math.PR 2026-03-17 0

Anisotropic Inviscid Limit for the Navier-Stokes Equations with Transport Noise Between Two Plates

We investigate an anisotropic vanishing viscosity limit of the 3D stochastic Navier-Stokes equations posed between two horizontal plates, with Dirichlet no-slip boundary condition. The turbulent viscosity is split into horizontal and vertical directi…

Authors: Daniel Goodair

Anisotropic In viscid Limit for the Na vier-Stok es Equations with T ransp ort Noise Bet w een Tw o Plates Daniel Go odair ∗ Marc h 18, 2026 Abstract W e inv estigate an anisotropic v anishing viscosity limit of the 3D sto c hastic Na vier-Stokes equations p osed betw een t wo horizon tal plates, with Dirichlet no-slip boundary condition. The turbulen t viscosit y is split into horizontal and v ertical directions, each of whic h approac hes zero at a different rate. The underlying Cylindrical Brownian Motion driving our transp ort- stretc hing noise is decomp osed into horizon tal and v ertical comp onen ts, which are scaled by the square ro ot of the resp ectiv e directional viscosities. W e prov e that if the ratio of the vertical to horizon tal viscosities approac hes zero, then there exists a sequence of weak martingale solutions con vergen t to the strong solution of the deterministic Euler equation on its lifetime of existence. A particular c hallenge is that the anisotropic scaling ruins the divergence-free prop ert y for the spatial correlation functions of the noise. Con ten ts 1 In tro duction 1 1.1 Deterministic Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.2 Structure of the Noise . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 1.3 Asp ects of the Pro of . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 2 Preliminaries 7 2.1 F unctional Analytic Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 2.2 Sto c hastic Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 2.3 T ransport-Stretching Noise . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 3 Anisotropic In viscid Limit 12 3.1 Definitions and Main Result . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 3.2 Selection of Martingale W eak Solutions . . . . . . . . . . . . . . . . . . . . . . . . . 12 3.3 Pro of of the Main Result . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 References 29 Hello ∗ ´ Ecole Polytec hnique F´ ed ´ erale de Lausanne, Lausanne, Switzerland, daniel.go o dair@epfl.c h 1 In tro duction This work concerns the inviscid limit of an anisotropic sto c hastic Navier-Stok es equation, as the horizon tal and v ertical comp onents of the turbulen t viscosit y approach zero at different rates. W e consider a divergence-free solution u of the equation du t = − B ( u t , u t ) dt − ν h A h u t dt − ν z A z u t dt − ν 1 2 h P G h u t ◦ d W t − ν 1 2 z P G z u t ◦ d W t (1) whic h represents the v elo cit y of a fluid, p osed on a domain O = T 2 × (0 , 1) and supplemen ted with the no-slip b oundary condition u = 0 on ∂ O . Here P is the Leray Pro jector on to divergence- free v ector fields with zero normal comp onen t on ∂ O , B ( u t , u t ) = P (( u t · ∇ ) u t ) is the nonlinear con vectiv e term, A h = −P P 2 j =1 ∂ 2 j is the horizon tal Stokes Op erator with turbulent horizontal viscosit y ν h and A z = −P ∂ 2 3 is the vertical Stokes Op erator with turbulent vertical viscosit y ν z . In the Stratonovic h sto c hastic in tegrals, W denotes a Cylindrical Bro wnian Motion acted up on by op erators G h , G z in the sense that G h u t ◦ d W t = ∞ X i =1 G h i u t ◦ dW i t , G z u t ◦ d W t = ∞ X i =1 G z i u t ◦ dW i t where ( W i ) is a collection of indep enden t standard Brownian Motions comprising W , along with pre-assigned spatial correlation functions ( ξ i ) with resp ect to which ( G h i ), ( G z i ) are defined. Each ξ i is smo oth, satisfies ξ i · n = 0 on ∂ O where n is the out ward unit normal v ector, but is not assumed to b e div ergence-free. The op erators G h i , G z i are defined by G h i u t = 2 X j =1  ξ j i ∂ j u t + u j t ∇ ξ j i  , G z i u t = ξ 3 i ∂ 3 u t + u 3 t ∇ ξ 3 i (2) where the sup erscript j denotes the j th comp onen t mapping of the vector field. Broadly sp eaking, our main result is the construction of martingale w eak solutions to ( 1 ) whic h conv erge, as ν h and ν z ν h approac h zero, to the strong solution w of the deterministic Euler equation ∂ t w = − B ( w , w ) (3) p osed on O with b oundary condition w · n = 0. The conv ergence is in L 2 ω L ∞ t L 2 x , ov er the newly constructed probability space and on the lifetime of existence of w . The precise statement can be found in Theorem 3.2 . 1.1 Deterministic Theory F or a smo oth b ounded domain in t wo or three dimensions, whether or not w eak solutions of the Na vier-Stokes equations with no-slip boundary conditions con verge, as the viscosity is tak en to zero, to the strong solution of the Euler equation remains one of the outstanding problems of mathemat- ical fluid mec hanics. Positiv e results hav e b een limited to very sp ecific cases regarding analyticity of initial data or structure of the flow [ 51 , 52 , 62 , 63 ], whilst conditional results such as [ 41 , 43 , 67 ] c haracterise the conv ergence by energy dissipation in a boundary lay er which is not known to hold in general. W e also mention that the limit is known if one replaces the no-slip b oundary condition of the Navier-Stok es equations with a Navier friction condition [ 12 , 23 , 40 , 42 , 56 ]. 1 In the absence of noise, the present mo del was considered in [ 55 ]. Anisotropic viscosity , that is where ν h  = ν z , is a classical feature of geophysical fluid mechanics. When the fluid is turbulen t w e sp eak of the viscosity not as the molecular kinematic viscosity but rather a turbulen t viscosity , measured for example by the speed of diffusion of tracers. The Coriolis force amplifies horizontal diffusion, so that ν z is m uch smaller than ν h . Whilst many asp ects of a complete geophysical fluid mo del are neglected here, the limit as ν z ν h approac hes zero is a useful setting to exp ose geophysical fluid phenomena. F or a more complete discussion, we refer the reader to [ 10 , 37 , 60 ]. In the aforementioned [ 55 ], Masmoudi prov es precisely the result that we are aiming to estab- lish in sto c hastic analogy; the conv ergence of all Lera y-Hopf weak solutions of the Na vier-Stokes equation with no-slip b oundary condition, as ν h and ν z ν h approac h zero, to the strong solution of the Euler equation. W e describ e the simple heuristics of the problem; as there is no physical b oundary in the horizon tal direction, then the passage of ν h → 0 is harmless and w e are only concerned with the limit ν z → 0 alongside its formation of the b oundary la yer at the horizon tal plates. F ollow- ing the work of Kato [ 41 ] in establishing the equiv alence b et ween in viscid con vergence and energy dissipation in a b oundary la yer, we an ticipate that the problem b oils down to sho wing that lim ν z → 0 ν z Z T 0 ∥∇ u s ∥ 2 L 2 Γ δ ds = 0 (4) where Γ δ = T 2 × [(0 , δ ) ∪ (1 − δ, 1)] is a b oundary strip of width δ shrinking with ν z . Ho w ever, more can b e said. The fundamental problem in v erifying the inviscid limit is the disparity b etw een the b oundary conditions, as the tangential component of u is prescrib ed to b e zero at the b oundary whilst there are no restrictions on the tangential component of w . On the other hand, u · n = 0 = w · n