ( τ )  ρ. This yields | R 3 ( τ , ρ ) | ≤ C ρ X i h δ i ( τ ) M 1 ( g ( τ )) + δ i ( τ ) 2 M 1 ( g ( τ )) 2 i ≤ C ρe − τ . W e no w turn to R 4 and use the previous estimate for h i 1 . This yields | R 4 ( τ , ρ ) | ≤ C ρ X i δ i ( τ ) Z R + × ∆ Z R + × ∆ r ( r ′ ) 2 G ( τ , dr, dω ) G ( τ , dr ′ , dω ′ ) ≤ C ρ X i δ i ( τ ) M 1 ( g ( τ )) ≤ C ρe − τ . LONG-TIME BEHA VIOUR OF R OULEA U F ORMA TION MODELS 35 F or R 5 w e use the previous estimate for h i 2 to get | R 5 ( τ , ρ ) | ≤ C ρ X i δ i ( τ ) Z R + × ∆ Z R + × ∆   r r ′ δ i ( τ )  r + r ′ − δ i ( τ )    G ( τ , dr, dω ) G ( τ , dr ′ , dω ′ ) ≤ C ρ X i h δ i ( τ ) M 1 ( g ( τ )) + δ i ( τ ) 2 M 1 ( g ( τ )) 2 i ≤ C ρe − τ . Finally , for R 6 w e ﬁrst observ e that | h 3 ( τ , ρ, r, r ′ ) | ≤ C ( r r ′ ) 2 ρ and   K i ( ω , ω ′ ) − K i ( ω θ , ω θ )   ≤ C ∥ ω − ω θ ∥ + C   ω ′ − ω θ   . W e th us obtain, making use of the symmetry ( r, ω ) ↔ ( r ′ , ω ′ ) , | R 6 ( τ , ρ ) | ≤ C ρ Z R + × ∆ Z R + × ∆ r 2 ( r ′ ) 2 ∥ ω − ω θ ∥ G ( τ , dr , dω ) G ( τ , dr ′ , dω ′ ) ≤ C ρ Z R + × ∆ r 2 G ( τ , dr, dω ) ! 3 / 2 Z R + × ∆ r 2 ∥ ω − ω θ ∥ G ( τ , dr , dω ) ! 1 / 2 ≤ C ρe − τ / 2 . In the last inequality we used Lemma 5.16 . This concludes the proof. □ Lemma 5.19. Under the assumptions of The or em 5.14 we have | ∂ ρ R ( τ , ρ ) | ≤ C (1 + ρ ) e − τ / 2 , for al l τ ≥ 0 , ρ ≥ 0 and some c onstant C > 0 . Pr o of. W e use the same splitting R = P 6 j =1 R k as in the pro of of Lemma 5.18 . W e estimate term by term. First of all, we hav e ∂ ρ R 1 = − ˙ Z ( τ ) Z ( τ ) ∂ ρ ˆ g +  Z ( τ ) | θ | 2 − 1   ( ∂ ρ ˆ g ) 2 + ˆ g ∂ 2 ρ ˆ g  . Hence we obtain | ∂ ρ R 1 | ≤ C e − τ . F or the remaining terms we ﬁrst observe ∂ ρ h i 1 = − 1 { r + r ′ − δ i ≥ 0 } h r ′ r 2 ( r + r ′ ) e − ρ ( r + r ′ )  1 − e δ i ρ  + r ′ r 2 δ i e − ρ ( r + r ′ − δ i ) i , ∂ ρ h i 2 = 1 { r + r ′ − δ i ≥ 0 } 1 2 r r ′ δ i  r + r ′ − δ i  e − ( r + r ′ − δ i ) ρ , ∂ ρ h 3 = − r ′ r 3 e − rρ  1 − e − r ′ ρ  − ( r ′ r ) 2 e − ( r + r ′ ) ρ . W e then obtain the estimates | ∂ ρ h i 1 | ≤ C ρδ i  r ′ r 3 + ( r ′ r ) 2 + r ′ r 2  , | ∂ ρ h i 2 | ≤ 1 2 r r ′ δ i  r + r ′ − δ i  , | ∂ ρ h 3 | ≤ ρ ( r ′ ) 2 r 3 + ( r ′ r ) 2 . 