MATHEMATICAL JUSTIFICATION OF A SHALLOW WATER MODEL

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1 METHODS AND APPLICATIONS OF ANALYSIS. c 7 International Press Vol. 14, No., pp , June 7 1 MATHEMATICAL JUSTIFICATION OF A SHALLOW WATER MODEL DIDIER BRESCH AND PASCAL NOBLE Abstract. The shallow water equations are widely used to model the flow of a thin layer of fluid submitted to gravity forces. They are usually formally derived from the full incompressible Navier-Stokes equations with free surface under the modeling hypothesis that the pressure is hydrostatic, the flow is laminar, gradually varied and the characteristic fluid height is small relative to the characteristics flow length. This paper deals with the mathematical justification of such asymptotic process assuming a non zero surface tension coefficient and some constraints on the data. we also discuss relation between lubrication models and shallow water systems with no surface tension coefficient necessity. Key words. Navier-Stokes, shallow water, lubrication models, thin domain, free surface, asymptotic analysis, Sobolev spaces AMS subject classifications. 35Q3, 35R35, 76A, 76B45, 76D8 1. Introduction. The shallow water description through the Saint-Venant system is usual for many applications rivers flow, tidal waves, oceans. Although Saint- Venant derived it from elementary principle in 1871, one understands only from less than few years how to derive it formally from the Navier-Stokes equations with free surface and various bottom boundary conditions assuming the depth to be small enough, see for instance [16] and [8] [11] for flat bottom. We also referred the reader to [4] and [5] for shallow water type equation with arbitrary topography. The flow these equations describe is the horizontal flow caused by changes in the height of the pressure surface of the fluid. Whereas the derivation of shallow water equations from the incompressible Navier-Stokes system with free surface, through a depth integration of continuity and momentum equations, is now understood, there is still a lack of mathematical justification of such approaches. In that paper, we present the derivation of a shallow water system, inspired from [16], that we justify rigorously. Note that the rigorous derivation is performed under further technical assumptions that are necessary to have smooth solutions. First, after a suitable scaling of the Navier-Stokes equations with free surface boundary conditions and no slip condition at bottom, the solutions are formally epanded with respect to the aspect ratio ε = H/L between the characteristic fluid height H and the characteristic wavelength L of flow variables. Substituting the asymptotic epansions into the depth averaged continuity and momentum equations, we obtain a shallow water system in a closed form plus a remainder that formally tends to as ε. To justify rigorously such asymptotic, see Theorem 1 and, we establish uniform estimates in classical Sobolev spaces on solutions of the full Navier-Stokes system with respect to ε and we obtain that the remainder in the shallow water system tends to as ε as a consequence. Such shallow water approimate system reads, in a Received November 17, 7; accepted for publication February 18, 8. Université de Savoie, Laboratoire de Mathématiques, UMR CNRS 517, Campus Scientifique, Le Bourget du Lac, France Didier.Bresch@univ-savoie.fr. Université de Lyon, Université Lyon 1, UMR CNRS 58, Institut Camille Jordan, Batiment du Doyen Jean Braconnier, 43, blvd du 11 novembre 1918, F Villeurbanne Cede, France noble@math.univ-lyon1.fr. 87

2 88 D. BRESCH AND P. NOBLE non-dimensional form, h t + hv =, hv t h v + c h Re s 75 h5 ε Wh h = 1 3 v s h εre h where h denotes the height of the free surface, v is the vertical average of the horizontal velocity component. The other quantitites c, s, W, ε, Re are respectively cosθ, sin θ where θ is the slope angle, the Weber coefficient, the aspect ratio of the domain and the Reynolds number given in 7. Remark that, to the authors knowledge, such mathematical justification of shallow water system seems to be new. In our mathematical study Reynolds number is assumed to be fied and some assumptions on various physical quantities sizes are done such as sin θ/ εκ and initial data small enough in some sense, see Theorem 1. The paper is divided in three parts: The first section, inspired from the recent derivation in [16], concerns the formal derivation of a shallow water model. The second section deals with uniform estimates, with respect to the aspect ratio, of solutions to Navier-Stokes equations combining similar process than those used in [1], [14] and similar estimates than those in [13]. Using the previous sections, the third section concludes on the rigourous derivation of the shallow water model. We also give some comments around lubrication models which are included in the shallow water mathematical derivation with no surface tension coefficient necessity. The reader, interested by lubrication models, is referred for instance to [15], [9], [3].. Formal derivation of a shallow water model. In what follows, we derive formally a shallow water model from the incompressible Navier-Stokes equations with a free surface. The fluid flows downward to an inclined plane under the effect of gravity. The Navier Stokes system comes with boundary conditions: we assume a no slip condition at the bottom and continuity of the fluid stress at the free surface, the fluid being submitted here to surface tension forces. Moreover, assuming that the layer of fluid is advected by the fluid velocity, we obtain an evolution equation for the fluid height. We first scale the equations in the shallow water setting. Then, we calculate formally an asymptotic apansion of the flow variables with respect to the film parameter ε. Inserting the epansion in depth averaged continuity and momentum equation, we obtain a shallow water model and identify a remainder that formally tends to as ε..1. Scaling the incompressible Navier-Stokes system. In that paper, we consider a relatively thin layer of fluid flowing down an inclined plane, at an angle θ with respect to the horizontal, under the effect of gravity. Remark that we consider a flat bottom for simplicity. The reader is referred to [5] for formal derivation with arbitrary topography. The fluid is incompressible and the fluid density is constant and set to 1. The flow is supposed to be -dimensional. A coordinate system, z is defined as the -ais downslope along and the z-ais upward normal to the plane bed. The longitudinal and transverse velocity are denoted u, w, the pressure by p and the total flow depth by h. The fluid layer Ω t is the set Ω t = {, z R : < z < ht, }.

3 MATHEMATICAL JUSTIFICATION OF A SHALLOW WATER MODEL 89 The incompressible Navier-Stokes equations read, for all, z Ω t : u t + u u + w u z + p = g sin θ + ν u, w t + u w + w w z + p z = g cosθ + ν w, u + w z =. The ν constant denotes the fluid viscosity and g is the gravity constant. The Navier- Stokes equations is supplemented with boundary conditions. More precisely, we assume at the bottom a no-slip condition on a flat bottom: 1 u z= = w z= =. At the free boundary, we assume that the atmospheric pressure p atm is constant and set to. The fluid is submitted to surface tension forces. Hence, the continuity of the fluid stress at the free surface, written σn = κn with σ = pid+ ν u + t u, yields the following conditions: p z=h + κ h 1 + h h = ν 1 h u z=h, u z z=h + w z=h = 4 h 1 h w z z=h, where κ measures the capillarity the curvature. The layer fluid is advected by the speed u = u, w, giving the evolution equation for the fluid height the usual kinematic condition 3 h t + h u z=h = w z=h. 4 This set of equations possesses a classical steady solution, so called Nusselt flow. The height of the fluid h, t = H is constant, the transverse component velocity w is equal to and the longitudinal component speed u has a parabolic dependence with respect to z U, z = g ν H z z sin θ. 5 Moreover, the pressure is hydrostatic: p, z = gh zcosθ. 6 Adimensionalization and rescaling. We shall use the Nusselt flow in the sequel to scale the Navier Stokes equations. With the notation U = gh /ν, we define the rescaled variables z = Hz, h = Hh, = L, ε = H L, u = U u, w = ε U w, p = ghp, t = L t, Re = HU U ν, W = κ gh. In what follows, we denote c = cos θ, s = sin θ. Under the shallow water scaling and dropping the, the Navier-Stokes equations read for all, z in Ω t : u t + u u + w u z + Re p = s εre + 1 εre ε u + u zz, w t + u w + w w z + ε Re p z = c ε Re + 1 εre ε w + w zz, 8 u + w z =. 7