so the normal comp onen ts of u and w match at the b oundary . This suggests to us that the wild b eha viour of u in the boundary lay er concerns only the normal deriv ative in tangen tial directions; it is therefore very interesting that Kato’s criterion was extended b y W ang, in [ 67 ] and with T emam in [ 66 ], to a consideration of only tangential deriv atives in a b oundary lay er. This replaces the sufficien t condition ( 4 ) with lim ν z → 0 ν z Z T 0 2 X j =1 ∥ ∂ j u s ∥ 2 L 2 Γ δ ds = 0 and b y rewriting ν z Z T 0 2 X j =1 ∥ ∂ j u s ∥ 2 L 2 Γ δ ds =  ν z ν h  ν h Z T 0 2 X j =1 ∥ ∂ j u s ∥ 2 L 2 Γ δ ds then, as ν h R T 0 P 2 j =1 ∥ ∂ j u s ∥ 2 L 2 Γ δ ds is b ounded due to the energy inequality of Leray-Hopf solutions, if ν z ν h approac hes zero the condition holds. Rigorously , the analysis of [ 55 ] relies on the construction of a b oundary corrector B such that w + B is zero on the b oundary , whilst B is only supp orted near the b oundary and of L 2 x norm v anishing with ν z . An in tegration by parts is now facilitated in energy estimates on u − w − B , and the result is achiev ed through a careful analysis of the man y terms inv olv ed. 1.2 Structure of the Noise There are several considerations to b e made when in tro ducing noise into this system. At the first lev el we ignore an y coupling with the viscosit y and ask what form the noise should tak e. W e hav e c hosen a transport-stretching noise, follo wing the principle of Sto chastic A dve ction by Lie T r ansp ort 2 in tro duced by Holm in [ 38 ]. This yields a Stratono vich in tegral in the velocity equation of fluid flo w, given by ∞ X i =1 G i u t ◦ dW i t , G i u t = 3 X j =1  ξ j i ∂ j u t + u j t ∇ ξ j i  where the ( ξ i ) are spatial correlation functions of the driving noise as previously discussed. In [ 38 ] the noise is derived through geometric v ariational principles and is shown to preserve Kelvin’s Circulation Theorem. This theory has bee n expanded upon across [ 16 , 39 , 64 ], and has run in tandem with deriv ations of transp ort noise in fluids through a Lagrangian Reynolds Decomp osition and T ransp ort Theorem giv en by M´ emin [ 57 ], which has b een further dev elop ed in [ 9 , 19 , 61 ]. The theory is b olstered by n umerical analysis and data assimilation presented throughout [ 8 , 13 , 15 , 21 , 22 ] amongst many others. Stratono vich transport noise has also b een derived follo wing a sto c hastic mo del reduction sc heme in [ 20 , 26 , 49 , 54 ]. All of this recen t progress supp orts the classical ideas of [ 4 , 44 , 58 , 59 ], and w e suggest [ 7 , 25 ] for a review of the topic. Secondly , we could choose for the limiting Euler equation to b e sto c hastic or to inv estigate the v anishing noise limit with viscosit y . Both regimes are of indep enden t interest, ho wev er the former is immediately limited b y the av ailable well-posedness theory of the sto chastic Euler equation on a b ounded domain. Indeed, whilst lo cal well-posedness is known for a Lipsc hitz m ultiplicative noise as shown in [ 28 ], or for transp ort noise on the torus demonstrated in [ 14 , 30 ], the case of transp ort noise on a b ounded domain remains op en. W eak solutions were prov en to exist in [ 32 ], how ev er these lack the regularit y required to conduct energy estimates akin to [ 55 ]. Consequen tly , we only consider the regime of v anishing noise. The next task is to decide on a viscous scaling sending the noise to zero, for whic h we use ν 1 2 ha ving b een motiv ated in [ 46 ] as the only noise scaling whic h leads to non-trivial limiting measures (in the limit t → ∞ and ν → 0) for an additiv e noise in t wo dimensions in the absence of a b oundary . The significance of this