Compressible primitive equations: formal derivation and stability of weak solutions

Mehmet Ersoya,b, Timack Ngoma,c, Mamadou Syc a LAMA, UMR 5127 CNRS, Université de Savoie, 73376 Le Bourget du lac cedex, France. b BCAM - Basque Center for Applied Mathematics, Bizkaia Technology Park 500, 48160, Derio, Basque Country, Spain. c Laboratoire d’Analyse Numérique et Informatique(LANI), Université Gaston Berger de Saint-Louis, UFR SAT BP 234 Saint-Louis, Sénégal.

Abstract We present a formal derivation of Compressible Primitive Equations (CPEs) for atmosphere modeling. They are obtained from the 3-D compressible Navier- Stokes equations with an anisotropic viscous stress tensor depending on the density. Then, we study the stability of weak solutions to this problem by introducing an intermediate model obtained by a suitable change of variables. This intermediate model is more simpler and practical to achieve the main result. Keywords: Compressible primitive equations, Compressible viscous fluid, Anisotropic viscous tensor, A priori estimates, Stability of weak solutions.

Contents

1 Introduction 2

2 Formal derivation of the atmosphere model 4 2.1 FormalderivationoftheCPEs ...... 5 2.2 Themainresult...... 7

3 Stability of weak solutions for the CPEs 8 3.1 A model problem; an intermediate model ...... 9 3.2 Mathematical study of the model problem ...... 9 3.2.1 Energy and entropy estimates ...... 10 3.2.2 Convergence of pξn ...... 14 3.2.3 Bounds of pξnun and pξnwn ...... 15 3.2.4 Convergence of ξnun ...... 16 3.2.5 Convergence of pξnun and ξnwn ...... 17 3.2.6 Convergencestep...... 19 3.2.7 ProofofTheorem1 ...... 21

4 Perspectives 22

Email addresses: [email protected] (Mehmet Ersoy), [email protected] (Timack Ngom), [email protected] (Mamadou Sy)

Preprint submitted to NonLinearity October 26, 2010 1. Introduction

Among equations of geophysical fluid dynamics (see Buntebarth [5]), classi- cally the equations governing the motion of the atmosphere are the Primitive Equations (PEs). In the hierarchy of geophysical fluid dynamics models, they are situated between non hydrostatic models and shallow water models.

Derivation of the Compressible PEs CPEs are obtained from the hydrostatic approximation (see, for instance, Pedlowski [11] or Temam et al. [12]) of the full 3 dimensional set of Navier- Stokes equations for atmosphere modeling. Neglecting phenomena such as the evaporation and solar heating, the Primitive Equations read:

d ρ + ρdivU =0,  dt d   ρ u + xp = , (1)  dt ∇ D ∂yp = gρ, −2  p(ρ)= c ρ   where d = ∂ + u + v∂ dt t ∇x y with x = (x1, x2) the horizontal and y the vertical coordinate. U is the three dimensional velocity vector with component u = (u1,u2) for the horizontal velocity and v for the vertical one. The terms ρ, p, g stand for the density, the barotropic pressure and the gravity vector (0, 0,g). The constant c2 is usually set to where is the specific gas constant for the air and the temperature. RT R T In the present paper, the diffusion term reads: D =2div (ν (t,x,y)D (u)) + ∂ (ν (t,x,y)∂ u) . D x 1 x y 2 y It is obtained by introducing an anisotropic viscous tensor in the initial Navier- Stokes equations where div stands for ∂ + ∂ , D = ( + t )/2 and x x1 x2 x ∇x ∇x ν1(t,x,y) = ν2(t,x,y) represent the anisotropic pair of viscosity depending on the density ρ. The main difference with respect to the classical viscous term found in the litterature (see for instance Temam et al. [12]) is that viscosities depend on the density.

Mathematical analysis of CPEs The mathematical analysis of PEs for atmosphere modeling was first carried out by Lions et al. [9]. These authors have taken into account evaporation and solar heating with constant viscosities. They produced the mathematical formulation in 2 and 3 dimensions based on the works of J. Leray and obtained

2 the existence of weak solutions for all time (see also Temam et al. [12] where the result was proved by different means). Following Temam et al. [12], Ersoy et al. [6] showed the global weak exis- tence for the 2-D version of model (1) by a useful change of vertical coordinates. Currently, up to our knowledge, there is no way to prove an existence or stability result for Model (1). One of the difficulties encountered is to obtain energy estimates. Indeed, proceeding by standard techniques, multiplying the conservation of the momentum equations of System (1) by (u, v), we get:

d 2 2 2 (ρ u +ρ ln ρ ρ+1) dxdy + 2ν1 Dx(u) +ν2 ∂ u dxdy + ρgv dxdy dt | | − | | y Ω Ω Ω where the sign of the integral ρgv dxdy is unknown. There is no way to Ω control, prima facie, the integral term ρgv dx introduced by the hydrostatic Ω equation ∂yp = gρ. To overcome this problem, we make a change of variables and we study an− intermediate problem. Following Ersoy et al. [6], setting

2 2 z =1 e−g/c y and w(t,x,z)= e−g/c yv(t,x,y) − and assuming

2y ν1(t,x,y)=¯ν1ρ(t,x,y) and ν2(t,x,y)=¯ν2ρ(t,x,y)e with ν¯i > 0, we obtain the following model: d ξ + ξ(div u + ∂ w)=0,  dt x z d   ρ u + xp = z, (2)  dt ∇ D ∂zξ =0,  p(ξ)= c2 ξ   d  where denotes dt d = ∂ + u + w∂ dt t ∇x z and =2div (ν (t,x,z)D (u)) + ∂ (ν (t,x,z)∂ u) . (3) Dz x 1 x z 2 z Consequently, in the computation of the energy the integral term vanishes since the right hand side of the equation ∂zξ becomes 0. Thus, we can obtain prelim- inary estimates. In order to show the weak stability, the additional required estimates are provided by the BD-entropy (see, for instance, Bresch [2, 4, 3, 1]) by adding a regularizing term to Equations (2). In this paper, we have added a quadratic friction source term. Combining this term to the viscous one (3) brings reg- ularity on the density which is required to pass to the limit in the non linear

3 terms (e.g. for the term ξu u where typically a strong convergence of ξu is needed). Finally, energy and⊗ BD-entropy estimates are enough to show a weak stability result for Model (2) and by the reverse change of variables for Model (1). Currently, the question of existence of weak solutions remains an open ques- tion for Model (2) (so, also for the Model (1)). This paper is organized as follows. In Section 2, starting from the 3-D compressible Navier-Stokes equations with an anisotropic viscous tensor, we formally derive the Model (1). Then, we present the main result in Section 2.2. We provide a complete proof in Section 3.2.