4 9 D. BRESCH AND P. NOBLE The no-slip condition u z= = w z= = and the evolution equation for h h t + h u z=h = w z=h, 9 are unchanged whereas the continuity of the fluid stress reads p z=h + ε W h 1 + ε h ε h = ε 1 ε h u z=h, u z z=h + ε w z=h = 4ε h 1 ε h w z z=h. 1 We shall prove that in the asymptotic regime ε, the solutions of the Navier- Stokes equations 8 with rescaled boundary conditions are close to the stationnary Nusselt flow and have an asymptotic epansion with respect ε. This is done in the net section... Asymptotic epansion of solutions. Under the shallow water scaling, the flow variables are close to a Nusselt flow. More precisely, we write the Navier- Stokes equations as a differential system in the crosstream variable z with boundary conditions at z = and z = h. First, the longitudinal component velocity u satisfies u zz + s = ε p + εre u t + u u + w u z ε u = Ψ u u, w, p t,, z, u z= =, u z z=h = ε 4ε w z=h + 1 ε h h u z=h = Π u u, w t,. 11 The functions Ψ u u, w, p, Πu u, w satisfy formally the estimates Ψ u u, w, p = O ε + εre, Πu u, w = O ε. 1 Integrating 11 with respect to z, we obtain ut,, z = sht, z z + F u u, w, p t,, z, 13 where the function F u u, w, p is defined by F u u, w, p t,, z = zπ u u, w t, z z Ψ 14 u u, w, p t,, ydy dz. As a consequence, the longitudinal velocity component u epands in the form ut,, z = sht, z z + O ε + εre. 15 The profile for u is then closed to a Nusselt type flow. We show in a similar way that the pressure is close to a hydrostatic distribution. More precisely, the pressure p

5 MATHEMATICAL JUSTIFICATION OF A SHALLOW WATER MODEL 91 satisfies the ordinary differential equation with respect to z: p z + c = ε w zz + ε w ε Re w t + u w + w w z = Ψ p u, wt,, z, p z=h = κ Re h 1 + ε h 3 = κ Re h + Π p u t,, ε u z=h 1 + ε h 1 ε h where the constant κ, defined by κ = ε W, is assumed to be of order O1 in order to see the capillarity effects. The functions Ψ p u, w, Π p u satisfy the formal estimates We integrate 16 with respect to z and find 16 Ψ p u, w = O ε, Π p u = O ε. 17 p, z, t = κ Re h + ch z + F p u, w, p, z, t, 18 where the function F p u, w, p is defined by F p u, w, p = Π p u, t Ψ p u, wt,, ydy. 19 z The pressure p has, in a first approimation, an hydrostatic distribution and epands in the form p, z, t = κ Re h + ch z + Oε. The transverse component velocity w is determined, using the free divergence condition w z + u = and the no-slip condition w, =. More precisely, we find wt,, z = z u t,, ydy. We deduce that w has the asymptotic epansion wt,, z = z u t,, ydy = s h z + O ε + εre. 1 In the following, we describe an iterative scheme to compute a formal epansion of Navier-Stokes solutions to any order in ε. First, we define the functions u and p as u t,, z = sht, z z, p t,, z = κreh / + c ht, z. We write the equations 13,18 in the form u u = p p + Fu, p, ε, 3 where the function Fu, p, ε is defined by Fu u, w, p Fu, p, ε =, wt,, z = F p u, w, p z u t,, ydy. 4

6 9 D. BRESCH AND P. NOBLE Then, any solution of the Navier-Stokes equations appears as a fied point of an operator that is formally Oε + εre-lipschitz on any bounded set. Following the proof of the fied point theorem, we define the sequence of functions u n, w n, p n n 1 such that u n+1 u = + Fu n, p n, ε, 5 z and w n t,, z = p n+1 u n t,, ydy. p We assume that all the derivatives of u, w, p remain bounded and the sequence u n, w n, p n is bounded. We can prove by induction the formal estimate n 1 ma u u n, w w n, p p n = O ε + εre n+1. 6 As a consequence, we clearly see that the n-th term of the sequence of functions u n, w n, p n is a formal approimation of a solution of the full Navier-Stokes n 1 equations up to order O ε + εre n+1. In what follows, we use that approimation of the solutions to close the depth average continuity and momentum equations and obtain a shallow water model..3. The shallow water model. In the sequel, we write a shallow water model for the fluid height h and the total discharge rate hv = u., y,.dy. Here v represents the mean velocity along the depth flow. On the one hand, integrating the divergence free condition u + w z = along the fluid height and using the kinematic equation for h namely h t + h u z=h = w z=h, we find h t + u, zdz = h t + hv =. 7 Note that the equation is eact and already in a closed form. Let us now write an evolution equation for hv = u,, ζdζ. For that purpose, we integrate the evolution equation on the velocity component u along the flow depth and substitute the boundary conditions 1 into the resulting equation: u dz t + u dz + Re 1 εre s h u z, h p dz + κ h 1 + ε h 3 = + ε Re u dz. 8 In order to write a momentum equation in a closed form, we shall calculate an epansion of the averaged quantities in 8 with respect to ε, h, v. First, let us note that hvt, = t, = s h3 3 + ut,, ζdζ = t, t, δ ut,, ζdζ. u t,, ζdζ + t, δ ut,, ζdζ,

7 MATHEMATICAL JUSTIFICATION OF A SHALLOW WATER MODEL 93 Here, the function δu is simply δu = u u. From the previous section, we deduce the formal estimate δu = O ε + εre. We compute the average quantities and t, p: u t,, ζdζ = = t, We substitute s h = 3v 1 h t, u t,, ζdζ + 8ht, s h t, + 15 where the function R 1 1 is defined by R 1 1 t, = t, v t, t, u u u t,, ζdζ, u u t,, ζdζ. 9 δu into 9. The average quantity u reads u t,, ζdζ = 6 5 hv t, + R 1 1, 3 u u t,, ζdζ t,, δut,, ζdζ + t, 15 ht, δut,, ζdζ. Inserting the estimate u u = O ε + εre into the epression for R 1, we find that, formally, R 1 = O ε + εre. We easily compute the average quantity p: with p,, ζdζ = κ Re h h + c h + R 1, 31 R 1 t, = t, p p t,, ζdζ. We easily prove that, formally, R 1 satisfies the estimates R 1 = Oε + εre. Substituting 3,31 into 8, one obtains hv t hv + Re ch κh h = 1 s h uz, + R 1, 3 εre where the function R 1 is defined by R 1 = R 1 1 R 1 + ε 1 Re u,, ζdζ + κ 1 + ε h 3 h h and satisfies the formal estimate 33 R 1 = O εre Re 1.