scaling for energy balance is further underlined in [ 45 , 47 , 48 ] and has b een used to study the inviscid limit problem in [ 5 , 6 , 27 , 36 , 53 ]. The main nov elt y and challenge arises due to the third consideration, which is ho w to introduce the anisotrop y into the noise scaling. As far as we are a ware, suc h a problem has not b een considered. Our approac h is to split the spatial correlation functions ( ξ i ) of the driving Cylindrical Bro wnian Motion in to their horizontal and v ertical components, scaling by ν 1 2 h and ν 1 2 z resp ectiv ely . T o express this let us fix a generalised notation G as an op erator on vector fields φ , f defined b y G ϕ f = 3 X j =1  φ j ∂ j f + f j ∇ φ j  (5) suc h that G i = G ξ i . The horizon tal and v ertical comp onen ts of ξ i are given by ξ h i =  ξ 1 i , ξ 2 i , 0  , ξ z i =  0 , 0 , ξ 3 i  whic h we scale and com bine for the effectiv e spatial correlation function ˜ ξ i = ν 1 2 h ξ h i + ν 1 2 z ξ z i =  ν 1 2 h ξ 1 i , ν 1 2 h ξ 2 i , ν 1 2 z ξ 3 i  . (6) F or simplicit y let us define ˜ G by ˜ G i = G ˜ ξ i , then our anisotropic sto chastic Navier-Stok es equation reads as du t = − B ( u t , u t ) dt − ν h A h u t dt − ν z A z u t dt − P ˜ G u t ◦ d W t . (7) 3 The op erator ˜ G i has the explicit expression ˜ G i u t = 3 X j =1  ˜ ξ j i ∂ j u t + u j t ∇ ˜ ξ j i  = ν 1 2 h 2 X j =1  ξ j i ∂ j u t + u j t ∇ ξ j i  + ν 1 2 z  ξ 3 i ∂ 3 u t + u 3 t ∇ ξ 3 i  = ν 1 2 h G h i u t + ν 1 2 z G z i u t for G h i , G z i as defined in ( 2 ). Thus, we arrive at equation ( 1 ). W e remark that G h i = G ξ h i and G z i = G ξ z i , hence the horizontal and v ertical sup erscripting on G h i and G z i refers to transp ort and stretc hing along the horizon tal and vertical comp onen ts, resp ectively , of the spatial correlation functions. In addition, whilst we ha ve c hosen to incorp orate the noise splitting in to the op erator G , one could equiv alen tly do this at the lev el of W . Indeed if one considers W = ∞ X i =1 ξ i W i and decomp oses in to its horizontal and v ertical comp onen ts W = W h + W z where W h = ∞ X i =1 ξ h i W i , W z = ∞ X i =1 ξ z i W i , (8) then one has the equiv alen t representation ν 1 2 h G h u t ◦ d W t + ν 1 2 z G z u t ◦ d W t = ν 1 2 h G u t ◦ d W h t + ν 1 2 z G u t ◦ d W z t . The splitting ( 8 ) also appeared in [ 24 ] for a T a ylor-Proudman mo del with transp ort-stretc hing noise, a consequence of the 2D-3C nature of the equation. 1.3 Asp ects of the Pro of T o motiv ate a discussion on the main elemen ts of the pro of, let us mention some existing results on in viscid limits for the sto chastic Navier-Stok es equations. As a reminder, the precise statement of our result is giv en in Theorem 3.2 . Sto c hastic versions of Kato’s Criterion hav e been established for a v anishing additiv e noise in 2D [ 53 ], v anishing transp ort t yp e noise in 3D [ 36 ], and non-v anishing additiv e noise in 2D [ 69 ]. F or Navier b oundary conditions in 2D, the limit for non-v anishing addi- tiv e noise [ 11 ] and transp ort-stretc hing noise [ 32 ] has b een prov en. Results on inv arian t measures in 2D for an additive noise without ph ysical b oundary are giv en in [ 1 , 5 , 27 ]. All of these results are for the isotropic case; with anisotropy we are only aw are of one result, given in [ 68 ], dealing with a non-v anishing horizontal viscosit y and a vertical viscosity v anishing as the rotation sp eed of an additional Coriolis term is tak en to infinit y . The authors consider an additive noise acting only in the t wo horizontal directions, and the limit is a damp ed 2D sto chastic Navier-Stok es equation. Therefore, the structure of the noise in ( 1 ) and its inviscid limit is fundamen