On the strong solutions of the primitive equations in 2D domains

On the strong solutions of the Primitive Equations in 2D domains. F. Guill´en-Gonz´alez1& M.A. Rodr´ıguez-Bellido2 Departamento de Ecuaciones Diferenciales y An´alisis Num´erico, Universidad de Sevilla. c/ Tarfia s/n, 41012 Sevilla (SPAIN). e-mails: [email protected], [email protected] Correspondence and proofs for corrections: F. Guill´en-Gonz´alez Key words: hydrostatic pressure, mixed boundary conditions, local and global existence, uniqueness, asymptotic behaviour. 1 Introduction Some geophysical fluids can be modelled through the so-called “primitive equations” [1], [2]. This model is obtained formally from the Navier-Stokes equations, with anysotropic (eddy) viscosity, assuming two important simplifications: hydrostatic pressure (depending linearly on the depth) and the rigid lid hypothesis (fix water surface) [3] . For simplicity, we take constant density and assume that the effects due to the temperature (and salinity) can be decoupled from the dynamic of the flow. Then, we have a three-dimensional flow induced by the wind tension on the surface and by the centripetal and Coriolis forces. When the Earth curvature is not considered, we can use cartesian coordinates instead of spherical coordinates (see Lions-Teman-Wang [2] for the model with spherical coordinates), hence the domain is given by Ω={(�x , z )∈IR 3;�x∈ω,−D(�x)<z<0},(1) where ω⊆IR 2is an open domain and D:ω→IR +is the depth function. The different boundaries of Ω(surface, bottom and sidewalls) are respectively: Γs={(�x , 0); �x∈ω}, Γb={(�x , −D(�x)); �x∈ω}and Γl={(�x , z ); �x∈∂ω,−D(�x)<z<0}. Including, as it is usual ([4]), centripetal effects into the pressure term, the three-dimensional model is: (EP)                      ∂t�u+(�u·∇)�u+u3∂z�u−νh∆�u−νv∂2 zz�u+α�u⊥+∇ps=� Fin (0,T)×Ω, ∇·��0 −D(�x )�u(t;�x , z )dz�=0 in(0,T)×ω, �u|t=0 =�u0in Ω, νv∂z�u|Γs=�τ ,�u|Γb∪Γl=� 0in(0,T). 1Partially supported by C.I.C.Y.T project MAR98-0486 2Supported by C.I.C.Y.T project MAR98-0486 1 Here, we denote �x=(x, y), ∇=(∂x,∂ y)and∆=∂2 xx +∂2 yy. The unknowns are the horizontal component of flow velocity �u=(u1,u 2):(0,T)×Ω→IR 2and the surface pressure ps:(0,T)×ω→IR, whereas the vertical component of the flow velocity is u3(t;�x , z )=−�z −D(�x )∇·�u(t;�x , s )ds, ∀t∈(0,T),∀(�x , z )∈Ω.(2) Moreover, νhand νv>0 are positive constants, representing horizontal and vertical (eddy) viscosity coefficients respectively, � F:(0,T)×Ω→IR 2is an horizontal external force field (depending on temperature and salinity, for instance) and �τ :(0,T)×Γs→IR 2represents the horizontal stress on the surface produced by the wind. Finally, α�u⊥=α(−u2,u 1) models Coriolis effects, the no-slipt condition is assumed on the bottom and vertical slipting is permitted on the sidewalls. To give a variational formulation to problem (EP), let us define the following function spaces: C∞ b,l (Ω) = {�ϕ ∈C∞(Ω)2;supp(�ϕ )isacompactset ⊆Ω\(Γb∪Γl)}, H1 b,l(Ω) = C∞ b,l H1 (Ω) = {�v∈H1(Ω)2;�v=0onΓ b∪Γl},H −1 b,l (Ω) = dual of H1 b,l(Ω), V={�ϕ ∈C∞ b,l (Ω)2;∇·��ϕ �=0 inω},where ��ϕ �(�x)=�0 −D(�x )�ϕ (�x , z )dz, H=VL2 ={�v∈L2(Ω)2;∇·��v�=0 inω,��v�·�n|∂ω =0}, V=VH1 ={�v∈H1(Ω)2;∇·��v�=0 inω,�v|Γb∪Γl=� 0}. Definition 1.1 Let �u0∈H,� F∈L2(0,T;H−1 b,l (Ω)2)and �τ ∈L2(0,T;H−1/2(Γs)2).We say that �u:(0,T)×Ω→IR 2is a weak solution of (EP)in (0,T)if �u∈L∞(0,T;H)∩ L2(0,T;V), verifies the variational formulation: �T 0�Ω�−�u·(∂t�ϕ +(�u·∇)�ϕ +u3∂z�ϕ )+νh∇�u:∇�ϕ +νv∂z�u·∂z�ϕ +α�u⊥·�ϕ �dΩdt =�Ω�u0·�ϕ (0) dΩ+�T 0�� F,�ϕ�Ωdt +�T 0��τ ,�ϕ �Γsdt, ∀�ϕ ∈C1([0,T]; V)s.t. �ϕ (T)=� 0, and, moreover �usatisfies the energy inequality: 1 2��u�2 L2(Ω)+�t 0�νh�∇�u�2 L2(Ω)+νv�∂z�u�2 L2(Ω)�ds ≤1 2��u0�2 L2(Ω)+�t 0�� F,�u�Ωds +�t 0��τ ,�u�Γsds a.e. t ∈(0,T). (3) Here, �·,·�Ωdenotes duality between H−1 b,l (Ω)and H1 b,l(Ω), whereas �·,·�Γsdenotes duality between H−1/2(Γs)and H1/2(Γs). 2 Definition 1.2 Let �u0∈V,� F∈L2(0,T;L2(Ω)2),�τ ∈L2(0,T;H1/2(Γs)2)and ∂t�τ ∈ L2(0,T;H−3/2(Γs)2). Let �ube a weak solution of (EP)in (0,T), we say that �uis a strong solution if verifies the additional regularity conditions: �u∈L∞(0,T;V)∩L2(0,T;H2(Ω)2∩V),∂ t�u∈L2(0,T;H). The existence of a weak solution is well known, see Lewandovski [3] and Lions-TemanWang [2], always in domains with sidewalls (i.e. D≥Dmin >0inω). In these works, compactness method is used to obtain the velocity �uin a space with the restriction ∇·��u�= 0 and the pressure is recovered, in the latter part of the argument, by a specific De Rham’s lemma on the surface. In domains without sidewalls, the existence of a weak solution is obtained as a consequence of a limit process applied to the Navier-Stokes equations with anysotropic viscosity when the ratio depth over horizontal diameter (of the domain) tends to zero, see Besson-Laydi [5] for the stationary case and Azerad-Guill´en [6] for the evolution case. Finally, the existence of a weak solution in domains without sidewalls can be proved by internal approximation arguments: a mixed (velocity-pressure) variational formulation of the stationary problem is approximated by a conform Finite Element method in Chac´on-Guill´en [7] and a semi-discretization in time of the evolution problem is proved that converges to continuous problem in Guill´en-Redondo [8, 9]. However, to as far as we know, there are not results about the existence of strong solution of problem (EP), excepting the stationary linear case [10]. One of the principal problems in this study is the treatment of the boundary conditions; on the surface we have a non homogeneous Neumann condition, whereas the sidewalls and the bottom have homogeneous Dirichlet condition. On the other hand, the uniqueness of the solution of problem (EP) is also an open problem, even in the case of strong solutions. 