8 94 D. BRESCH AND P. NOBLE Dropping the small term R 1, we can see that the momentum equation 3 is almost in a closed form and yields a shallow water model for h, hv. There remains to eliminate u z,. Due to the presence of the singular factor 1/εRe, we cannot directly use the epansion u z = u 3v z + Oε + εre = + Oε + εre. h In that case, the remainder would be of order O1. As a consequence, we need an asymptotic epansion of u z and hv with respect to ε and function of h and its derivatives up to order 1. For that purpose, we use the function u 1 introduced in the previous section. Recall that we have the estimate u u 1 = O ε + εre. We aim to compute an epansion of hv and u z., up to order 1: let us epand u with respect to ε. We find, using the iterative scheme introduced previously that u = u where the notation z y z ε p + εre u t + u u + w u 1 z + εre R, 34 y z y f is used to defined the function of t,, z as f t, = Moreover the function R 1 is defined as R 1 = ε Re z y z t, y u u z y ft,, zdzdy. Re p p ε z w h + w zh Re 1 ε h w h + w z h 1 ε h + z y u u t + uu u u + wu z w u z. It is easily seen that R 1 = O ε1+re+re 1. In order to simplify the notations, we denote u 1 the function u 1 = p Re u t + u u + w u z. Then, the component velocity u epands in the form From 35, we deduce that u z z= = sh + ε z u 1 z= + εre z R 1 z=, h vt, = s h3 t, 3 u = u + ε u 1 + εrer ε t, u 1 t,, ζdζ + εre t, R 1 t,, ζdζ.

9 MATHEMATICAL JUSTIFICATION OF A SHALLOW WATER MODEL 95 Then, eliminating sh from the epansion of u z z=, one obtains the asymptotic epansion u z z= = 3 v h + ε z u 1 z= 3 h u 1 + εre z R 1 z= 3 h R We substitute the epansion 36 into the momentum equation 3 hv t h v + Re ch κh h = 1 3 v s h τ + R1 + R, 37 εre h with τ = 7 1 s h 4 h sh h t and R = z R 1 z= 3 h R 1. Dropping the small term R = R 1 +R = Oε1+Re+Re 1, we obtain a shallow water model in a closed form h t + hv =, hv t h v + Re ch κh h = 1 εre 3 v 38 s h τ. h The term τ can be written as the sum of a conservative term and a remainder of order Oε + εre. More precisely, we substitute the epansion hv = s h3 3 + u u, into the conservation law h t + hv =. As a result we find that h t = s h h u u, and the function τ reads τ = s 75 h5 sh 4 the notation R = R + sh 4 u u. If we introduce u u, the momentum equation 37 reads hv t h v + Re ch s 75 h5 κh h = 1 3 v s h + R, 39 εre h The remainder R is formally of order Oε1 + Re + Re 1. Dropping this small term, we obtain a conservative form of the shallow water model 38: h t + hv =, hv t h v + c h Re s 75 h5 4 κh h = 1 3 v s h. εre h This concludes the formal derivation of a shallow water model from the full Navier- Stokes equations with free surface. Formally, the remainder term R resp. R tends to as ε. In what follows, we compute a priori estimates in classical Sobolev norms on the solutions u, w, p of the full Navier-Stokes system. We shall prove rigorously that the convergence of R resp. R to holds true in a suitable norm and thus justify rigorously the formal derivation presented here. Finally, we can easily obtain a shallow water model without capillarity, setting the capillary coefficient κ to in 4. However, we shall see later on that capillarity is important to obtain energy estimates on solutions to Navier-Stokes equations and to justify the asymptotic process.

10 96 D. BRESCH AND P. NOBLE 3. Uniform estimates and Navier Stokes Eqs. In that section, we compute a priori estimates of solutions of the full Navier-Stokes equations in the shallow water scaling. The method is based on the energy estimates obtained by T. Nishida et al. [1] to prove the eistence of global solutions to the Navier-Stokes flows down an inclined plane. First, the fluid domain is transformed into a fied domain through a suitable change of variable. Then, the Navier-Stokes equations are written in the neighbourhood of a Nusselt flow and the authors derive a priori estimates on the perturbations of that steady solutions. Classically, for sufficiently small initial data, the solutions are proved to be global solutions and decays to as t. In order to simplify the discussion, the functions are periodic in the -direction. This is the point of view adopted in that paper. The main issue here is to obtain estimates in the shallow water asymptotic regime ε. The shallow water scaling make the diffusion term anisotropic: in order to adapt the method of T. Nishida et al., we have chosen to transform the fied domain into a thin domain of size ε in the z-direction. In that setting, the diffusion is isotropic and we can completely follows the approach of T. Nishida et al.. Their estimates are mainly based on estimates on the Stokes problem. At this stage, we follow ideas of R. Temam and M. Ziane [13] on the eistence of solutions to Navier-Stokes equations in thin and fied domain: in particular they obtain Sobolev inequalities and estimates on the Stokes problem in a thin domain. As a result of this analysis, provided that the perturbation of the Nusselt flow is small in a suitable weighted norm, we prove uniform estimates on solutions of the Navier- Stokes equations in the shallow water regime ε. In all the paper long, we will denote r and r the L r norm respectively in two-dimensional in space domains and in the one-dimensional torus T Preliminaries. In that section, we follow the approach initiated by J.T. Beale [1, ] to prove the eistence of solutions to the Navier-Stokes equations with a free boundary. We assume that the functions are periodic in the variable and consider small perturbations of the Nusselt flow: for a suitable change of variable, we write the Navier-Stokes equations in a fied thin domain. Then, we recall briefly the fundamental inequalities in thin domains that are used in the paper: Sobolev inequalities and estimates for the Stokes problem in thin domains Reduction of N-S equations to a fied thin domain. The rescaled Navier-Stokes equations are given in Ω t by u t + u u + w u z + Re p = s εre + 1 εre ε u + u zz, w t + u w + w w z + ε Re p z = c ε Re + 1 εre ε w + w zz, 41 u + w z =, where Ω t = {, z T R : < z < ht, } is the fluid domain. The system is supplemented with boundary conditions: The no-slip condition on the bottom u z= = w z= =, and the kinematic equation at surface for h h t + h u z=h = w z=h.