tally new. The first step in treating ( 1 ) is to con v ert the Stratono vic h equation to Itˆ o form, where the anal- ysis is muc h more fav ourable. One can see that the cross-v ariation will in volv e cross-terms b et ween 4 the horizon tal and vertical viscosities. T o make the conv ersion w e use the compact expression ( 7 ), whic h by following [ 31 ] yields the equation du t = − B ( u t , u t ) dt − ν h A h u t dt − ν z A z u t dt + 1 2 ∞ X i =1 P ˜ G 2 i u t dt − P ˜ G u t d W t (9) up to a ‘cost of a deriv ativ e’. Note that the Itˆ o-Stratono vich corrector has the form ( P ˜ G i ) 2 u t a priori, how ev er w e use the prop ert y that P ˜ G i = P ˜ G i P shown in [ 34 ] Lemma 2.7 to rewrite it in the form of ( 9 ). W e stress that this prop erty is a not consequence of ˜ ξ i b eing divergence-free, as indeed it is not here. In full, equation ( 9 ) reads as du t = − B ( u t , u t ) dt − ν h A h u t dt − ν z A z u t dt − ν 1 2 h P G h u t d W t − ν 1 2 z P G z u t d W t + ν h 2 ∞ X i =1 P G h i G h i u t dt + ν 1 2 h ν 1 2 z 2 ∞ X i =1 P  G h i G z i u t + G z i G h i u t  dt + ν z 2 ∞ X i =1 P G z i G z i u t dt. (10) Ev en the existence of martingale weak solutions to ( 10 ) is unclear. T ypically , the key step in sho wing suc h existence for transp ort noise mo dels is using the divergence-free prop ert y of ξ i to obtain that for vector fields f , g and the inner pro duct in L 2 x , ⟨ ( ξ i · ∇ ) f , g ⟩ = − ⟨ f , ( ξ i · ∇ ) g ⟩ (11) hence in energy estimates, as one meets the term com bining the Itˆ o-Straton voic h corrector and quadratic v ariation, ⟨ ( ξ i · ∇ )( ξ i · ∇ ) u, u ⟩ + ∥ ( ξ i · ∇ ) u ∥ 2 = − ⟨ ( ξ i · ∇ ) u, ( ξ i · ∇ ) u ⟩ + ∥ ( ξ i · ∇ ) u ∥ 2 = 0 . (12) With the additional stretc hing term one has to work a bit more in con trolling comm utators, but ev entually a bound b y ∥ u ∥ 2 is ac hieved primarily due to the same fact. Without the div ergence-free prop ert y we cannot come to the same conclusion; whilst w e could very happily assume that ξ i is div ergence-free, this is ruined b y the viscous scaling in the effective spatial correlation function ˜ ξ i defined in ( 6 ). T o circum ven t the issue one could recognise that it is sufficient to hav e the hori- zon tal and vertical comp onen ts of ξ i div ergence-free, that is ∂ 1 ξ 1 i + ∂ 2 ξ 2 i = 0 = ∂ 3 ξ 3 i . Ho wev er ξ i m ust also satisfy the imp ermeabilit y condition ξ i · n = 0 on ∂ O , whic h simply sa ys that ξ 3 i = 0 on ∂ O = T 2 × ( { 0 } ∪ { 1 } ). Combining with the assumption ∂ 3 ξ 3 i = 0 implies that ξ 3 i = 0 everywhere in O , hence the underlying Cylindrical Brownian Motion would only act in tw o dimensions so the phenomenon that we are inv estigating is trivialised. The analysis of transp ort noise where the spatial correlation functions are not divergence-free app ears completely absent in the literature. Even in the cases where transp ort noise is in tro duced in to a compressible fluid [ 3 , 17 ], the div ergence-free assumption still app ears to facilitate the anal- ysis. W e also mention that for the 2D-3C mo del considered in [ 24 ] the authors assume that ξ i is div ergence-free, which also implies that ∂ 1 ξ 1 i + ∂ 2 ξ 2 i = 0 = ∂ 3 ξ 3 i as ξ i is indep enden t of the third v ariable, ho wev er as their domain is the torus then ξ 3 i do es not need to b e trivial. W e progress our analysis by recognising that in place of ( 11 ), we hav e D ( ˜ ξ i · ∇ ) f , g E = − D f , ( ˜ ξ i · ∇ ) g E − * f ,   3 X j =1 ∂ j ˜ ξ j i   g + 5 and therefore, revisiting ( 12 ), D ( ˜ ξ i · ∇ )( ˜ ξ i · ∇ ) u, u E +    ( ˜ ξ i · ∇ ) u    2 = − * ( ˜ ξ i · ∇ ) u,   3 X j =1 ∂ j ˜ ξ j i   u + ≤      ˜ ξ h i    L ∞ 2 X j =1 ∥ ∂ j u ∥ +    ˜ ξ z i    L ∞ ∥ ∂ 3 u ∥      ˜ ξ i    W 1 , ∞ ∥ u ∥ ≤   ν 1 2 h    ξ h i    L ∞ 2 X j =1 ∥ ∂ j u ∥ + ν 1 2 z ∥ ξ z i ∥ L ∞ ∥ ∂ 3 u ∥   ( ν 1 2 h + ν 1 2 z ) ∥ ξ i ∥ W 1 , ∞ ∥ u ∥ ≤ c δ ( ν h + ν z ) ∥ ξ i ∥ 2 W 1 , ∞ ∥ u ∥ 2 + δ ν h    ξ h i    2 L ∞ 2 X j =1 ∥ ∂ j u ∥ 2 + δ ν z ∥ ξ z i ∥ 2 L ∞ ∥ ∂ 3 u ∥ 2 b y Y oung’s Inequalit y with an y small parameter 0 < δ . In particular δ can b e chosen sufficien tly small suc h that the latter t wo terms can b e hidden in the viscous smo othing, and with δ fixed the first term will v anish with ν h and ν z . Whilst this illustrates an imp ortan t p oin t, in the pro of of our main result one m ust b e muc h more precise in dealing with v arious cross-terms of the t w o directions. An exp ected consequence of the lack of uniqueness for weak solutions of the 3D Navier-Stok es equations is that our solutions to ( 9 ) are only probabilistically weak, meaning that for every ν h and ν z w e can find a probability space and Cylindrical Brownian Motion supp orting a solution, ho wev er a priori the choice of that space ma y w ell dep end on ν h and ν z themselv es. T o consider the exp ectation of the difference of the solutions we will need a single probability space supp orting all solutions, whic h w e construct by first fixing a sequence ( ν k h , ν k z ) and then taking the infinite pro duct of the probability spaces supp orting the corresp onding solutions u k . Our main result is therefore stated for sequences of viscosities. Note that w e do not obtain or need a uniform Cylindri- cal Brownian Motion. W e pro ceed by considering E  sup t ∈ [0 ,T ]   u k t − w t − B t   2 L 2 x  where B is the b oundary corrector from [ 55 ]. T o work with the evolution equation for u k w e m ust lo ok at the lev el of its Galerkin approximation and pass to the limit, given that the nonlinear term do es not b elong to L 2 t H − 1 x . Whilst the same is necessary in [ 55 ] it is only men tioned as a formality , how ev er this do es not app ear trivial. Strong solutions of the Euler equation w are required as H γ b ounds are used for it with 5 2 < γ , but if one passes the Galerkin pro jections P n on to w and tries to conclude b y using H γ b ounds on P n w then this will fail as ∥P n w ∥ H γ explo des with n if w  = 0 on ∂ O (see [ 33 ] App endix D). W e must remov e the pro jections b efore estimating term b y term, generating quan tities of the form ∥ ( I − P n )( w + B ) ∥ H 1 x with some careful b ounds, which do conv erge to zero as w + B satisfies the Diric hlet b oundary condition. W e close this section with a brief comment on the necessity of features of the noise scaling for our method. Indeed if we were to scale b y ν α for some 0 < α < 1 2 , or to a void decomp osing the noise and supp ose that the vertical component only deca ys with ν 1 2 h , then our arguments w ould fail. This is a consequence of the ν 1 2 h , ν 1 2 z scaling for the first order transp ort term matc hing the ν h , ν z scaling of the Laplacian. 6 2 Preliminaries 2.1 F unctional Analytic Preliminaries W e recall that O = T 2 × (0 , 1), and denote the usual Sob olev Spaces W s,p ( O ; R 3 ), H γ ( O ; R 3 ) by simply W s,p , H γ . W e shall use ⟨· , ·⟩ to represen t the L 2 inner pro duct and similarly for the norm, whilst also emplo ying subscripts L p h , L p z as shorthand for L p  T 2 ; R 3  and L p  (0 , 1); R 3  resp ec- tiv ely . This shorthand will also apply for general Euclidean target spaces, which shall b e clear from the con text. Let C ∞ 0 ,σ b e the space of smo oth, compactly supp orted, divergence-free functions from O into R 3 . Then we define L 2 σ , W 1 , 2 σ as the completion of C ∞ 0 ,σ in L 2 and W 1 , 2 resp ectiv ely; W 1 , 2 σ is precisely the subspace of W 1 , 2 consisting