1.1 The 2Dproblem In this work, we are going to consider mainly the two-dimensional problem (with only one horizontal direction). In this case, Coriolis forces have not sense. Now, the model is: (EP2)                ∂tu+u∂xu+u3∂zu−νh∂2 xxu−νv∂2 zzu+∂xps=Fin (0,T)×Ω, ∂x�u�=0 in(0,T)×ω, u|t=0 =u0in Ω, νv∂zu|Γs=τ, u|Γb∪Γl=0 in(0,T). Then, all the unknowns are scalar: the horizontal component of the flux velocity u: (0,T)×Ω→IR a n d t h e s u p e r fi c i a l p r e s s u r e ps:(0,T)×ω→IR. The vertical component of the flux velocity is u3(t;x, z)=−�z −D(x)∂xu(t;x, s)ds. One important differencerespect to 3D case is that now ω⊆IR is an interval, which changes the function spaces of free divergence. Now, we have the following simpler characterizations: V={ϕ∈C∞ b,l (Ω); �ϕ�=0 inω}, 3 H={v∈L2(Ω); �v�=0 inω}, V={v∈H1(Ω); �v�=0 inω, v|Γb∪Γl=0}. Finally, definitions of weak and strong solutions are similar to the 3D case (changing vectorial notation by scalar notation in uand x, and vanishing the Coriolis term). Remark 1.1 Now, the 2nd. equation in (EP2)means that �u�only depends on t. 1.2 Main results In this paper, we will obtain the following main results, all in the 2D case and in domains with sidewalls. Theorem 1.3 (Strong global solution for small data.) Let ω⊆IR an interval and D∈C2(ω)such that D≥Dmin >0in ω. We assume u0∈V,F∈L2(0,T;L2(Ω)) and τ∈L2(0,T;H1/2+ε 0(Γs)), for some ε>0, with ∂tτ∈L2(0,T;H−1/2(Γs)). If the following “smallness restriction” is assumed: ∀t∈[0,T], (H)           exp �−1 4Ct+�t 0a(s)ds��2��u0�2 V+C2�τ(0)�2 H−1/2(Γs)� +�t 0exp �1 4Cs−�s 0a(σ)dσ�b(s)ds�<M 2, where Mis a small enough positive constant (see Lemma 5.2), Cis a constant that appears in (11) and a,bare functions depending on the data τand F(see (30) and (31)), then there exists a unique strong solution (u, ps)of (EP2)in (0,T)(psis unique up to a function of t). Corollary 1.4 (Asymptotic behaviour when t↑+∞.) Let ω⊆IR an interval and D∈C2(ω)such that D≥Dmin >0in ω. We assume u0∈V,F∈L2(0,+∞;L2(Ω)) and τ∈L2(0,+∞;H1/2+ε 0(Γs)), for some ε>0, with ∂tτ∈L2(0,+∞;H−1/2(Γs)). If the “smallness restriction” (H)is assumed ∀t∈[0,+∞),then there exist a unique strong solution u∈L2(0,+∞;H2(Ω)∩V)) ∩L∞(0,+∞;V),∂tu∈L2(0,+∞;H). Moreover, if �+∞ 0exp �1 4Ct���τ(t)�2 H1/2+ε 0(Γs)+�∂tτ(t)�2 H−1/2(Γs)+�F(t)�2 L2(Ω)�dt < +∞,(4) there exists two constants K1,K2>0such that: �∇u(t)�2 L2(Ω)≤exp �−1 4Ct�K1�2�u0�2 V+C2�τ(0)�2 H−1/2(Γs)+K2�∀t≥0,(5) (i.e. the solution vanishes exponentially in the H1(Ω)-norm, as tincreases). 4 Theorem 1.5 (Strong local solution for any data.) Under hypotheses of Theorem 1.3, changing the restriction (H)by Dmax(= max ωD)small enough, then there exists T∗∈(0,T]and a unique strong solution (u, ps)of (EP2)in (0,T ∗). Theorem 1.6 (Uniqueness of strong/weak solution.) Let ube a weak solution of (EP2)in (0,T). If there exists another weak solution ¯uof (EP2)in (0,T), such that verifies the additional regularity: ∂z¯u∈L4(0,T;L4(Ω)),(6) then both solutions coincide in (0,T). Remark 1.2 The arguments to prove all these main results, will not be valid in the 