11 MATHEMATICAL JUSTIFICATION OF A SHALLOW WATER MODEL 97 The continuity of the fluid stress reads p z=h + ε W h 1 + ε h ε h = ε u z=h 1 ε h, u z z=h + ε w z=h = 4ε h 1 ε h w z z=h. In order to simplify the N-S equations, we introduce the hydrostatic correction of the pressure p = p c z. We also assume that h = 1 + h and h is small. We clearly see that, in the shallow water scaling, the diffusion is anisotropic: in order to recover an isotropic viscosity and applies the idea of T. Nishida et al. [1], we write the equations in a thin domain and introduce the functions ũ, w and P so that 4 ut,, z = ũt,, ε z, ε wt,, z = wt,, εz, P = p ε. 43 We drop the for the sake of simplifying the notations. The divergence free equation reads u + w z =, whereas the momentum equations are given, for all, z T, ε1 + h, by u t + u u + w u z + µ P = µ ε s + µ u, w t + u w + w w z + µ P z = µ w, where µ is the rescaled viscosity µ = ε. In that setting, we write the continuity of Re the fluid stress at the boundary z = ε1 + h in the form P c ε 1 + h + ε W h 1 + ε h ε h = u 1 ε h, u z + w = 4ε h 1 ε h w z. The no-slip condition is unchanged and the kinematic equation for h reads in T h t + h u z=ε1+h = w z=ε1+h. 46 ε In that framework, the steady solution, so called Nusselt flow, is defined by h = and U, z = s ε ε z z, P = c ε. 47 We consider the Navier-Stokes equations in the neighbourhood of the Nusselt flow and introduce the functions ũ, P so that u = U + ũ, P = c ε + P. 48 We remove the to simplify the notation. The momentum equations read: u t + U + uu + w U z + u z + µ P = µ u, The divergence free equation is written w t + U + uw + w w z + µ P z = µ w. 49 u + w z =. 5

12 98 D. BRESCH AND P. NOBLE The boundary conditions at the free surface z = ε1 + h are given by P c ε h + ε W h 1 + ε h ε h = u 1 ε h, u z + w s ε h = 4ε h 1 ε h w z. 51 We obtain for the evolution equation for h h t + s1 h + u., ε1 + h h = w., ε1 + h. 5 ε In the sequel, following the approach initiated by J.T. Beale [1, ], we aim to write the N-S equations 49,5, in the fied thin domain Ω = T, ε. For that purpose, let us first remark that, integrating the divergence free condition along the flow depth and using the boundary condition 5, one finds ε1+h h t + u =. As a consequence, we obtain d h, td = and the spatial mean of h is constant. dt T In the sequel, we assume < h >= T h, td =. Under that hypothesis, we define h the etension of h to Ω as: ht,, z = k h k t 1 + k z ε eik, T. 53 Now, we define the change of variable from Ω to Ω t in the form: Θ t : T, ε T, ε1 + h,, z, z1 + ht,, z. 54 The Jacobian matri of the transformation is 1 DΘ t = zh 1 + z zh. 55 The function Θ t is then a diffeomorphism between Ω and Ω t provided that J = 1 + z zh > h is supposed to be small. In order to preserve the divergence free condition, we introduce the Jacobian decomposition of the fluid velocity u w = 1 J DΘ t Θ 1 ũ t w Θ 1 t. 56 Dropping the from the equations, the momentum equations transform into u t + U u + U z w + µ p = µ u + F 1, w t + U w + µ p z = µ w + F, 57

13 MATHEMATICAL JUSTIFICATION OF A SHALLOW WATER MODEL 99 whereas the divergence free condition u +w z = is preserved. The functions F i, i = 1, in 57 contain the nonlinear terms: F 1 = J t u J + h t z z u J u u J u h z u z J J J z u J + U J u + z h z u J z h u U z + 1 J z u J w z u J + µh z p z + 1 Jp + µ J u 1 J + J u 1 J J h z J z u J h z z u J + h z z h z J z u J z 1 J z u J + J 1 zz u + J z 1 J u z + u J zz 1 J, F = h t z u J + h z J U u + h z J U z w J J u w h z J u + u w J + h z J U z u u w J h z J U u h z U z u J J + µ zh z h z z p + h z p + µ 1 J u z J + h z J u h z J u z + µ h z J u J + h z J J u J h z z u J + h z u J + h z u J + h z J u h z J z h z J u + h z J J J h z J z J J u J + h z J J z u J + h z J J z u + J z u J J + J J z u J J J z u J In that setting, the continuity of the fluid stress at the fied boundary z = ε reads, for all in T: u z z=ε + w z=ε s ε h = H 1, P z=ε c ε h + ε W h + u z=ε = H, 59 where the functions H i, i = 1, contain the nonlinear terms and are given by 4ε u H 1 = 1 ε h 1 + h h 1 + h zz u + ε h h zz u 1 + h ε h u 1 + h + ε h wz + ε h u z 1 + h 1 + h + u z h z J + ε h zz u 1 + h 3, H = ε W h ε h 3 h u h + u 1 + h ε h 1 ε h ε h h 1 ε h 1 + h zz u ε h h zz u 1 + h 3. 6 Finally, the time evolution for h is described by the equation in T. h t + s1 h h = w z=ε. 61 ε

14 1 D. BRESCH AND P. NOBLE The N-S equations with free surface are now formulated in a fied thin domain. Following the approach of T. Nishida et al., we shall compute estimates in Sobolev norms on solutions of the Stokes problem. In order to estimate the nonlinear terms, we shall also write some fundamental Sobolev inequalities in thin domain with a particular attention on the optimal dependence of Sobolev constant with respect to ε. This is done in the net section Fundamental inequalities in thin domain. In what follows, we write the classical Poincaré inequalities and Sobolev injections for the thin domain Ω = T, ε. The resulting inequalities are summarized in the following lemma. Lemma 1. Assume that u z= = or u z=ε =, there eists a constant C independent of ε so that the following estimates hold u C ε z u, u L 6 Ω C u, u L Ω C ε z u + z u, In the general case, these inequalities read: u L 6 Ω C ε 1 3 u 1, u L Ω C ε 1 u, For a proof of these Sobolev-type inequalities, see the paper of R. Temam and M. Ziane [13] on Navier-Stokes equations in thin domain. Those inequalities are deduced from Poincaré inequalities Proposition.1, Agmon s inequalities in thin domain Proposition. and anisotropic Ladhyzhenskaya s inequality Proposition.3. A straightforward application of this lemma yields the further Sobolev-type inequalities. We easily prove the following result. Lemma. Denote H s the subspace of Sobolev space Hs Ω vanishing either on z = or z = ǫ. There eists a constant C, independent of ε so that the following inequalities hold true: u v C ε u + u v, u, v H L, u v C ε 1 u v, u, v H L, Moreover, we can also prove that u v C ε 1 6 u v, u, v H 1 H1, u v C ε 1 6 u v 1, u, v H 1 H1, u v C ε 1 u 1 v 1, u, v H 1 H 1. The proof of the first set of inequalities is obtained using the classical Sobolev embedding H Ω L Ω, with the optimal Sobolev constant defined in Lemma 1. The second set of inequalities results from the embedding L 6 Ω L 6 Ω L Ω together with the estimate with C is independent of ε. u v L Ω Cε 1 6 u L 6 Ω v L 6 Ω,