2. Formal derivation of the atmosphere model

We consider the Navier-Stokes model in a bounded three dimensional domain with periodic boundary conditions on Ωx and free conditions on the rest of the boundary. More exactly, we assume that the motion of the medium occurs in a 2 domain Ω= (x, y); x Ωx, 0

∂tρ + div(ρu)=0, (4) ∂ (ρu)+ div(ρu u) divσ ρf =0, (5) t ⊗ − − p = p(ρ) (6) where ρ is the density of the fluid and u = (u, v)t stands for the fluid velocity t with u = (u1,u2) the horizontal component and v the vertical one. σ is the total asymmetric stress tensor. The pressure law is given by the equation of state: p(ρ)= c2ρ (7) for some given positive constant c. The term f regroups the quadratic friction source term and the gravity strength:

2 2 t f = R u1 + u2 (u1,u2, 0) gk − − where R is a positive constant, g is the gravitational constant and k = (0, 0, 1)t (where Xt stands for the transpose of tensor X). Remark 1. As we will see later, the friction term is a mathematical remedy to ensure the stability of weak solutions of the problem. The total stress tensor is: σ = pI + 2Σ.D(u)+ λdiv(u) I − 3 3 where the term Σ.D(u) reads: 2µ D (u) µ (∂ u + v) 1 x 2 y ∇x µ (∂ u + v)t 2µ ∂ v 3 y ∇x 3 y

4 with I3 the identity matrix. In the definition above, the term Σ = Σ(t,x,y) stands for the following non constant anisotropic viscous tensor (see, for in- stance, [8, 7, 6]): µ1 µ1 µ2  µ1 µ1 µ2  . µ µ µ  3 3 3  The term Dx(u) is the strain tensor with respect to the horizontal variable x, i.e. 2D (u)= u + t u = ∂ u + ∂ u . x ∇x ∇x xi j xj i 16i,j62 The last term λdiv(u) is the classical normal stress tensor where λ is the volu- metric viscosity.

Remark 2. Let us remark that, if we play with the magnitude of viscosity µi, the matrix Σ will be useful to set a privileged flow direction. The Navier-Stokes system is closed with the following boundary conditions on ∂Ω: periodic conditions on ∂Ωx, v|y=0 = v|y=H =0, (8) ∂yu|y=0 = ∂yu|y=H =0. We also assume that the distribution of the horizontal component of the velocity u and the density distribution are known at the initial time t =0:

u(0,x,y)= u0(x, y), 2 −g/c y (9) ρ(0,x,y)= ξ0(x)e where ξ0 is a bounded positive function: 0 6 ξ (x) 6 M < + . 0 ∞ Remark 3. The expression of ρ at time t = 0 is quite natural since in the atmosphere the density is stratified, i.e. for each altitude y, the density has the profile of the given function ξ0. Moreover, it is also mathematically justified at the end of Section 2.1, more precisely see Equation (13).

2.1. Formal derivation of the CPEs Taking advantages of the shallowness of the atmosphere, we assume that the characteristic scale for the altitude H is small with respect to the characteristic length L. In this context, we also assume that the vertical movements and variations are very small compared to the horizontal ones which justifies the following approximation: let ε be a “small” parameter such as: H V ε = = L U where V and U are respectively the characteristic scale of the vertical and L horizontal velocity. We introduce the characteristic time T such as: T = and U

5 the pressure unit P = ρU 2 where ρ is a characteristic density. Finally, we note the dimensionless quantities of time, space, fluid velocity, pressure, density and viscosities: t x y u v t = , x = , y = , u = , v = , T L H U V

p ρ λ µj p = 2 , ρ = , λ = , µj = , j =1, 2, 3 . ρU¯ ρ¯ λ¯ µ¯j With these notations, the Froude number Fr, the Reynolds number associated to the viscosity µi , Rei, (i = 1, 2, 3), the Reynolds number associated to the viscosity λ, Reλ, and the Mach number Ma are respectively: U ρUL ρUL U Fr = , Rei = , Reλ = ,Ma = . (10) √gH µi λ c Applying this scaling to System (4)–(7), using the definition of the dimensionless number (10) and dropping “ . ” we get the following non-dimensional System:

∂tρ + divx (ρ u)+ ∂y (ρv)=0, 1  ∂ (ρ u)+ div (ρ u u)+ ∂ (ρ vu)+ ρ + rρ u u =  t x ⊗ y M 2 ∇x | |  a  2 1 1  divx (µ1Dx(u)) + ∂y µ2 ∂yu + xv  Re Re ε2 ∇  1 2  1  + x (λ divx(u)+ λ∂yv) ,  Reλ ∇  (11)  1 1 1 1 ∂ (ρ v)+ div (ρ u v)+ ∂ ρ v2 + ∂ ρ = ρ  t x y 2 2 y 2 2  ε Ma −ε Fr  1 1 2  + div µ ∂ u + v + ∂ (µ ∂ v)  x 3 2 y x 2 y 3 y  Re3 ε ∇ ε Re3  1  + ∂y (λ divx(u)+ λ∂yv)  ε2 Re  λ  r where we have noted R = . L Next, assuming the following asymptotic regime:

µ1 µi 2 λ 2 = ν1, = ε νi, i =2, 3 and = ε γ. Re1 Rei Reλ and dropping all terms of order O(ε), System (11) reduces to the following Compressible Primitive Equations (CPEs):

∂tρ + divx (ρ u)+ ∂y (ρv)=0, 1  u u u u u u ∂t (ρ )+ divx (ρ )+ ∂y (ρ v )+ 2 xρ + rρ =  ⊗ Ma ∇ | |  u u (12)  2divx (ν1Dx( )) + ∂y (ν2∂y ) , M 2 ∂ ρ = a ρ  y − F 2  r 

6 holding in the domain Ω= (x, y); x Ω R2, 0

2.2. The main result In order to define a weak solution of the CPEs, we introduce the set of function ρ (u, v; y,ρ ) which satisfy ∈ PE 0 ρ L∞(0,T ; L3(Ω)), √ρ L∞(0,T ; H1(Ω)), ∈ ∈ √ρu L2(0,T ; (L2(Ω))2), √ρv L∞(0,T ; L2(Ω)), ∈ ∈ √ρD (u) L2(0,T ; (L2(Ω))2×2), √ρ∂ v L2(0,T ; L2(Ω)), x ∈ y ∈ √ρ L2(0,T ; (L2(Ω))3) ∇ ∈ with ρ > 0 and where (ρ, √ρu, √ρv) satisfies:

∂tρ + divx(√ρ√ρu)+ ∂y(√ρ√ρv)=0, ρt=0 = ρ0. We also define the following integral operators for any smooth test function ϕ with compact support such as ϕ(T,x,y)=0 and ϕ0 = ϕt=0: T (ρ, u, v; ϕ, dy) = ρu∂tϕ dxdydt A − 0 Ω T + (2ν1(t,x,y)ρDx(u) ρu u): xϕ dxdydt 0 Ω − ⊗ ∇ T T + rρ u uϕ dxdydt ρdiv(ϕ) dxdydt 0 Ω | | − 0 Ω T u∂y(ν2(t,x,y)∂yϕ) dxdydt − 0 Ω T ρvu∂yϕ dxdydt − 0 Ω (15) T (ρ, u, v; ϕ, dy)= ρvϕ dxdydt B 0 Ω and

(ρ, u; ϕ, dy)= ρ|t=0u|t=0ϕ0 dxdy (16) C Ω Under these definitions, we consider weak solutions of the CPEs in sens of the distributions. More precisely, we will say that:

7 Definition 1. A weak solution of System (12) on [0,T ] Ω, with boundary conditions (8) and initial conditions (9), is a collection of× functions (ρ, u, v) such as ρ (u, v; y,ρ ) and the following equality holds for all smooth test ∈ PE 0 function ϕ with compact support such as ϕ(T,x,y)=0 and ϕ0 = ϕt=0:

(ρ, u, v; ϕ, dy)+ (ρ, u, v; ϕ, dy)= (ρ, u; ϕ, dy) . A B C Then, we can state the main result:

Theorem 1. Let (ρn, un, vn) be a sequence of weak solutions of System (12), with boundary conditions (8) and initial conditions (9), satisfying entropy in- equalities (23) and (40) such as

ρ > 0, ρn ρ in L1(Ω), ρnun ρ u in L1(Ω). n 0 → 0 0 0 → 0 0 Then, up to a subsequence,

ρ converges strongly in 0(0,T ; L3/2(Ω)), • n C √ρ u converges strongly in L2(0,T ; (L3/2(Ω))2), • n n ρ u converges strongly in L1(0,T ; (L1(Ω))2) for all T > 0, • n n (ρ , √ρ u , √ρ v ) converges to a weak solution of System (12), • n n n n n (ρn, un, vn) satisfies the energy inequality (23), the entropy inequality (40) • and converges to a weak solution of (12)-(8). The proof of the main result is divided into three parts:

in Sections 3.1, we perform a change of variables using (ξ, u, w = e−yv) as • unknowns instead of (ρ, u, v) and we obtain an intermediate model, in Section 3.2.2-3.2.6, we prove the stability of weak solutions of the model • problem, in Section 3.2.7, by the reverse change of variables, we prove the main • result.

3. Stability of weak solutions for the CPEs

As pointed out in Section 1, the classical techniques fails. To overpass this difficulty, following Ersoy et al. we perform a useful change of variables which transform the initial problem into a more simpler and more practical for math- ematical analysis.

8 3.1. A model problem; an intermediate model Let us first remark that the structure of the density ρ, defined as a tensorial product (see Equation (13)), suggests the following change of variables:

z =1 e−y (17) − where the vertical velocity in the new coordinates becomes:

w(t,x,z)= e−yv(t,x,y). (18)

d Since the new vertical coordinate z is defined as z = e−y, multiplying by dy ey system (12) and using the viscosity profile (14) and the change of variables (17)-(18) provides the following model, called model problem:

∂tξ + divx (ξ u)+ ∂z (ξ w)=0,  ∂t (ξ u)+ divx (ξ u u)+ ∂z (ξ u w)+ xξ + rξ u u =  ⊗ ∇ | | (19)  2¯ν1divx (ξDx(u))+ν ¯2∂z(ξ∂zu), ∂zξ =0   ′ ′ ′ −1 T2 holding in the domain is Ω = (x, z); x Ωx, 0

′ Definition 2. A weak solution of System (19) on [0,T ] Ω , with boundary (20) and initial conditions (21), is a collection of functions× (ξ, u, w), if ξ (u, w; z, ξ ) and the following equality holds for all smooth test function ∈ϕ PE 0 with compact support such as ϕ(T,x,y)=0 and ϕ0 = ϕt=0:

(ξ, u, w; ϕ, dz)= (ξ, u; ϕ, dz) A C where and are given by (15) and (16). A C We then have the following result:

9 Theorem 2. Let (ξn, un, wn) be a sequence of weak solutions of System (19), with boundary conditions (20) and initial conditions (21), satisfying entropy inequalities (23) and (40) such as