3D case. On the other hand, the additional regularity (6) that implies uniqueness is verified by the strong solutions of (EP2)(and not by only weak solutions). Applying this uniqueness argument to the 3Dcase, it is necessary an additional regularity that is not verified by the strong solutions. To make the study about existence of strong solutions, it will be convenient to decompose the problem (EP) in two: one linear problem (L) with nonhomogeneous boundary conditions on the surface, and a nonlinear problem (P) with homogeneous boundary conditions. This paper is organized as follows. In Section 2, we prove some technical inequalities that we will use in the following. The linear problem (L) is studied in Sections 3 and 4, whereas the study of (P)(by means of a Galerkin method) is made in Section 5, where the proof of Theorem 1.3is finished. Indeed, in Section 3, using the known results ([10]) about strong solution of the linear stationary problem (Lst), we deduce some properties of the differential operator associated, that we apply in Section 4, arriving at the existence and uniqueness of strong solution of (L) (all these results are valid in any space dimension). In Section 6, we present the proof of Theorem 1.5, based in a fixed point argument (in particular, it is not possible to make a Galerkin argument as in Theorem 1.3). Finally, the uniqueness of weak solution assuming that a strong solution exists (Theorem 1.6), is proved in Section 7. 2 Some technical results First, we see three technical lemmas that we will used several times in this paper: Lemma 2.1 Let Ω⊆IR N(N=2or 3) be the domain considered in this work (defined by (1)). Then, for all �v∈W1,p(Ω)N−1(p>1), if we define v3(�x , z )=−�z −D(�x )∇·�v(�x , s )ds, one has: �v3�Lp(Ω)≤Dmax�∇ ·�v�Lp(Ω) 5 Proof: It is a consequence of Fubini’s Theorem: �v3�p Lp(Ω)=�Ω������z −D(�x )∇·�v(�x , s )ds����� p dΩdz ≤�Ω��z −D(�x )|∇·�v(�x , s )|pds�(z+D(�x))p/p�dΩdz =�ω�0 −D(�x )|∇·�v(�x , s )|p��0 s(z+D(�x))p/p�dz�dΩds ≤Dp max p�ω�0 −D(�x )|∇·�v(�x , z )|pdΩds =Dp max p�∇ ·�v�p Lp(Ω) Lemma 2.2 (Interpolation inequalities.) Let Ω⊆IR Nbe a Lipschitz-continuous domain. The following inequality holds: ��u�Lp(Ω)≤C��u�1−q/p W1,N (Ω)��u�q/p Lq(Ω),∀�u∈W1,N (Ω)N−1,(7) where N≤q≤p<+∞. Proof: It is taken from the Nirenberg’s paper [11], where is proved the result when Ω=IR N. Here, we adapt the proof to a Lipschitz-continuous domain Ω. For this, we pass these inequalities to Ωusing a prolongation operator [12] E:W1,1(Ω)N−1−→ W1,N (IRN)N−1, verifying E�u|Ω=�uand �E�u�W1,N (IRN)≤C��u�W1,N (Ω),∀�u∈W1,N (Ω)N−1,forsome C=C(Ω)>0. Nirenberg’s result says �E�u�Lp(IRN)≤C�E�u�1−q/p W1,N (IRN)�E�u�Lq(IRN). Therefore, since ��u�Lp(Ω)≤�E�u�Lp(IRN)and �E�u�Lq(IRN)≤C��u�Lq(Ω), we arrive at (7). An easy application to the above Lemma and the Poincar´e’s inequality, give us the following: Corollary 2.3 Let Ω⊆IR Nbe the domain considered in this work. The following inequality holds: ��u�Lp(Ω)≤C�∇�u�1−q/p LN(Ω)��u�q/p Lq(Ω)∀�u∈W1,N b,l (Ω)N−1,(8) where N≤q≤p<+∞. Remark 2.1 The main advantage of the 2D case is to consider (7) and (8) for N=2. In the following, we will call Gagliardo-Nirenberg’s inequality to (7) or (8) in the case N=2,p=4and q=2, i.e. �u�L4(Ω)≤C�u�1/2 L2(Ω)�u�1/2 H1(Ω)∀u∈H1(Ω),(9) �u�L4(Ω)≤C�u�1/2 L2(Ω)�∇u�1/2 L2(Ω)∀u∈H1 b,l(Ω).