36 E. FRANCO AND B. KEPKA This yields then | R 2 | ≤ C ρ X i δ i  M 1 ( g ( τ )) M 3 ( g ( τ )) + M 2 ( g ( τ )) 2 + M 1 ( g ( τ )) M 2 ( g ( τ ))  ≤ C ρe − τ , | R 3 | ≤ C X i δ i M 1 ( g ( τ )) M 2 ( g ( τ )) + C X i δ 2 i M 1 ( g ( τ )) 2 ≤ C e − τ . Here, w e used the estimates on the momen ts, see Corollary 5.11 together with the deﬁnition of g . The ab o ve estimates yield in the same wa y | R 4 | ≤ C ρ X i δ i  M 1 ( g ( τ )) M 3 ( g ( τ )) + M 2 ( g ( τ )) 2 + M 1 ( g ( τ )) M 2 ( g ( τ ))  ≤ C ρe − τ , | R 5 | ≤ C X i δ i M 1 ( g ( τ )) M 2 ( g ( τ )) + C X i δ 2 i M 1 ( g ( τ )) 2 ≤ C e − τ . Finally , w e hav e | R 6 | ≤ C Z R + × ∆ Z R + × ∆  ρ ( r ′ ) 2 r 3 + ( r ′ r ) 2   ∥ ω − ω θ ∥ +   ω ′ − ω θ    G ( τ , dr, dω ) G ( τ , dr ′ , dω ′ ) ≤ C ρ Z R + × ∆ ( r 2 + r 3 ) ∥ ω − ω θ ∥ G ( τ , dr, dω ) + C Z R + × ∆ r 2 ∥ ω − ω θ ∥ G ( τ , dr, dω ) ≤ C ρe − τ / 2 + C e − τ / 2 . Here, we used Lemma 5.16 . Putting all estimates together concludes the pro of. □ W e no w give the proof of Theorem 5.14 . Pr o of of The orm 5.14 . As men tioned b efore it suﬃces to show that ∂ ρ ˆ g ( τ , ρ ) conv erges p oint wise to 1 / √ 1 + 2 K 0 ρ . In fact, b elo w w e prov e the following. There are constants C , κ > 0 such that for all M ≥ 1 and all τ ≥ 0 we hav e sup ρ ∈ [0 ,M ]     ∂ ρ ˆ g ( τ , ρ ) − 1 √ 1 + 2 K 0 ρ     ≤ C M κ e − τ / 10 . Let us giv e a short o verview of the proof. The idea is to use the equation satisﬁed b y ˆ g in Lemma 5.17 . Since ˆ g is a classical solution to this equation, ˆ g can b e quan tiﬁed using the metho d of characteristics. In order to turn this into a p erturbative analysis, w e do not study the dynamics starting at τ = 0 but at τ = T large. This ensures that the formal initial condition ˆ g ( T , ρ ) is arsing from a distribution that is almost lo calized on the concentration line { λθ : λ ≥ 0 } with controlled v alues of the third momen ts. On the level of the desingularized Laplace transform this means that ∂ 2 ρ ˆ g ( T , 0) = 1 Z ( τ ) Z R + × ∆ r 3 G ( T , dr, dω ) = 1 Z ( τ ) Z R 2 + | η | 3 F ( T , dη ) ≈ c 0 | θ | = K 0 . Recall that by Corollary 5.11 we ha ve R R 2 + | η | 3 F ( T , dη ) → c 0 | θ | 3 and Z ( τ ) → | θ | 2 . Observ e that the v alue K 0 c haracterizes the limiting self-similar distribution, see the formula for g ∞ in Theorem 5.14 . Note that ∂ ρ ˆ g ( T , 0) = 1 for all T ≥ 0 by deﬁnition of g . In addition, using the starting p oint τ = T also ensures that the reminder R ( T , ρ ) is small, i.e. of order e − T / 2 . This allows to prov e that the c haracteristics of the Burger’s t yp e equation and hence also the solution can b e controlled on a time in terv al of size cT , c > 0 . And w e obtain for all τ ≤ cT ∂ ρ ˆ g ( T + τ , ρ ) − 1 √ 1 + 2 K 0 ρ = O  e − τ  + O  e − T / 10  . Setting τ = cT will then pro vide the ab o ve estimate. W e no w provide the full details and to this end we