′ ′ ξ > 0, ξn ξ in L1(Ω ), ξnun ξ u in L1(Ω ). (22) n 0 → 0 0 0 → 0 0 Then, up to a subsequence,

′ ξ converges strongly in 0(0,T ; L3/2(Ω )), • n C ′ ξ u converges strongly in L2(0,T ; (L3/2(Ω ))2), • n n ′ ξ u converges strongly in L1(0,T ; (L1(Ω ))2) for all T > 0, • n n

(ξn, ξnun, ξnwn) converges to a weak solution of System (19), • (ξn, un, wn) satisfies the energy inequality (23), the entropy inequality (40) • and converges to a weak solution of (19)-(20). We divide the proof of Theorem 2 into three steps:

in Section 3.2.1, we obtain suitable a priori bounds on (ξ, u, w), • in Sections 3.2.2-3.2.5, we show the compactness of sequences (ξn, un, wn) • in apropriate space function, in Section 3.2.6, we prove that we can pass to the limit in all terms of • System (19) which ends the proof of Theorem 2.

3.2.1. Energy and entropy estimates A part of a priori bounds on (ξ, u, w) are obtained by the physical energy inequality which is obtained in a classical way by multiplying the momentum equation by u, using the mass equation and integrating by parts. We obtain the following inequality:

2 d u 2 2 ξ + (ξ ln ξ ξ + 1) dxdz + ξ(2¯ν1 Dx(u) +¯ν2 ∂zu ) dxdz dt ′ 2 − ′ | | | | Ω Ω +r ξ u 3 dxdz 6 0 ′ Ω | | (23) which provides the uniform estimates:

′ ξu is bounded in L∞(0,T ; (L2(Ω ))2), (24) 1 ′ ξ /3u is bounded in L3(0,T ; (L3(Ω ))2), (25) ′ 2 2 2 ξ∂zu is bounded in L (0,T ; (L (Ω )) ), (26) ′ 2 2 2×2 ξDx(u) is bounded in L (0,T ; (L (Ω )) ), (27) ′ ξ ln ξ ξ +1 is bounded in L∞(0,T ; L1(Ω )). (28) −

10 The strong convergence of ξu required to pass to the limit in the non linear term ξu u is obtained by the mathematical BD-entropy. To this end, we first ⊗ take the gradient of the mass equation, then we multiply by 2¯ν1 and write the term ξ as ξ ln ξ to obtain: ∇x ∇x ∂ (2¯ν ξ ln ξ)+div (2¯ν ξ ln ξ u)+∂ (2¯ν ξ ln ξw)+div 2¯ν ξ t u t 1 ∇x x 1 ∇x ⊗ z 1 ∇x x 1 ∇x + ∂ (2¯ν ξ w)=0 . (29) z 1 ∇x Next, we sum Equation (29) with the momentum equation of System (19) to get the equation:

∂ (ξ(u +2¯ν ln ξ)) + div (ξ(u +2¯ν ln ξ) u)+ ∂ (ξwu)+2¯ν ∂ (ξw) t 1∇x x 1∇x ⊗ z 1 z∇ 2¯ν div (ξA (u)) ν¯ ∂ (ξ∂ u)+ rξ u u + ξ =0, (30) − 1 x x − 2 z z | | ∇x u t u where A (u) = ∇x − ∇x is the vorticity tensor. The mathematical BD- x 2 entropy inequality is then obtained by multiplying the previous equation by u +2¯ν1 x ln ξ and by integrating by parts. To this end, multiplying Equation ∇ ′ (30) by the term u + 2¯ν1 x ln ξ and integrating over Ω , we have to compute each term of the following∇ integral:

∂ (ξ(u +2¯ν ln ξ))(u +2¯ν ln ξ) dxdz ′ t 1 x 1 x Ω ∇ ∇ + div (ξ(u +2¯ν ln ξ) u)(u +2¯ν ln ξ) dxdz ′ x 1 x 1 x Ω ∇ ⊗ ∇ + ∂ (ξwu)(u +2¯ν ln ξ)+2¯ν ∂ (ξw)(u +2¯ν ln ξ) dxdz ′ z 1 x 1 ′ z 1 x Ω ∇ Ω ∇ ∇ 2¯ν div (ξA (u))(u +2¯ν ln ξ) dxdz + r ξ u u(u +2¯ν ln ξ) dxdz 1 ′ x x 1 x ′ 1 x − Ω ∇ Ω | | ∇ ν¯ ∂ (ξ∂ u)(u +2¯ν ln ξ) dxdz + ξ(u +2¯ν ln ξ) dxdz =0 . 2 ′ z z 1 x ′ x 1 x − Ω ∇ Ω ∇ ∇ (31)

The two first one reads as follows:

∂ (ξ(u +2¯ν ln ξ))(u +2¯ν ln ξ) dxdz ′ t 1 x 1 x Ω ∇ ∇ + div (ξ(u +¯ν ln ξ) u)(u +2¯ν ln ξ) dxdz ′ x 1 x 1 x Ω ∇ ⊗ ∇ 1 = ξ∂ u +2¯ν ln ξ 2 dxdz + (u +2¯ν ln ξ)2∂ ξ dxdz ′ t 1 x ′ 1 x t 2 Ω | ∇ | Ω ∇ 1 + (ξu ) u +2¯ν ln ξ 2 dxdz + (u +2¯ν ln ξ)2div (ξu) dxdz ′ 1 x ′ 1 x x 2 Ω ∇ | ∇ | Ω ∇

11 which is also:

∂ (ξ(u +2¯ν ln ξ))(u +2¯ν ln ξ) dxdz ′ t 1 x 1 x Ω ∇ ∇ + div (ξ(u +¯ν ln ξ) u)(u +2¯ν ln ξ) dxdz ′ x 1 x 1 x Ω ∇ ⊗ ∇ 1 1 = ξ∂ u +2¯ν ln ξ 2 dxdz div(ξu) u +2¯ν ln ξ 2 dxdz ′ t 1 x ′ 1 x 2 Ω | ∇ | − 2 Ω | ∇ | + (u +2¯ν ln ξ)2(∂ ξ + div (ξu)) dxdz . (32) ′ 1 x t x Ω ∇ Remarking that ∂ (ξwu)= ∂ (ξw(u +2¯ν ln ξ)) 2¯ν ∂ w ξ, z z 1∇x − 1 z ∇x we have:

∂ (ξw(u +2¯ν ln ξ))(ξwu)(u +2¯ν ln ξ) dxdz ′ z 1 x 1 x Ω ∇ ∇ 2¯ν ξ∂ w(u +2¯ν ln ξ) dxdz 1 ′ x z 1 x − Ω ∇ ∇ 1 = ξw∂ u +2¯ν ln ξ 2 dxdz + (u +2¯ν ln ξ)2∂ (ξw) dxdz+ ′ z 1 x ′ 1 x z 2 Ω | ∇ | Ω ∇ 2¯ν w ξ∂ u dxdz 1 ′ x z Ω ∇ which is finally:

ξwu∂ (ξw(u +2¯ν ln ξ))(u +2¯ν ln ξ) dxdz ′ z 1 x 1 x Ω ∇ ∇ 2¯ν ξ∂ w(u +2¯ν ln ξ) dxdz 1 ′ z 1 x − Ω ∇ ∇ 1 = u +2¯ν ln ξ 2∂ (ξw) dxdz + (u +2¯ν ln ξ)2∂ (ξw) dxdz ′ 1 x z ′ 1 x z −2 Ω | ∇ | Ω ∇ +2¯ν w ξ∂ u dxdz . (33) 1 ′ x z Ω ∇ Summing (32) and (33), we obtain:

∂ (ξ(u +2¯ν ln ξ))(u +2¯ν ln ξ) dxdz ′ t 1 x 1 x Ω ∇ ∇ + div (ξ(u +¯ν ln ξ) u)(u +2¯ν ln ξ) dxdz ′ x 1 x 1 x Ω ∇ ⊗ ∇ + ∂ (ξwu)(u +2¯ν ln ξ) dxdz ′ z 1 x Ω ∇ 1 d = ξ u +2¯ν ln ξ 2 dxdz +2¯ν w ξ∂ u dxdz . (34) ′ 1 x 1 ′ x z 2 dt Ω | ∇ | Ω ∇

12 The fourth term in Equation (31), i.e. 2¯ν ∂ (ξw)(u + 2¯ν ln ξ) dxdz 1 ′ z 1 x Ω ∇ ∇ gives:

∂ (ξw)(u +2¯ν ln ξ) dxdz = (ξw)∂ u dxdz ′ z x 1 x ′ x z Ω ∇ ∇ − Ω ∇ = ξw∂ div (u) dxdz ′ z x Ω = (w∂ div (ξu) w ξ∂ u) dxdz . ′ z x x z Ω − ∇ Differentiating the equation of the conservation of the mass with respect to z,

∂ div (ξu)= ξ∂2w, z x − z we get:

2¯ν ∂ (ξw)(u +2¯ν ln ξ) dxdz = 2¯ν (ξw∂2w w ξ∂ u) dxdz 1 ′ z x 1 x 1 ′ z x z Ω ∇ ∇ − Ω − ∇ = 2¯ν ξ ∂ w 2 dxdz 2¯ν w ξ∂ u) dxdz . (35) 1 ′ z 1 ′ x z Ω | | − Ω ∇

In order to compute the term 2¯ν div (ξA (u))(u + 2¯ν ln ξ) dxdz in 1 ′ x x 1 x − Ω ∇ Equation (31), we have just to remark that thanks to periodic conditions, we have div (ξA (u)) ln ξ dxdz =0 ′ x x x Ω ∇ which leads to:

2¯ν div (ξA (u))(u +2¯ν ln ξ) dxdz = 2¯ν ξ A (u) 2 dxdz (36) 1 ′ x x 1 x 1 ′ x − Ω ∇ Ω | | The fifth and sixthterm in Equation (31) simply read:

r ξ u u(u+2¯ν ln ξ) dxdz = r ξ u 3 dxdz+2¯ν r u u ξ dxdz (37) ′ 1 x ′ 1 ′ Ω | | ∇ Ω | | Ω | | ∇ and ν¯ ∂ (ξ∂ u)(u +2¯ν ln ξ) dxdz =ν ¯ ξ ∂ u 2 dxdz . (38) 2 ′ z z 1 x 2 ′ z − Ω ∇ Ω | |

The last term ξ(u +2¯ν ln ξ) dxdz gives ′ x 1 x Ω ∇ ∇

d ξ(u +2¯ν ln ξ) dxdz = (ξ log ξ ξ + 1) dxdz ′ x 1 x ′ Ω ∇ ∇ dt Ω − +8¯ν ξ 2 dxdz . (39) 1 ′ x Ω |∇ |

13 Finally, summing the terms (34) to (39), we obtain the entropy inequality:

1 d 2 ξ u +2¯ν1 x ln ξ + 2(ξ log ξ ξ + 1) dxdz 2 dt ′ | ∇ | − Ω + 2¯ν ξ ∂ w 2 +2¯ν ξ A (u) 2 +¯ν ξ ∂ u 2 dxdz ′ 1 z 1 x 2 z Ω | | | | | | + rξ u 3 +2¯ν r u u ξ +8¯ν ξ 2 dxdz =0. (40) ′ 1 x 1 x Ω | | | | ∇ |∇ | which gives the following estimates:

′ ξ is bounded in L∞(0,T ; (L2(Ω ))3), (41) ∇ ′ 2 2 ξ∂zw is bounded in L (0,T ; L (Ω )), (42) ′ 2 2 2×2 ξAx(u) is bounded in L (0,T ; (L (Ω )) ) . (43) This finishes the first step of the proof of Theorem 2. Remark 5. Estimate (41) is a straightforward consequence of estimates ′ ′ ∞ 2 2 ∞ 2 2 ξ(u+2¯ν1 x ln ξ) L (0,T, (L (Ω )) ) and ξu L (0,T, (L (Ω )) ) since ∇ ∈ ∈ ξ ξ(u +2¯ν ln ξ)= ξu +2¯ν ∇x . 1 x 1 √ ∇ ξ

To show the compactness of sequences (ξn, un, wn) in apropriate space func- tion we follow the work of Mellet et al. [10]. To this end, we divid this second step of the proof of Theorem 2 into 4 parts :

1. in Section 3.2.2, we show the convergence of the sequence ξn,

2. in Section 3.2.3, we seek bounds of ξnun and ξnwn, 3. in Section 3.2.4, we prove the convergence of ξnun, 4. in Section 3.2.5, we prove the convergence of ξnun. 3.2.2. Convergence of ξn Let us first prove the following

Lemma 1. For every ξn satisfying the mass equation of System (19), we have:

′ ∞ 1 ξn is bounded in L (0,T,H (Ω )),

′ 2 −1 ∂t ξn is bounded in L (0,T,H (Ω )). Then, up to a subsequence,′ the sequence ξn converges almost everywhere and′ strongly in L2(0,T ; L2(Ω )). Moreover, ξ converges to ξ in 0(0,T ; L3/2(Ω )). n C Proof of Lemma 1: ′ ∞ 1 ξn is bounded in L (0,T,H (Ω )) since we have 2 n ′ 1 ′ ξn(t) L2(Ω ) = ξ0 L (Ω ) || || || || 14 from the continuity equation and by Estimate (41). Using again the mass conservation equation, we write: 1 ∂ ( ξ ) = ξ div (u ) u . ξ ξ ∂ w t n −2 n x n − n ∇x n − n z n 1 = ξ div (u ) div (u ξ ) ξ ∂ w . 2 n x n x n n n z n − − Then, from Estimates (27), (41), (42) and (43), we get:

′ 2 −1 ∂t ξn is bounded in L (0,T,H (Ω )).

′ We have then the compactness of ξ in 0(0,T,L2(Ω ) by Aubin’s Lemma, n C i.e. ′ 0 2 ξn converges strongly to ξ in (0,T,L (Ω )). C ′ ∞ p We also have, by Sobolev embeddings, bounds of ξn in spaces L (0,T,L (Ω )) for all p [1, 6]. Consequently, for p =6, we get bounds of ∈ ′ ∞ 3 ξn in L (0,T,L (Ω )) and we deduce that:

′ ∞ 3/2 2 ξnun = ξn ξnun is bounded in L (0,T,L (Ω ) ).

′ ∞ −1,3/2 It follows that ∂tξn is bounded in L (0,T,W (Ω )) since

∂ ξ = div (ξ u ) ξ ∂ w t n − x n n − n z n and Estimate (42) holds. To conclude, writing

′ ξ =2 ξ ξ L∞(0,T ; L3/2(Ω )2), ∇x n n∇x n ∈ ′ ∞ 1,3/2 we deduce bounds of ξn in L (0,T ; W (Ω )). Then, using again Aubin’s ′ 3/2 lemma provides compactness of ξn in the intermediate space L (Ω ):

′ 0 3/2 compactness of ξn in (0,T ; L (Ω )). C 

3.2.3. Bounds of ξnun and ξnwn To prove the convergence of the momentum, we have to control bounds of ξnun and ξnwn. Thus, we have to prove the following Lemma 2. We have

′ ∞ 2 2 ξnun bounded in L (0,T ; (L (Ω )) ) and ′ 2 2 ξwn bounded in L (0,T ; L (Ω )).

15 Proof of Lemma 2: We have already bounds of ξn (see Estimates (24)). ′ 2 2 There is left to show bounds of ξnwn in L (0,T ;L (Ω )). As ξn = ξn(t, x) and Estimates (42) holds, by the Poincaré inequality, we have:

h 2 h 2 ξnwn dz 6 c ∂z( ξnwn) dz. 0 0 Consequently, the following inequality

ξ w 2 dxdz 6 c ξ ∂ w 2 dxdz ′ n n ′ n z n Ω | | Ω | |

′ gives bounds of ξ w in L2(0,T ; L2(Ω )). n n 

3.2.4. Convergence of ξnun As bounds of ξnun and ξnwn are provided by Lemma 2, we are able to show the convergence of the momentum.

Lemma 3. Let mn = ξnun be a sequence satisfying the momentum equation (19). Then we have:

′ ξ u m in L2(0,T ; (Lp(Ω ))2) strong , 1 6 p< 3/2 n n → ∀ and ′ ξ u m a.e. (t,x,y) (0,T ) Ω . n n → ∈ × Proof of Lemma 3: Writing (ξ u ) as: ∇x n n

x(ξnun)= ξn ξn xun +2 ξnun x ξn ∇ ∇ ⊗ ∇ provides ′ (ξ u ) bounded in L2(0,T ; (L1(Ω ))2×2). (44) ∇x n n Next, we have

′ 2 3/2 2 ∂z(ξnun)= ξn ξn∂z(un) is bounded L (0,T ; (L (Ω )) ). (45) Then, from bounds (44) and (45), we deduce:

′ 2 1,1 2 ξnun is bounded L (0,T ; (W (Ω )) ). (46)

On the other hand, we have:

∂ (ξ u ) = div (ξ u u ) ∂ (ξ u w ) ξ t n n − x n n ⊗ n − z n n n − ∇x n

+2¯ν1divx (ξnDx(un))+ν ¯2∂z (ξn∂zun)

rξ u u . − n | n| n

16 As ξnun un = ξun ξun, (47) ⊗ ⊗ we deduce bounds of

′ ξ u u in L∞(0,T ; (L1(Ω ))2×2). n n ⊗ n Particularly, we have

′ div (ξ u u ) bounded in L∞(0,T ; (W −2,4/3(Ω ))2). x n n ⊗ n ′ Similarly, as ξ u w = ξu ξw (L1(Ω ))2, we also have: n n n n n ∈ ′ ∞ −2,4/3 2 ∂z(ξnunwn) bounded in L (0,T ; (W (Ω )) ).