of divergence-free and zero-trace functions, whilst H γ ∩ L 2 σ for 1 ≤ γ is the subspace of H γ of div ergence-free functions f satisfying f · n = 0 on ∂ O , where n is the outw ard unit normal vector at ∂ O . The geometry of the domain means that f · n = 0 is equiv alent to f 3 = 0 on ∂ O . W e recommend [ 65 ] for a pro of of these facts. Henceforth we fix a deterministic initial condition u 0 ∈ H γ ∩ L 2 σ for some 5 2 < γ , as the classical regime for existence and uniqueness of lo cal strong solutions of the Euler equation. Namely , referring to [ 2 ] Theorem 1 for example, there exists some 0 < T < ∞ and a unique w ∈ C  [0 , T ]; H γ ∩ L 2 σ  ∩ C 1  [0 , T ] × ¯ O ; R 3  suc h that the iden tity w t = u 0 − Z t 0 B ( w s , w s ) ds (13) holds for all 0 ≤ t ≤ T in L 2 σ . W e recall the b oundary corrector function constructed in [ 55 ] Subsection 2.1. Lemma 2.1. L et 0 < θ b e an arbitr ary p ar ameter and fix 0 < ν z . Ther e exists a function B ∈ C  [0 , T ]; H γ ∩ L 2 σ  such that w t + B t ∈ W 1 , 2 σ for every t ∈ [0 , T ] , of the form B t ( x, y , z ) = M ( z )   w 1 t ( x, y , 0) + w 1 t ( x, y , 1) w 2 t ( x, y , 0) + w 2 t ( x, y , 1) ∂ 3 w 3 t ( x, y , 0) + ∂ 3 w 3 t ( x, y , 1)   : = M ( z ) A t ( x, y ) wher e M ( z ) ∈ R 3 × 3 and M is only supp orte d on  0 , 1 4  ∪  3 4 , 1  . Mor e over the matrix value d function M is c omp ose d of just an upp er left matrix M h ∈ R 2 × 2 and a lower right sc alar M z ∈ R , satisfying the b ounds ∥ M ∥ L 2 z + ∥ z ∂ 3 M ( z ) ∥ L 2 ( 0 , 1 4 ) + ∥ (1 − z ) ∂ 3 M ( z ) ∥ L 2 ( 3 4 , 1 ) ≤ c ( θ ν ) 1 4 ∥ M z ∥ L ∞ z +   z 2 ∂ 3 M ( z )   L ∞ ( 0 , 1 4 ) +   (1 − z ) 2 ∂ 3 M ( z )   L ∞ ( 3 4 , 1 ) ≤ c ( θ ν ) 1 2 ∥ ∂ 3 M ∥ L 2 z ≤ c ( θ ν ) − 1 4 ∥ M ∥ L ∞ z + ∥ ∂ 3 M z ∥ L ∞ z ≤ c for a c onstant c indep endent of θ and ν z . W e remark that the boundary corrector from [ 55 ] is constructed for the domain T 2 × [0 , ∞ ) so only one horizon tal plate is considered. On O we first build the Masmoudi corrector near the b ottom horizontal plate T 2 × { 0 } with supp ort in T 2 × [0 , 1 4 ], and then identically establish the corrector near T 2 × { 1 } with supp ort in T 2 × [ 3 4 , 1] b y replacing z with 1 − z . Our B is then the sum of these correctors. W e also establish b ounds on A below. 7 Lemma 2.2. The function A fr om L emma 2.1 satisfies the b ounds ∥ A s ∥ L ∞ h + 2 X j =1 ∥ ∂ j A s ∥ L 2 h + ∥ ∂ t A s ∥ L 2 h ≤ c ∥ w s ∥ H γ . for some c onstant c indep endent of w . Pr o of. Con trol on the first term follo ws simply from the fact that ∥ A s ∥ L ∞ h ≤ c ∥ w s ∥ W 1 , ∞ ≤ c ∥ w s ∥ H γ b y the usual Sob olev Embedding H γ  → W 1 , ∞ . F or the second we observe that ∥ ∂ j A s ∥ L 2 h ≤ c ∥ w s ∥ H 2 ∂ O ≤ c ∥ w s ∥ H γ b y the trace inequalit y , as 1 2 < γ − 2. In the third term, ∥ ∂ t A s ∥ L 2 h ≤ c ∥ ∂ t w s ∥ H 1 ∂ O ≤ c ∥ ∂ t w s ∥ H γ − 1 using the trace inequality again as 1 2 < ( γ − 1) − 1. Now w e can substitute in the explicit form ∂ t w s = B ( w s , w s ), and using that H γ − 1 is an algebra as 3 2 < γ − 1 the result swiftly follows. W e conclude this subsection with a final inequalit y to b e used in the main pro of. Lemma 2.3. F or 1 < p ≤ ∞ , let f ∈ W 1 ,p ( O ; R ) b e such that f = 0 on ∂ O . Then     f ( x, y , z ) z     L p +     f ( x, y , z ) 1 − z     L p ≤ c ∥ ∂ 3 f ∥ L p for some c onstant c indep endent of f . Pr o of. Starting with the first term, then f = 0 on ∂ O implies that f ( x, y , 0) = 0 and as f ∈ W 1 ,p ( O ; R ) w e may write f ( x, y , z ) = Z z 0 ∂ 3 f ( x, y , η ) dη . Suppressing the dep endence on x, y , note that     f ( z ) z     ≤ 1 z Z z 0 | ∂ 3 f ( η ) | dη ≤ 1 z Z 2 z 0 | ∂ 3 f ( η ) | dη ≤ sup 0

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