(10) 6 3 The stationary linear case In this Section, we will see some preliminary results about the linear stationary system (also called hydrostatic Stokes system): (Lst)         −νh∆�u−νv∂2 zz�u+∇ps=�gin Ω, ∇·��u�=0inω, νv∂z�u=�aon Γs, �u=� 0onΓ b∪Γl. 3.1 Known results about existence and uniqueness Lemma 3.1 (Weak solution of (Lst)) Let ω⊆IR d(d=1or 2) and let Ω⊆IR d+1, defined as in (1), be a Lipschitz-continuous domain. If �g∈H−1 b,l (Ω)dand �a∈H−1/2(Γs)d, then the problem (Lst)has a unique solution �u∈H1(Ω)d. Moreover, one has the continuous dependence, i.e. there exists a constant C=C(Ω,ν h,ν v)>0such that ��u�V≤C���a�H−1/2(Γs)+��g�H−1 b,l (Ω)�.(11) In [5], [7] and [3], there are different proofs of this result (even in the nonlinear case). Lemma 3.2 (Strong solution of (Lst)) Let ω⊆IR d(d=1or 2)beaC2domain and D∈C2(ω)with D≥Dmin >0in ω.If�g∈L2(Ω)dand �a∈H1/2+ε 0(Γs)d, for some ε>0, then the unique solution �uof the problem (Lst)belongs to H2(Ω)d∩V. Moreover, we have the continuous dependence, i.e. there exists a constant C=C(Ω,ν h,ν v)>0such that: ��u�H2(Ω)≤C���a�H1/2+ε 0(Γs)+��g�L2(Ω)�.(12) See [10] for the proof of regularity. The continuous dependence can be deduced following the construction of the auxiliary problems made by Ziane in [10]. 3.2 The hydrostatic Stokes operator We define A, that it will call “hydrostatic Stokes operator”, as the resolvent operator related to the homogeneous Neumann boundary conditions on the surface and Dirichlet boundary conditions on the bottom and sidewalls, i.e. A:V→V�such that �A�u,�v�V�,V =�Ω(νh∇�u:∇�v+νv∂z�u·∂z�v)dΩ∀�u , �v∈V. (13) Then, if we denote �g=A�u∈V�,fromLemma3.1, �uis the unique weak solution of the hydrostatic Stokes problem (Lst), with �a=� 0. Moreover, taking into account Lemma 3.2, Ais a self-adjoint isomorphism from H2(Ω)2∩Vto H. In particular, if A�u=�gwith �g∈H,�uis characterized as the unique strong solution of the problem (Lst), with �a=� 0. Finally, the domain of A,definedby D(A)={�u;�u∈Vand A�u∈H}. can be characterized as follows: 7 Lemma 3.3 Let ω⊆IR d(d=1or 2)beaC2domain and D∈C2(ω)with D≥Dmin >0 in ω. Then D(A)={�u;�u∈H2(Ω)d∩Vand ∂z�u=� 0on Γs}.(14) Moreover, there exists C=C(Ω,ν h,ν v)>0sucht that ��u�H2(Ω)≤C�A�u�L2(Ω)∀�u∈D(A).(15) Proof: Let Ybe the right hand side of (14). a) D(A)⊂Y:Let �u∈D(A). If we denote �g=A�u,then�uis the weak solution of (Lst)with�a=� 0. As �g∈H, from the Ziane’s regularity results [10], we deduce that �u∈Y,andthecontinuousdependence(12)says: ��u�H2(Ω)≤C��g�L2(Ω)=C�A�u�L2(Ω) b) Y⊂D(A):Let �u∈Y.Ifwedenote� f=−νh∆�u−νv∂2 zz�u,then� f∈L2(Ω)dand A�u=P� f,wherePis the ortogonal projection from L2(Ω)donto H. Hence A�u∈H,i.e. �u∈D(A). 