′ 2 3/2 2 Moreover, as ξn ξn∂zun L (0,T ; (L (Ω )) ) and ξn ξnDx(un) ′ ∈ ∈ L2(0,T ; (L3/2(Ω))2×2), we get bounds of

′ ∂ ( ξ ξ ∂ u ), div ( ξ ξ D (u )) L2(0,T ; (W −1,3/2(Ω ))2). z n n z n x n n x n ∈ ′ ∞ −1,3/2 2 We also have bounds of xξn L (0,T, (W (Ω )) ). ′ ∇ ∈ ′ Using W −1,3/2(Ω ) W −1,4/3(Ω ), we obtain ⊂ ′ 2 −2,4/3 2 ∂t(ξnun) bounded in L (0,T ; (W (Ω )) ). (48)

Using Aubin’s Lemma with the bounds (46), (48) provides the compactness of

′ 2 p 2 ξnun L (0,T ; (L (Ω )) ), p [1, 3/2[. ∈ ∀ ∈ 

3.2.5. Convergence of ξnun and ξnwn Let us note that, up to Section 3.2.4, we can always define u = m/ξ on the set ξ > 0 , but we do not know, a priori, if m equals zero on the vacuum set. To this{ end,} we need to prove the following lemma: Lemma 4.

1. The sequence ξnun satisfies ′ m ξ u converges strongly in L2(0,T ; (L2(Ω ))2) to . n n √ • ξ We have m =0 almost everywhere on the set ξ =0 and there exists • a function u such that m = ξu and { }

′ ξ u ξu strongly in L2(0,T ; (Lp(Ω ))2) for all p [1, 3/2[, n n → ∈ ′ ξ u ξu strongly in L2(0,T ; (L2(Ω ))2). n n → ′ 2 2 2. The sequence ξnwn converges weakly in L (0,T ; L (Ω )) to ξw.

17 To prove Lemma 4, we adapt the proof of Mellet et al. [10]. As already pointed out by Bresch et al. [3], the presence of the term rξ u u simplify also this proof. Proof of Lemma 4: | | To start, we set mn = ξnun. m ′ Since n is bounded in L∞(0,T ; (L2(Ω ))2) Fatou’s lemma yields: √ξn

m2 lim inf n < . ′ Ω ξn ∞ In particular, we have m(t,x,z)=0 almost everywhere on the set ξ(t, x)=0 . So, if we define the limit velocity u(t,x,z) by setting { }

m(t,x,z) if ξ(t, x) =0, u(t,x,z)=  ξ(t, x)  u(t,x,z)=0 if ξ(t, x)=0,  then, we have m(t,x,z)= ξ(t, x)u(t,x,z) and m2 dxdz = ξu2 dxdz < . ′ ′ Ω ξ Ω ∞

Next, since mn and ξn converge almost everywhere, it is readily seen that on the set ξ(t, x) =0 , { } m m ξ u = n converges almost everywhere to ξu = . n n √ √ ξn ξ Moreover, for a constant M > 0, we have: 1 1 ξnun |un|≤M ξu |u|≤M almosteverywhere. (49) → As a matter of fact, the convergence holds almost everywhere on the set

ξ(t, x) =0 ξ(t, x)=0 { }∪{ } and we have 1 ξnun |un|≤M M ξ. ≤ To complete the proof, we cut the L2 norm as follows

2 2 ξ u ξu dxdz ξ u 1 u ξu1 u dxdz ′ n n ′ n n | n|≤M | |≤M Ω − ≤ Ω − 2 +2 ξ u 1 u dxdz ′ n n | n|≥M Ω 2 +2 ξu1 u dxdz. ′ | |≥M Ω

18 ′ 1 ∞ 2 2 It is obvious that ξnun |un|≤M is uniformly bounded in L (0,T ; (L (Ω )) ), then using (49) gives the convergence of the first integral:

2 ξ u 1 u ξu1 u dxdz 0. (50) ′ n n | n|≤M | |≤M Ω − → Finally, writing

2 1 3 ξ u 1 u dxdz ξ u dxdz, (51) ′ n n | n|≥M ′ n n Ω ≤ M Ω | | 2 1 3 ξu1 u dxdz ξ u dxdz. (52) ′ | |≥M ′ Ω ≤ M Ω | | and putting together (50), (51) and (52), we deduce:

2 C lim sup ξ u ξu dxdz , M > 0 ′ n n n→+∞ Ω − ≤ M ∀ which ends the first point of the lemma by taking M + . → ∞ The second part of the theorem is done by weak compactness. As ξnwn is ′ 2 2 bounded in L (0,T ; L (Ω )), there exists, up to a subsequence, ξnwn which ′ converges weakly to some limit l in L2(0,T ; L2(Ω )). Next, we define w as:

l if ξ > 0, w =  √ξ  0 a.e. if ξ =0  l where the limit l is written: l = ξ = ξw. √ξ 

This finishes the second point of the proof of Theorem 2.