divide the pro of in to several steps. LONG-TIME BEHA VIOUR OF R OULEA U F ORMA TION MODELS 37 Step 1. Setup. Let us ﬁx M ≥ 1 and T ≥ 1 . W e deﬁne H T ( τ , ρ ) = ˆ g ( T + τ , ρ ) , R T ( τ , ρ ) = R ( T + τ , ρ ) . W e kno w that H T satisﬁes the equation ∂ τ H T = H T + ( H T − 2 ρ ) ∂ ρ H T + R T . (5.19) W e also hav e b y T a ylor’s theorem H T (0 , ρ ) = ρ − 1 2 K 0 ρ 2 + h T ( ρ ) . Here, we deﬁned the function h T ( ρ ) = 1 2  ∂ 2 ρ H T (0 , 0) − K 0  ρ 2 + ρ 3 2 Z 1 0 ∂ 3 ρ H T (0 , sρ )(1 − s ) 2 ds. (5.20) This satisﬁes the estimates | h T ( ρ ) | ≤ C e − T ρ 2 + C ρ 3 , | ∂ ρ h T ( ρ ) | ≤ C e − T ρ + C ρ 2 . (5.21) Step 2. Char acteristics. The c haracteristics of the equation ( 5.19 ) are deﬁned by the follo wing ODE d dτ P T ( τ , ρ ) = 2 P T ( τ , ρ ) − H T ( τ , P T ( τ , ρ )) , P T (0 , ρ ) = ρ. (5.22) Since the righ t-hand side is a contin uously diﬀeren tiable function of P T ( τ , ρ ) , the function P T ( τ , ρ ) is w ell-deﬁned and diﬀerentiable in b oth argumen ts. Since 0 ≤ ∂ ρ H T ( τ , ρ ) ≤ 1 we hav e 0 ≤ H T ( τ , ρ ) ≤ ρ . This implies e τ ρ ≤ P T ( τ , ρ ) ≤ e 2 τ ρ (5.23) for all τ , ρ ≥ 0 . W e diﬀerentiate equation ( 5.22 ) and obtain that d dτ ∂ ρ P T ( τ , ρ ) = 2 ∂ ρ P T ( τ , ρ ) − ∂ ρ H T ( τ , P T ( τ , ρ )) ∂ ρ P T ( τ , ρ ) . Notice that since ∂ ρ H T ( τ , P T ( τ , ρ )) = ∂ ρ ˆ g ( τ + T , P T ( τ , ρ )) , w e hav e that ∂ ρ H T ( τ , P T ( τ , ρ )) ∈ (0 , 1) , hence e τ ≤ ∂ ρ P T ( τ , ρ ) ≤ e 2 τ . (5.24) The equation ( 5.19 ) implies d dτ [ H T ( τ , P T ( τ , ρ ))] = H T ( τ , P T ( τ , ρ )) + R T ( τ , P T ( τ , ρ )) . Hence, we obtain H T ( τ , P T ( τ , ρ )) = e τ H T (0 , ρ ) + e τ Z τ 0 e − σ R T ( σ, P T ( σ, ρ )) dσ. (5.25) F rom ( 5.22 ) we obtain P T ( τ , ρ ) = e 2 τ ( ρ − H T (0 , ρ )) + e τ H T (0 , ρ ) + e 2 τ Z τ 0  e − σ − e − τ  e − σ R T ( σ, P T ( σ, ρ )) dσ = K 0 2 e 2 τ ρ 2 + e τ ρ + ¯ E T ( τ , ρ ) , ¯ E T ( τ , ρ ) = − e 2 τ h T ( ρ ) − K 0 2 e τ ρ 2 + e τ h T ( ρ ) + e 2 τ Z τ 0  e − σ − e − τ  e − σ R T ( σ, P T ( σ, ρ )) dσ. 38 E. FRANCO AND B. KEPKA Here, w e used ( 5.20 ). The goal now is to giv e a asymptotic form ula for in verse ρ 7→ P T ( τ , ρ ) . Note that b y ( 5.24 ) we already kno w that this function is in v ertible on R + . T o this end, it is con v enien t to deﬁne Q T ( τ , ρ ) := P T ( τ , e − τ ρ ) = Q 0 ( ρ ) + E T ( τ , ρ ) , Q 0 ( ρ ) := K 0 2 ρ 2 + ρ, E T ( τ , ρ ) := − e 2 τ h T ( e − τ ρ ) − K 0 2 e − τ ρ 2 + e τ h T ( e − τ ρ ) + e 2 τ Z τ 0  e − σ − e − τ  e − σ R T  σ, P T ( σ, e − τ ρ )  dσ. Observ e that Q 0 : R + → R + is bijective and Q − 1 0 ( ρ ) = 2 ρ 1 + √ 1 + 2 K 0 ρ , ∂ ρ Q − 1 0 ( ρ ) = 1 √ 1 + 2 K 0 ρ . Step 3. Estimates on E T and Q T . W e no w give estimates on E T and ∂ ρ E T . F or the ﬁrst we make use of Lemma 5.18 and ( 5.21 ), ( 5.23 ) to get | E T ( τ , ρ ) | ≤ C  e − T ρ 2 + e − τ ρ 3 + e − τ ρ 2  + C e 2 τ Z τ 0 e − 2 σ e 2 σ e − τ e − T / 2 e − σ / 2 ρ dσ ≤ C  e − T ρ 2 + e − τ ρ 3 + e − τ ρ 2 + e τ e − T / 2 ρ  . F or the deriv ative we note that ∂ ρ E T ( τ , ρ ) = − e τ ∂ ρ h T ( e − τ ρ ) − K 0 e − τ ρ + ∂ ρ h T ( e − τ ρ ) + e τ Z τ 0  e − σ − e − τ  e − σ ∂ ρ R T ( σ, P ( σ, e − τ ρ )) ∂ ρ P T ( σ, e − τ ρ ) dσ. W e then use Lemma 5.19 and ( 5.21 ), ( 5.24 ) | ∂ ρ E T ( τ , ρ ) | ≤ C  e − T ρ + e − τ ρ 2 + e − τ ρ  + C e τ Z τ 0 e − 2 σ  1 + e 2 σ e − τ ρ  e − T / 2 e − σ / 2 e 2 σ dσ ≤ C  e − T ρ + e − τ ρ 2 + e − τ ρ + e τ e − T / 2 + e 3 τ / 2 e − T / 2 ρ  . In particular, for τ ∈ [ 0 , T / 8] w e obtain for all ρ ≥ 0 | E T ( τ , ρ ) | ≤ C  1 + ρ 2  ρe − T / 4 , | ∂ ρ E T ( τ , ρ ) | ≤ C  1 + ρ 2  e − T / 4 . (5.26) This implies in particular | Q T ( τ , ρ ) | ≤ C (1 + ρ 3 ) , | ∂ ρ Q T ( τ , ρ ) | ≤ C (1 + ρ 2 ) (5.27) Step 4. Inverting char acteristics. W e now study the inv erse of Q T ( τ , ρ ) = Q 0 ( ρ ) + E T ( τ , ρ ) for τ ∈ [0 , T / 8] and ρ ∈ [ 0 , M ] . T o this end, we mak e use of the estimates in Step 3, in particular T will b e c hosen larger than T 0 ( M ) . W e sho w b elo w that in this case sup ρ ∈ [0 ,M ] sup τ ∈ [0 ,T / 8]    Q − 1 T ( τ , ρ ) − Q − 1 0 ( τ , ρ )    ≤ C M 3 / 2 e − T / 4 , sup ρ ∈ [0 ,M ] sup τ ∈ [0 ,T / 8]    ∂ ρ Q − 1 T ( τ , ρ ) − ∂ ρ Q − 1 0 ( τ , ρ )    ≤ C M 3 / 2 e − T / 4 . (5.28) T o this end, let use deﬁne U T ( τ , ρ ) = Q T  τ , Q − 1 0 ( ρ )  = ρ + E T  τ , Q − 1 0 ( ρ )  . Observ e that Q − 1 0 ( ρ ) ≤ C √ 1 + ρ , ∂ ρ Q − 1 0 ( ρ ) ≤ C √ 1 + ρ . LONG-TIME BEHA VIOUR OF R OULEA U F ORMA TION MODELS 39 W e hence obtain with ( 5.26 ) sup ρ ∈ [0 , 2 M ] sup τ ∈ [0 ,T / 8] | U T ( τ , ρ ) − ρ | ≤ C M 3 / 2 e − T / 4 , sup ρ ∈ [0 , 2 M ] sup τ ∈ [0 ,T / 8] | ∂ ρ U T ( τ , ρ ) − 1 | ≤ C M 3 / 2 e − T / 4 . In particular, for T ≥ T 0 ( M ) = c ln(1 + M ) , where c > 0 is some large constant, we get 1 2 ≤ ∂ ρ U T ( τ , ρ ) ≤ 3 2 and thus for ρ ∈ [0 , M ] 3 2 ρ ≤ U − 1 T ( τ , ρ ) ≤ 2 ρ ≤ 2 M . This then yields sup ρ ∈ [0 ,M ] sup τ ∈ [0 ,T / 8]    U − 1 T ( τ , ρ ) − ρ    ≤ C M 3 / 2 e − T / 4 . sup ρ ∈ [0 ,M ] sup τ ∈ [0 ,T / 8]    ∂ ρ U − 1 T ( τ , ρ ) − 1    ≤ C M 3 / 2 e − T / 4 . (5.29) W e hence obtain Q − 1 T ( τ , ρ ) − Q − 1 0 ( τ , ρ ) = Q − 1 0 ( τ , U − 1 T ( τ , ρ )) − Q − 1 0 ( τ , ρ ) = Z U − 1 T ( τ ,ρ ) ρ ∂ ρ Q − 1 0 ( τ , ξ ) dξ , ∂ ρ Q − 1 T ( τ , ρ ) − ∂ ρ Q − 1 0 ( τ , ρ ) = ∂ ρ Q − 1 0 ( τ , U − 1 T ( τ , ρ )) ∂ ρ U − 1 T ( τ , ρ ) − ∂ ρ Q − 1 0 ( τ , ρ ) . Using the estimates ( 5.29 ) and ∂ 2 ρ Q − 1 0 ( ρ ) ≤ C yields ( 5.28 ) for T ≥ T 0 ( M ) . Recalling that P T ( τ , ρ ) = Q T ( τ , e τ ρ ) yields with ( 5.28 ) sup ρ ∈ [0 ,M ] sup τ ∈ [0 ,T / 8]    P − 1 T ( τ , ρ ) − e − τ Q − 1 0 ( τ , ρ )    ≤ C M 3 / 2 e − T / 4 , sup ρ ∈ [0 ,M ] sup τ ∈ [0 ,T / 8]    ∂ ρ P − 1 T ( τ , ρ ) − e − τ ∂ ρ Q − 1 0 ( τ , ρ )    ≤ C M 3 / 2 e − T / 4 (5.30) for T ≥ T 0 ( M ) . Step 5. Conclusion. W e no w obtain from ( 5.25 ) ∂ ρ H T ( τ , ρ ) = ∂ ρ H T  0 , P − 1 T ( τ , ρ )  e τ ∂ ρ P − 1 T ( τ , ρ ) + e τ Z τ 0 e − σ ∂ ρ R T  σ, P T  σ, P − 1 T ( τ , ρ )  ∂ ρ P T  σ, P − 1 T ( τ , ρ )  ∂ ρ P − 1 T ( τ , ρ ) dσ = ∂ ρ H T  0 , P − 1 T ( τ , ρ )  e τ ∂ ρ P − 1 T ( τ , ρ ) + Z τ 0 ∂ ρ R T  σ, Q T ( σ, e σ e − τ Q − 1 T ( τ , ρ ))  ∂ ρ Q T ( σ, e σ e − τ Q − 1 T ( τ , ρ )) ∂ ρ Q − 1 T ( τ , ρ ) dσ Note that 1 √ 1 + 2 K 0 ρ = ∂ ρ H T (0 , 0) ∂ ρ Q − 1 0 ( τ , ρ ) . Th us, using Lemma 5.19 and ( 5.30 ) w e get for ρ ∈ [ 0 , M ] and T ≥ T 0 ( M )     ∂ ρ H T ( τ , ρ ) − 1 √ 1 + 2 K 0 ρ     ≤    ∂ 2 ρ H T    ∞ P − 1 T ( τ , ρ ) ∂ ρ Q − 1 0 ( τ , ρ ) + ∥ ∂ ρ H T ∥ ∞    e τ ∂ ρ P − 1 T ( τ , ρ ) − ∂ ρ Q − 1 0 ( τ , ρ )    + e − T / 2 Z τ 0  1 + Q T ( σ, e σ e − τ Q − 1 T ( τ , ρ )) 2  ∂ ρ Q T ( σ, e σ e − τ Q − 1 T ( τ , ρ )) ∂ ρ Q − 1 T ( τ , ρ ) dσ. 40 E. FRANCO AND B. KEPKA In order to estimate the ﬁrst t w o terms we use ( 5.30 ). T o estimate the last term we use ( 5.27 ) and ( 5.28 ) for ρ ∈ [0 , M ] Q T ( σ, e σ e − τ Q − 1 T ( τ , ρ )) ≤ C  1 +  e σ e − τ Q − 1 T ( τ , ρ )  3  ≤ C M 9 / 2 , | ∂ ρ Q T ( σ, e σ e − τ Q − 1 T ( τ , ρ )) | ≤ C  1 +  e σ e − τ Q − 1 T ( τ , ρ )  2  ≤ C M 3 , | ∂ ρ Q − 1 T ( τ , ρ ) | ≤ C M 3 / 2 . W e th us obtain for τ ∈ [0 , T / 8] sup ρ ∈ [0 ,M ]     ∂ ρ H T ( τ , ρ ) − 1 √ 1 + 2 K 0 ρ     ≤ C  e − τ + M 3 / 2 e − T / 4 + M 14 τ e − T / 2  . This implies when setting τ = T / 8 sup ρ ∈ [0 ,M ]     ∂ ρ ˆ g ( T + T / 8) , ρ ) − 1 √ 1 + 2 K 0 ρ     ≤ C M 14 e − T / 8 , for T ≥ T 0 ( M ) = c ln(1 + M ) . Since the left-hand side is b ounded w e can c hange constants to obtain for all τ ≥ 0 sup ρ ∈ [0 ,M ]     ∂ ρ ˆ g ( τ , ρ ) − 1 √ 1 + 2 K 0 ρ     ≤ C M κ e − τ / 10 , where κ = 14 + c/ 8 . This concludes the pro of. □ W e no w give the proof of Theorem 2.4 . Pr o