15 MATHEMATICAL JUSTIFICATION OF A SHALLOW WATER MODEL 11 Applying Lemma 1, we can also obtain estimates of the etension h of h on the domain Ω. The proof of the following lemma is straightforward. Lemma 3. There eists a constant C independent of ε such that for any α = α 1, α and n = α, the etension h of h on Ω satisfies the estimates α j h C ε n h, α j h C h n 1, Moreover, using the fact that z k+1 h, ǫ = and Lemma 1, one obtains the uniform estimates: h C h, h z C ε h 5, h z C ε h. Proof. We prove the estimates on α j h and start with the derivatives in the -direction. It is an easy computation to show that n h = k h k k n ε With a change of variable in 6, one proves that n h = k h k k n 1 fε k, f = dz 1 + k z ε. 6 du 1 + u. The function f is bounded and f,, +. We easily prove that In a similar way, one can prove that n h ε n h, n h f h n 1. n zh = k h k k n+1 gε k, g = u du 1 + u 4 The function g is bounded and g,, +. We easily deduce the estimates on n z h. The bounds on the higher order derivatives of h follows similarly. Following the method of T. Nishida et al. [1] to obtain a priori estimates on the solutions of the full Navier-Stokes equations, we shall write energy inequalities involving the fluid velocity u, w, the time and transverse derivatives t α β u, w and their gradients. We estimate the other derivatives of u, w and the pressure p using the regularity in the Stokes problem. The main issue here is to obtain the dependence of the regularity constant with respect to ε. In what follows, we consider the Stokes problem on the thin domain Ω = T, ε with Dirichlet boundary conditions. We shall prove the following result. Ω: Lemma 4. Let f H 1 Ω, φ H 3 T and consider the Stokes problem in u + p = f, div u =,

16 1 D. BRESCH AND P. NOBLE with the boundary conditions u z= =, u z=ε = φ. There eists a constant C independent of ε so that for i =, 1: u +i + p i C f i + φ 1 1+i + φ i + u ε 3 ε 5 + u. 63 Proof. We first prove the case i =. More precisely, we show that there eists a constant C independent of ε so that: u + p C f + φ φ + u ε 3 ε The case i = 1 follows immediately from that particular case: we derive the Stokes equations in the -direction and use the case i = to estimate the functions α β z u, p for any α 1, α + β 3. The other derivatives of w and z p are estimated using the divergence free condition whereas the derivatives of u are estimated using the Stokes equations and in particular u zz = f 1 + p u. Now we prove 64 and write the function u in the form u = u, w. In the sequel, we shall note C a constant independent of ε. Using the Poincaré inequality and the divergence free condition, we easily prove that u + w + u + w C z u + u. From the Stokes equation, we deduce that u zz C f + p + u. 65 Assume for the moment that p satisfies the estimate p C f + φ 1 1 ε 3 We want to estimate z u. An integration by parts yields z u = u, εu z, εd u u zz. Ω T + φ. 66 ε 5 Using the boundary conditions and Cauchy Schwarz inequalities, one finds z u φ 1 u z, ε + ε z u u zz. Ω It is easily proved that u z ε,. C 1 ε z u + ε u zz. Using Young s inequality and the estimate 65 on u zz, we obtain z u C f + φ 1 ε 3 Ω + φ. 67 ε 5

17 MATHEMATICAL JUSTIFICATION OF A SHALLOW WATER MODEL 13 We prove the estimate 66 on p. From the Stokes equation and the divergence free condition, we easily prove that Let v H Ω so that z p f + u. v z= =, v z=ε = φ, div v =. Such a function eists due to potential theory [7]. Let us write u in the form u = v+ũ. Then we find that ũ, p satisfies the Stokes problem ũ + p = f + v, div ũ =, ũ z= = ũ z=ε =. 68 With a symmetry argument, R. Temam and M. Ziane [13] Lemma.6 proved that in such a case, there eists C independent of ε so that ũ + p C f + ṽ. The estimates obtained in [7] for ṽ would give u + p C f + φ 1 3 ε 3 + φ ε 5 The inequality 69 is not optimal here. In the sequel, we improve that estimate provided that we add a correction term u. Using the divergence free condition, the lemma is proved provided that min u φ1 1 C u X ε 3 where X is the functional space { X = u H Ω : u z= =, u z=ε = φ 1, In order to prove 7, we decompose u and φ into Fourier series: u, z = k Z + φ, 7 ε 5 ǫ u k ze ik, φ = k Zφ 1,k, φ,k e ik. u, zdz = φ }. Note that φ shall satisfy the compatibility condition φ, =. Then the problem reduces to find the optimal solution u k, k Z so that ε under the constraints u k z k u k z dz = min v z= =, v z=ε = φ 1,k, ε v z k vz dz, ε vzdz = iφ,k k. 71

18 14 D. BRESCH AND P. NOBLE It is easily proved that if u k is a solution of the minimization problem, then for any φ C c, ε, the function u k satisfies the Euler-Lagrange equation ε u 4 k k u k + k 4 u k φzdz =. As a consequence, the function u k satisfies u 5 k k u 3 k + k 4 u k written = and can be u k z = A + B coshkz + C sinhkz + Dz coshkz + Ez sinhkz. The minimization problem then reduces to finding A k,..., E k so that ε k A k k D k sinhkz + E k coshkz whereas the constraints 71 read A + B =, = min ε k A k D sinhkz + E coshkz, A + B coshkε + C sinhkε + Dε coshkε + Eε sinhkε = φ 1,k, εa + B sinhkε + C coshkε 1 k k 1 coshkε + εk sinhkε εk coshkε sinhkε + D k + E k = iφ,k k. 7 We eliminate A from the constraints 7. Furthermore, we can consider, without loss of generality, the minimization problem in R: ε k B k + k D k sinhkz + E k coshkz under the constraints = min ε k B + k D sinhkz + E coshkz, Bcoshkε 1 + C sinhkε + Dε coshkε + Eε sinhkε = φ 1,k, Bsinhkε kε + Ccoshkε 1 +Dε sinhkε coshkε 1 + Eε coshkε sinhkε = φ,k, 73 k k where φ i,k, A k,..., E k R. Here, we choose D = E =. In that case, we find B = coshkε 1 1 coshkε + kε sinhkε φ sinhkε 1,k + 1 coshkε + kε sinhkε φ,k. In order to simplify the notations, we introduce A k 1, A k so that B = A k 1 φ 1,k + A k φ,k