3.2.6. Convergence step Gathering the previous results, we show straightforwardly that we can pass to the limit in all terms of System (19) in the sense of Theorem 2. To this end, let (ξn, un, wn) be a weak solution of System (19) satisfying Lemma 1 to 4 ′ and let φ ∞([0,T ] Ω ) be a smooth function with compact support such ∈ Cc × as φ(T,x,z)=0 and φ(0,x,z)= φ0(x, z). Then, writing each term of the weak formulation of System (19), we have: For the first integral, we have: • T T ∂ (ξ u )φ dxdzdt = ξ u ∂ φ dxdzdt ′ t n n ′ n n t 0 Ω − 0 Ω ξnunφ dxdz. ′ 0 0 0 − Ω

19 Using convergences (22) and Lemma 3, we get

T ξ u ∂ φ dxdzdt ξnunφ(0,x,z) dxdz ′ n n t ′ 0 0 − 0 Ω − Ω → T ξu∂ φ dxdzdt ξ u φ(0,x,y) dxdz. ′ t ′ 0 0 − 0 Ω − Ω For the following integral, we write: • T T div (ξ u u ) φ dxdzdt = ξ u u : φ dxdzdt ′ x n n n ′ n n n x 0 Ω ⊗ − 0 Ω ⊗ ∇ then, from Equality (47) and Lemma 4, we have:

T T ξ u u : φ dxdzdt ξu u : φ dxdzdt. ′ n n n x ′ x − 0 Ω ⊗ ∇ →− 0 Ω ⊗ ∇ Writing • T T ∂ (ξ u w ) φ dxdzdt = ξ u w ∂ φ dxdzdt, ′ z n n n ′ n n n z 0 Ω − 0 Ω

as ξnunwn = ξnun ξnwn, by Lemma 4, we get: T T ξ u w ∂ φ dxdzdt ξuw ∂ φ dxdzdt. ′ n n n z ′ z − 0 Ω →− 0 Ω For the following integral, we write: • T T ξ φ dxdzdt = ξ div (φ) dxdzdt ′ x n ′ n x 0 Ω ∇ − 0 Ω then, Lemma 1 provides: T T ξ div (φ) dxdzdt ξdiv (φ) dxdzdt ′ n x ′ x − 0 Ω →− 0 Ω We write the integral as follows: • T T div (ξ D (u )) φ dxdzdt = ξ D (u ): φ dxdzdt. ′ x n x n ′ n x n 0 Ω − 0 Ω ∇ 1 Since D (u )= ( u + t u ), expanding the term in the last integral x n 2 ∇x n ∇x n gives: T ξ D (u ): φ dxdzdt ′ n x n x − 0 Ω ∇ 1 T = (ξ u ∆ φ + φ ( ξ ) ξ u dxdzdt ′ n n x x x n n n 2 0 Ω ∇ ∇ T 1 t t + (ξnun divx( φ)+ ξn xφ ξnun) dxdzdt. 2 ′ x x 0 Ω ∇ ∇ ∇

20 From Estimates (41), the sequence x ξn weakly converges, and using Lemma 1, Lemma 3 and 4, we obtain:∇

1 T (ξnun ∆xφ + xφ x( ξn) ξnun dxdzdt 2 ′ 0 Ω ∇ ∇ T 1 t t + (ξnun divx( φ)+ ξn xφ ξnun) dxdzdt 2 ′ x x 0 Ω ∇ ∇ ∇ → 1 T (ξu ∆xφ + xφ x( ξ) ξu dxdzdt 2 ′ 0 Ω ∇ ∇ T 1 t t + (ξu divx( xφ)+ x ξ xφ ξu) dxdzdt. 2 ′ 0 Ω ∇ ∇ ∇ Hence

T T ξ D (u ): φ dxdzdt ξD (u): φ dxdzdt. ′ n x n x ′ x x − 0 Ω ∇ →− 0 Ω ∇

We have straightforwardly • T T ∂2(ξ u ) φ dxdzdt ξ u ∂2(φ) dxdzdt. ′ z n n ′ n n z 0 Ω → 0 Ω Using Lemma 3 provides the following convergence:

T T ξ u ∂2(φ) dxdzdt ξu ∂2(φ) dxdzdt ′ n n z ′ z 0 Ω → 0 Ω

The convergence of the integral • T T rξ u u φ dxdzdt rξ u u φ dxdzdt ′ n n n ′ 0 Ω | | → 0 Ω | | is obtained by Lemma 4, and finishes the proof of Theorem 2. 

3.2.7. Proof of Theorem 1 Following Ersoy et al. [6], to finish the proof of Theorem 1 we consider a sequence (ξn, un, wn) of weak solution of System (19). All obtained estimates in steps 3.2.2-3.2.6 hold if we replace ξn by ρn and wn by vn, since

ρ(t,x,y)= ξ(t, x)e−y and w(t,x,z)= v(t,x,y)e−y d where z = e−y. Moreover, by the change of variables z =1 e−y in integrals, dy − we have the following properties:

ρ 2 = α ξ 2 ′ , • || ||L (Ω) || ||L (Ω )

21 ρ 2 = α ξ 2 ′ , • ||∇x ||L (Ω) ||∇x ||L (Ω )

∂ ρ 2 = α ξ 2 ′ • || y ||L (Ω) || ||L (Ω ) where −1 1−e α = (1 z) dz < + . 0 − ∞ We deduce then, ′ ρ 1,2 = α ξ 1,2 || ||W (Ω) || ||W (Ω ) which provides ρ L∞(0,T ; W 1,2(Ω)) ∈ and ∂ ρ L2(0,T ; L2(Ω)). t ∈ Again, by the change of variable in integrals, the fact that v L2(0,T ; L2(Ω)) is obtained from the inequality: ∈

1 2 v L2(Ω) = v(t,x,y) dy dx || || Ω 0 | | x −1 1−e 1 3 = w(t,x,z) 2 dz dx ′ 1 z | | Ωx 0 3 − < e w 2 ′ . || ||L (Ω ) Finally, all estimates on u remaining true, Theorem 1 is proved. 

4. Perspectives

In this paper, we have presented a Compressible Primitive Equations where viscosities are anisotropic and density dependent. We have established a sta- bility result for weak solutions by introducing a useful change of variable. The question of the existence of weak solutions for these equations remains an open question. However, with the obtained estimations, it may be possible to con- struct an approximate sequence of solutions, as Faedo-Galerkin approach and to adapt the technique presented by Va˘ıgant et al. [13]. Although, their models does not take into account the anisotropy and the dynamical viscosity is con- stant, useful additional estimates can be derived, particularly, to show that the density is bounded. The work is actually in progress.

Acknowledgment

The first author was supported by the ERC Advanced Grant FP7- 246775 NUMERIWAVES.

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