of of The or em 2.4 . The statemen ts in (i) and (ii) are exactly as in Corollary 5.11 and Prop osition 5.12 , resp ectiv ely . F or (iii) we argue as follo ws. Let F ∞ ( dη ) = | η | F s ( | η | ) δ  η | η | − θ | θ |  dη , ˜ F ( τ , dη ) = | η | 2 F ( τ , dη ) Z ( τ ) . Here, F s is giv en in ( 2.4 ) with K 0 = c 0 | θ | . Note that F s ( r ) = g ∞ ( r ) /r 2 with g ∞ as in Theorem 5.14 . Consequen tly , F ∞ is a probabilit y measure. Recall that dη = rdr dω using the change of v ariables T ( η ) = ( r , ω ) . F or φ ∈ C 1 c ( R 2 + ) we hav e      Z R 2 + φ ( η ) ˜ F ( τ , dη ) − Z R 2 + φ  | η | θ | θ |  ˜ F ( τ , dη )      ≤ ∥∇ φ ∥ ∞ Z R 2 + | η |     η | η | − θ | θ |     ˜ F ( τ , dη ) . The latter goes to zero as τ → ∞ by Prop osition 5.12 and the momen t b ounds in Corollary 5.11 . F urthermore, w e hav e Z R 2 + φ  | η | θ | θ |  ˜ F ( τ , dη ) − Z R 2 + φ  | η | θ | θ |  F ∞ ( τ , dη ) = Z ∞ 0 φ  r θ | θ |  r 2 g ( τ , dr ) − Z ∞ 0 φ  r θ | θ |  g ∞ ( dr ) , with g as deﬁned in Deﬁnition 5.13 . The latter expression conv erges to zero by Theorem 5.14 . Using the fact that the family of probability measures ˜ F ( τ ) is tigh t by Corollary 5.11 , we hence obtain ˜ F ( τ ) → F ∞ as τ → ∞ in the sense of measures. This concludes the pro of. □ W e no w collect our results to yields Theorem 2.4 . Pr o of of The or em 2.4 . Part (i) is a result of Corollary 5.11 , while part (ii) is just Proposition 5.12 . Finally , statemen t (iii) is a consequence of Theorem 5.14 . □ LONG-TIME BEHA VIOUR OF R OULEA U F ORMA TION MODELS 41 A cknowledgments B. Kepka gratefully ac kno wledges support of the SNSF through grant PCEFP2_203059 and the NCCR SwissMAP . E. F ranco gratefully ac knowledge the supp ort of the gran t CR C 1720 "Analysis of Criticalit y: from Complex Phenomena to Mo dels and Estimates" (Pro ject-ID 539309657) of the Univ er- sit y of Bonn funded through the Deutsche F orsch ungsgemeinschaft (DF G, German Research F oundation) and Germany’s Excellence Strategy-EX C-2047/2-390685813. The funders had no role in study design, analysis, decision to publish, or preparation of the man uscript. References [1] J. Bertoin. T wo solv able systems of coagulation equations with limited aggregations. In A nnales de l’Institut Henri Poinc aré C, A nalyse non Linéair e , volume 26, pages 2073–2089. Elsevier, 2009. [2] P . Blatz and A. T ob olsky . 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Long-time behaviour of rouleau formation models

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