19 MATHEMATICAL JUSTIFICATION OF A SHALLOW WATER MODEL 15 and we note u, z = k Z u kze ik the solution associated to that Fourier decomposition, we find that u k ε k 4 A k 1 φ 1,k + A k φ,k. We estimate k ε k 4 A k 1 φ 1,k : separating the different cases kε = Oε, kε = O1 and kε 1, we obtain k ε k 4 A k 1 φ 1,k C φ 1 1 ε 3, with C, a constant independent of ε. Similarly, we prove that As a consequence, we have proved that k ε k 4 A k φ,k C φ ε 5. p C f + φ 1 1 ε 3 This completes the proof of the lemma. + φ. ε 5 We shall also give a formulation of a Körn s inequality in the thin domain setting. H. Ito has proved, in [1], the following result Lemma 5. Let u H 1 Ω so that Then the following Körn s inequality holds u z= =, div u =. Du L Ω = u + u t L Ω 3 u L Ω. 3.. Classical estimates in the shallow water scaling. In the following, we shall prove that the energy estimates obtained by T. Nishida and coauthors in [1] still hold true in the shallow water scaling. We recall the Navier-Stokes system, written in the fied domain Ω = T, ε. u t + U u + U z w + µ p = µ u + F 1, w t + U w + µ p z = µ w + F, u + w z =. 74 The continuity of the fluid stress at the fied boundary z = ε reads in T, u z z=ε + w z=ε s ε h = H 1, P z=ε c ε h + ε W h + u z=ε = H, 75

20 16 D. BRESCH AND P. NOBLE whereas h satisfies the equation in T: h t + s1 h h = w z=ε. 76 ε We recall that the full epressions of the nonlinear terms F 1, F in the Navier-Stokes equations and H 1, H in the boundary conditions are given by 58,6. The nondimensional constant µ is given by µ = ε/re. In what follows, we shall denote βh the symmetric matri βh = bh Id bh 1 with z z h bh = z h z h z z h z h 1 + z z h The purpose of that section is to prove the following result. Theorem 1. There eists ε 1 > such that if. sinθ εκ ε 1, h ε + h + ε 1 4 h 5 + ε u, w H ε 1, 77 then there eists γ 1, so that the following energy estimate holds Here E, K, G denote the functions Eu, w = Eu, w + µ Kh + γ 1 µ Gu, w. t j u, w + Id + βh t u, t u, j= Kh = c j h + t h + κ j+1 h + t h Gu, w = j= j= u, j w + t u, w. j= Remark. As we shall see later, this estimate is sufficient to obtain estimates of the fluid speed u, w in the classical Sobolev spaces H Ω, H 3 Ω. Proof. Following the paper of T. Nishida and coauthors [1], we shall decompose the proof into several steps. First, we show the following result. Proposition 1. There eists ε 1 > so that for h + ε 1 3 h 5 + ε u, w ε 1, the solution u, w, p of Navier-Stokes system satisfies for i =, 1 u +i + p i C µ u t i + s + Re 1 u + u + u. 78 In order to prove that proposition, we consider the Stokes problem

21 MATHEMATICAL JUSTIFICATION OF A SHALLOW WATER MODEL 17 µ u + µ p = F u t Uu U w, t, div u =, u z= =, u z=ε = u z=ε, w z=ε. 79 We apply Lemma 4 to the Stokes problem 79 and obtain µ u +i +µ p i C The boundary terms are bounded by F u t Uu U w, t i +µ u z=ε 1+i ε 3 u z=ε 1+i ε u + u, w z=ε i ε ε u + u. + w z=ε i. 8 ε 5 We obtain the second inequality using the divergence free condition in the form w z = u and a Poincaré inequality. Similarly, the linear term Uu +U w, t satisfies for i =, 1 Uu + U w, t i C s u + u + u, whereas u t i C u t. Inserting those inequalities into 8, we obtain µ u +i + µ p i C F i + u t + s + Re 1 u + u + u. 81 Now we consider the nonlinear terms F = F 1, F. We can write F in the form F = µ bh p + F where F contains only derivatives of h and u. We easily prove that µ bh p i C h + ε 1 3 h 5 p i. Under the hypothesis of the proposition and for ε 1 sufficiently small, the inequality 81 still holds with F replaced by F. The second order terms of F have the form µ a j,k h, h z j k u with j + k = and are estimated by µ a j,k h, h j z k u i µ Cε 1 h + ε 1 3 h 5 u +i. Net, we consider the nonlinear terms involving a third order derivative in h. More precisely, those terms have the form µ b j,k h, h, zu j k zh with j +k = 3. We prove easily that b j,k h, h, zu j k z h Cε 1 ε 3 h 5 u + u, b j,k h, h, zu j k z h 1 Cε 1 ε u h 7. 8 We shall obtain a bound for h 7 : for that purpose, we consider the equation on the boundary z = ε: p z=ε c ε h + εwh + u z=ε = H. 83

22 18 D. BRESCH AND P. NOBLE We apply the operator 3 to 83 and take the scalar product with 7 h. Using Cauchy Schwarz inequality, we obtain c ε h 5 + εw h 7 h 7 3 p + 5 uε,. + 3 H We set κ = ε W. In the derivation of the shallow water model, we assumed that κ = O1: in particular, κ can be set to. In what follows, we assume for the moment that κ >. We deduce from 84 that κ h 7 Cε p 1 + u + 3 H. 85 The Sobolev space H 3 T being an algebra, we easily prove that 3 H C u + u + ε u 3 + Cε u + h 5 h 5 h 7. Inserting that inequality into 85, one obtains κ Cε 3 u + h 5 h 5 h 7 Cε p 1 + ε u 3 + u + u. 86 Substituting 86 into 8, we easily obtain that for ε 1 sufficiently small, those terms have no influence in 78. Furthermore, the other nonlinear terms are of lower order and have no influence here. The proof of Proposition 1 is then complete. We are now in a position to obtain energy estimates: according to Proposition 1, we shall derive estimates for α u with α =, 1, and t u and their gradients. We first consider the derivatives in the -direction. We note γ a constant defined by γ = 3 sre. In the following, we shall denote u = u L Ω, u, v = u, v L Ω and h = h L T. One can prove the following result. Proposition. Setting u = u, w, the fluid velocity u and the fluid height h satisfy u t µ + c h + κ h + γ u s hu z=ǫ 87 ε T + H 1 u z=ǫ H w z=ǫ + κ s h h h + 1 F, u, µ T 84 u t µ + c h + κ h + γ u s h u z=ǫ 88 ε T + H 1 z=ǫ u H w z=ǫ sch κh h h + 1 µ F, u, T u t + c h + κ h + γ u s h u z=ǫ 89 µ ε T + H 1 u z=ǫ H w z=ǫ sch κh h h + 1 µ F, u. T

23 MATHEMATICAL JUSTIFICATION OF A SHALLOW WATER MODEL 19 Denote F = F +µ bh p and β = bhid bh 1. The function u t and h t satisfy the estimate 1 + βhu t, u t t + c h t + κ h t + γ u t 9 µ s h t u t z=ǫ + H 1 t u t z=ǫ H t w t z=ǫ sch t κh t t h h ε T T + 1 µ F t, u t + 1 t bh p z=ǫ, u t z=ǫ + t βhu t z=ǫ, u t z=ǫ + 1 βh Ft + µ t b p Uu t U w t, t z=ǫ, u t z=ǫ. µ Proof. The method for proving 87 9 is similar to the one employed by T. Nishida and coauthors: see [1] for more details. We shall mention here that we have applied Lemma 5 and used a Körn s inequality Du L Ω cω u L Ω, with a constant cω = /3 that is independent of ε. Let us prove Theorem 1: for that purpose, we sum the inequalities We obtain Eu, w + Kh + γ Gu, w R nl, 91 t µ where R nl is defined as R nl = s hu z=ǫ + h u z=ǫ + h u z=ǫ + h t u t z=ǫ ε T + 1 F, u + F, u + F, u + µ F t, u t + 1 βh Ft + µ t b p Uu t U w t, t z=ǫ, u t z=ǫ µ + 1 t bh p z=ǫ, u t z=ǫ + t βhu t z=ǫ, u t z=ǫ + H 1 u z=ǫ H w z=ǫ + κ s h h h T + H 1 u z=ǫ H w z=ǫ sch κh h h T + H 1 u z=ǫ H w z=ǫ sch κh h h T + H 1 t u t z=ǫ H t w t z=ǫ sch t κh t t h h. 9 T We show that under the hypothesis 77, and for ε 1 sufficiently small, R nl γ Gu, w: that proves Theorem 1 with the choice γ 1 = γ. Denoting I 1 the singular term in front of s/ε, we can prove that s ε I 1 s ε h + h t Gu, w 1.

24 11 D. BRESCH AND P. NOBLE Furthermore, we obtain an estimate on h with the boundary condition εp z=ε ch + κ h + εu z=ε = εh. 93 Mutiply 93 with h and integrate over T: for ε 1 sufficiently small, h C κ Gu, w1. Similarly, using the boundary condition 76, we prove that ht satisfies the same estimate. As a consequence, we have proved that s ε I 1 C s κ Gu, w ε and for ε 1 small enough, that term is absorbed in γ Gu, w. We consider the other boundary terms: one can show that H k 1 u k z=ǫ εgu, w, k =, 1,. 94 T The boundary terms T k H k w z=ǫ, T th 1 t u z=ǫ, T th t u z=ǫ satisfy the same estimate. We consider I = T sκh h h : I sκ h 7 h 5 C s ε h 5 Gu, w. 95 εκ The last inequality is obtained with the estimate 86. The other boundary terms are of lower order and can be bounded similarly. We finish the proof with the terms involving the nonlinear terms F. Using a Poincaré inequality and integration by parts in the -direction, one proves that I 3 = 1 µ F, u + F, u F u + u. It is a lenghtly but straightforward computation to prove that F Cε 1 Gu, w and I 3 is absorbed in the term γgu, w. We consider the term containing high order derivatives F, u : an integration by parts yields 1 µ F, u = 1 µ F, u. We have to deal with the term I 4 = Ω J 1u zz u : the other terms are of lower order and can be estimated it is easily proved that I 4 C h ε + ε h 5 µ u 3 u. 96 Then for ε 1 small enough, I 4 is absorbed by γ F δ u, w. Similarly, the only term containing time derivatives that is important here has the form I 5 = J 1u zzt u t Ω = J 1u zt z=ǫ u t z=ǫ J 1u t z u zt. 97 T We deduce u zt z=ǫ from the boundary condition 75: deriving the equation on u z z=ǫ with respect to t, one obtains J 1u zt z=ǫ u t z=ǫ = J 1H 1,t w t z=ǫ s ε h tu t. T T Ω

25 MATHEMATICAL JUSTIFICATION OF A SHALLOW WATER MODEL 111 Those terms have been treated before and we show that I 5 Cε 1 Gu, w. As a conclusion, we have proved that for ε 1 sufficiently small, This completes the proof of Theorem 1. R nl Cε 1 Gu, w γ Gu, w. 4. Rigorous derivation of a shallow water model. In that section, we justify rigorously the derivation of the shallow water model from the Navier-Stokes equations. Recall that the evolution system for the fluid height h and the discharge rate hv = v can be written, in the shallow water scaling as h t + hv =, 6 hv t + 5 h v + c h Re s 75 h5 κh h = 1 3 v 98 s h + R, εre h with R a function of h, u, w, p and their derivatives. Formally, that function R is order Oε1+Re+Re 1 and converges formally to as ε. In what follows, we prove that R converges to as ε in a suitable functional space. As a byproduct of the method of derivation, we also obtained lubrication models with a single equation on the fluid height h that can be justified rigorously Estimate of the remainder in the shallow water model. In that section, we shall prove that lim ε R = in a suitable functional space: we first recall the definition of R. Let ũ, w, p, h a solution of the Navier-Stokes system 74,75,76. We define u, w, p, h = 1 + h solution of the Navier-Stokes system 8, 9, 1 that was used to derive shallow-water equations associated to ũ, w, p, h through the change of variables 43, 48, 56. More precisely, the functions u, w, p reads u ε 1 w t,, z = U + 1 J DΘ t Θ 1 t, ε z ũ w t, Θ 1 t, εz,., pt,, z = c1 z + ε p t, Θ 1 t, εz, 99 with Uz = s z z and the function R is written The function R 1 is defined as R = R 1 + R + R 3. with R 1 = R 1 1 t, R 1 t, + ε t, 1 Re u t,, ζdζ + κ 1 + ε h 3 h h R 1 1 t, = t, v u u t,, ζdζ,t δu, y, tdy + 15 h t, R 1 t, = p p t,, ζdζ. δut,, ζdy, 1

26 11 D. BRESCH AND P. NOBLE The function R reads with R 1 defined as R = z R 1 3 h R 1, 11 R 1 = ε Re z ζ u u z ζ Re p p ε z w h + w zh Re 1 ε h w h + w z h 1 ε h + z whereas the function R 3 is written ζ u u t + uu u u + wu z w u z, R 3 = sh h 4 u u. 1 The purpose of that section is to prove the following result. Theorem. Assume that the capillary constant κ has the form κ = κ ε. Let us suppose that the initial conditions ũ, w, h satisfy the assumptions of Theorem 1 and that sinθ is sufficiently small. Then the function κ R satisfies the estimate R L C ε, ;L T with C, a constant, that does not depend on ε. Proof. In what follows, we shall epress the remainders R i as functions of ũ, w, p, p and use the estimates obtained previously. Let us first note that the functions u, w, p read ut,, z = sz z + 1 J ũθ 1 t wt,, z = ε w + z h J ũ t, Θ 1 t, εz, pt,, z = p c h + ε pt, Θ 1 t, εz, εz = u s hz + 1 J ũt, Θ 1 t, εz, For the sake of simplicity, denote z = θt 1, εz defined so that Θ 1 t, εz =, z and εz = z 1 + ht,, z, The proof of Theorem is done in three steps. We first prove the following result. Lemma 6. The function R 3 satisfies the estimate: R 3 L C ε., ;L T

27 MATHEMATICAL JUSTIFICATION OF A SHALLOW WATER MODEL 113 Proof. The function R 3 can be written in the form R 3 = sh 4 1+ h s hz + ũ J, θ 1 t t,, εzdz. 13 We make the change of variable z1 + ht,, z = εz in the integral 13: R 3 = sh 4 s hh + 1 ε ε = s h h h + 1 ε ε ũt,, zdz ũ t,, zdz. Recall that we have obtained an estimate of h in the form h C ε j κ ũ + ũ t. j= We deduce from Theorem 1 that h L, ;L T C. Furthermore, using Poincaré inequality, one finds 1 ε ũ ε ũ. ε As a consequence, the function R 3 satisfies R 3 L, ;L T C ε. This completes the proof of the lemma. We net show that the correction terms obtained from the averaged convection terms and pressure terms are order O ε. Lemma 7. The function R 1 satisfies the estimate: R 1 L C ε., ;L T Proof. After a change of variable and using the continuity of fluid stress, We write R 1 R 1 = ε h H ũ z=ε + ε It is a straightforward computation to prove that J pt,, z pt,, ε dz. R 1 Cε H 1 + u, ε + u, ε +C ε p p, ε + p p, ε. From the Poincaré inequality and estimates on p and p together with the estimates on H, we deduce that R 1 C ε ũ j + ũ t. j=

28 114 D. BRESCH AND P. NOBLE Following the method to obtain an estimate of R 3, one shows that R 1 1 satisfies Furthermore, one can prove that ε R 1 1 C ε ũ j + ũ t. j= u t,, ζdζ Cε 3 ũ + ũ. Then, there remains to deal with the capillary term I κ = κ 1 1+ε h 3 h h. Here we use the fact that h is a bounded function of time and h h : one can prove that I κ Cκε h. From the estimates on h, we obtain I κ Cε 3 ũ j + ũ t. j= This completes the proof of the lemma 7. We finish the proof of the Theorem with the estimate of R. Lemma 8. The function R satisfies the estimate R L C ε., ;L T Proof. The function R is written R = z R 1 + h first consider the function z R 1 defined as z R 1 = ε u u Re ε Re + Re p p w h + w zh 1 ε h w h + w z R 1 t,., ζdζ. We h 1 ε h u u t + uu u u + wu z w u z, First, following the proof of Lemma 7, one can show that ε h u u Re Re p p C ε ũ j + ũ t, j= uu u u + wu z w u z C ε ũ j + ũ t. Furthermore, we easily show, after a change of variable, that u u t C ε ε ũ ε + ũ t. j=

29 MATHEMATICAL JUSTIFICATION OF A SHALLOW WATER MODEL 115 Then, using Cauchy Schwarz and Poincaré inequalities, one obtains u u t C ε ũ + ũ t. The boundary term εw h w h/re is written, using divergence free condition and the no slip condition ε Re w h w h = ε Re u u. That term has been treated previously. Finally, we consider the boundary term ε Re 1 ε h 1 w z h w z h = ε Re u z u z. After a change of variable, we obtain ε Re 1 ε h 1 w z h w z h ε ε C ũ z + ũ + C ε ũ + ũ. ε ũ z Finally, the integral term R1 /h is estimated similarly. One can prove that and Lemma 8 is proved. 1 h R 1 C ε ũ j + ũ t j= As a consequence, from Lemmas 6, 7, 8, we deduce that the remainder term R in the shallow-water system 98 satisfies R L C ε.,,l T This concludes the proof of Theorem and the mathematical justification of the shallow water model Lubrication models. As a by product of the derivation of shallow water equations, we shall obtain a hierarchy of model with a single equation on the fluid height h: these are lubrication models. First let us recall that h, v satisfies the mass conservation equation As a first approimation, hv satisfies the epansion h t + hv =. 14 h v = s h3 3 + u u. 15 Substituting equation 15 into 14, one obtains the Burgers type equation h t + s h3 3 = u u. 16

30 116 D. BRESCH AND P. NOBLE We then deduce from the analysis of the previous section that u u = O ε in the L,, L R -norm and this justifies the fact that the Burgers type equation 16 is a first approimation of shallow water flows. Furthermore, we can obtain more accurate models, using the more precise epansion of hv: h vt, = s h3 t, 3 + ε t, u 1 t,, ζdζ + εre t, where u 1 is the function u 1 = p Re u t + u u + w u z. R 1 t,, ζdζ, 17 Then, it is a lengthly but straightforward computation that h satisfies the lubrication equation h 3 8s t h + s + εre 3 5 h3 ch + εκh = R lub, 18 and the function R lub satisfies the estimate 3 R lub L Cε, ;L. R This justifies the lubrication approimation for shallow water flows. h 3 8s t h + s + εre 3 5 h3 ch + εκh =. 19 At this stage, one can derive a viscous Burgers equation: let us write h in the form h = 1+εhε s t, ε τ. This is precisely the diffusive scaling used in [15], to derive a viscous Burgers equation from the full Navier-Stokes system. Then one proves that h satisfies, up to zeroth order in ε the equation t h + α h = β h, α, β >. 11 As a conclusion, we recover a result similar to that of H. Uecker [15] in the particular case of periodic functions in the streamwise variable: here we can also prove that the perturbation h decays eponentially fast whereas localized solutions of 11 have a self similar decay. 5. Conclusion. In this paper, we have rigourously justified the derivation of a shallow water system. Such proof has been possible due to the iterative scheme which is described in Subsection.. Mathematical justification of the shallow water derivation described in [8] and for which well posedness has been studied in [6] is an open problem. It corresponds to a Navier slip boundary condition assumption on the bottom instead of a no-slip condition with a particular relation between the Reynolds number and the aspect ratio coefficient. Remark also that our proof has been written from the D Navier-Stokes equations to a 1D shallow water systems but the generalization to the 3D-D case may be done in the same way with higher derivative estimates. The main hypothesis is a non zero surface tension coefficient. Looking at the vanishing surface tension limit is an interesting open problem that is postponed to a forthcoming work.