3.3 Construction of a special basis In this subsection, we will prove the following result: Lemma 3.4 Under the conditions of Lemma 3.3, there exists a sequence {λj}j≥1⊆IR with 0<λ 1≤λ2≤... ≤λj≤λj+1 ≤...,{λj}→+∞, and an orthonormal basis of H, {�wj}j≥1, where each �wjis an eigenfunction of Aassociated to eigenvalue λj. Proof: Let Λ:H−→ D(A)�→Hbe the operator that associates each �g∈Hto �u∈D(A), the unique strong solution of the problem (Lst)with�a=� 0 (i.e. A�u=�g). This is an compact (using Lemma 3.3 and the compact embedding of H2(Ω)2∩Vinto H)and self-adjoint operator (Λ�g1,�g2)=(�u1,�g2)=(�u1,A�u 2)=(A�u1,�u 2)=(�g1,Λ�g2). Then, as His separable, we can apply the Hilbert Schmidt’s Theorem (of spectral decomposition), and there exists an orthogonal basis of Hformed by eigenfunctions of Λ, {�vj}j≥1(Λ�vj=µj�vj,whereµj�0asj�+∞). Let λj=1/µjand �zj=µj�vj.Then A�zj=λj�zj,andthesequence�wj=�zj/�λjis the orthonormal basis of the Lemma. 4 The evolution linear case In this section we will study the strong solution of the nonstationary linear problem: (L)                      ∂t�v−νh∆�v−νv∂2 zz�v+∇qs=� fin (0,T)×Ω, ∇·��v�=0 in(0,T)×ω, �v|t=0 =�v0in Ω, νv∂z�v=�τ on (0,T)×Γs, �v=� 0on(0,T)×(Γb∪Γl). 8 Theorem 4.1 Let ω⊆IR d(d=1or 2) be a C2domain and D∈C2(ω)with D≥ Dmin >0in ω.If� f∈L2((0,T)×Ω)d,�v0∈V,�τ ∈L2(0,T;H1/2+ε 0(Γs)d), for some ε>0, with ∂t�τ ∈L2(0,T;H−1/2(Γs)d), then there exists a unique strong solution �vof (L) in (0,T). Moreover, there exists C>0sucht that ��v�2 L∞(V)+��v�2 L2(D(A)) +�∂t�v�2 L2(H)≤C���v0�2 V+��τ (0)�2 H−1/2(Γs) +�� f�2 L2(L2(Ω)) +��τ �2 L2(H1/2+ε 0(Γs)) +�∂t�τ �2 L2(H−1/2(Γs))�(16) Proof: Uniqueness can be easily deduced from the linearity of the problem (L). The proof of the existence will be separate in several steps. Step 1. Weak solution of (L).The weak solution �vof (L)in(0,T)canbeobtained as a limit of Galerkin approximations �vm∈C1([0,T]; Vm)(beingVmam-dimensional subspace of V)suchthat (L)m                  d dt �Ω�vm·�ϕ dΩ+νh�Ω∇�vm:∇�ϕ dΩ+νv�Ω∂z�vm·∂z�ϕ dΩ =�Ω � fm·�ϕ dΩ+�Γs �τ m·�ϕ |Γsdσ∀�ϕ ∈Vm, �vm(0) being the projection of �v0onto Vm, where � fm∈C0([0,T]; H−1 b,l (Ω)2)and�τ m∈C0([0,T]; H−1/2(Γs)2)arerespectivelyregular approximations to � fand �τ . Taking �vmas test function in (L)m,onecandeducethatthesequence�vmis bounded in L∞(0,T;H)∩L2(0,T;V). Passing to the limit in a standard way, we obtain the weak regularity for �v. Remark 4.1 (Weak solution of (EP)). Galerkin approximations of nonliner problem (EP)are similar to problem (L)m. The only differences are the nonlinear terms: �Ω�(�um·∇)�um+um3∂z�um�·�ϕ dΩ, where um3is defined from ∇·�umas in (2). But, these terms vanish when �umis taken as test function, hence we can also deduce that �umis bounded in L∞(0,T;H)∩L2(0,T;V). Now, by using a compactness result (estimating ∂t�umin a convenient space), we could pass to the limit and obtain a weak solution �uof (EP2)in (0,T). Step 2. “Lifting” of the Neumann boundary conditions. We define the operator B:�a∈H−1/2(Γs)d→�u=B�a∈V,where�uis the weak solution of the hydrostatic Stokes problem (Lst)with�g=� 0, i.e. �u∈Vsuch that �A�u, � ψ�V�,V =��a, � ψ�Γs∀� ψ∈V. 9 Lemma 5.2 Under the hypothesis (H)of the Theorem 1.3and supposing (29), let Mbe a constant such that : (a)1−C1DmaxM>1/2, (b)C2M2<1/(4C), (C1and C2are the constants that appear in (29) and C>0is the equivalence constant between �Au�L2and the H2-norm, see Lemma 3.3), then �wm(t)�V<M, ∀t∈[0,T]. Proof: Arguing by contradiction, we suppose there exists some instant in (0,T)where the bound Mis reached. Let t∗the smallest of these instants, i.e. �wm(t)�V<M, ∀t∈[0,t ∗)and�wm(t∗)�V=M.Then,∀t∈[0,t ∗], 1−C1Dmax�wm(t)�V≥1−C1DmaxM>1/2. In the last estimation, we have used hypothesis (a). If we denote y(t)=�wm(t)�2 V,using that 1 C�wm�2 V≤�Awm�2 L2(Ω)(see (15) in Lemma 3.3), (29) yields: y�(t)+ 1 2Cy(t)≤C2M2y(t)+a(t)y(t)+b(t),∀t∈[0,t ∗]. Now, from hypothesis (b), y�(t)+ 1 4Cy(t)≤a(t)y(t)+b(t),∀t∈[0,t ∗].(32) Integrating this differential inequality between 0 and t∗, we obtain: y(t∗)≤exp �−1 4Ct∗+�t∗ 0a(t)dt��y(0) + �t∗ 0exp �1 4Ct−�t 0a(s)ds�b(t)dt� Therefore, since y(0) = �wm0�2 V≤�w0�2 V≤2��u0�2 V+�e(0)�2 V�≤2��u0�2 V+C2�τ(0)�2 H−1/2(Γs)�, hypothesis (H) implies �wm(t∗)�V<M, hence we arrive at contradiction. Step 3. Proof of Theorem 1.3: From Lemma 5.2, wmis bounded in L∞(0,T;V). Moreover, applying hypothesis (a)ofLemma5.2 in (29), one has d dt�wm�2 V+1 2�Awm�2 L2(Ω)≤C2M4+M2a(t)+b(t),(33) hence, integrating in time, we deduce that wmis bounded in L2(0,T;H2(Ω)). On the other hand, taking ∂twm(t)∈Vmas a test function in (24), integrating in time and using the above regularity, one deduces that ∂twmis bounded in L2(0,T;H). Then, by a standard argument of passage to the limit, we obtain that w(and a surface pressure associated πs) is a strong global solution of (P). Finally, (u, ps)=(e+w,qs+πs) is a strong solution of (EP2)in(0,T). The uniqueness of strong solution of (EP2)stemsfromSection7. 16 Remark 5.1 In the 3D case, we cannot obtain the above strong estimates. It is because in the right hand side of (27), if we estimate the corresponding I2term, we obtain a bound of the form Dmax�Awm�5/2 L2(Ω)�wm�1/2 V which cannot be controlated with the left hand side of (27). 5.2 Proof of Corollary 1.4. Let us first prove existence of a strong solution of (EP2)in(0,+∞). The argument is based in Step 1 and 2of the proof of Theorem 1.3. In particular, it is not difficult to obtain the global weak estimations: wmis bounded in L2(0,+∞;V)∩L∞(0,+∞;H). Now, using hypothesis (H)in[0,+∞), we can deduce that �wm�V<M,∀t∈[0 + ∞). Let us change Step 3. Instead of (33), we rewrite (29) as: d dt�wm�2 V+1 2�Awm�2 L2(Ω)≤C2M2�wm�2 V+M2a(t)+b(t). Using that wmis bounded in L2(0,+∞;V)anda,b∈L1(0,+∞) (thanks to (31) and the global regularity of τ,∂tτand F), we have that wmis bounded in L2(0,+∞;H2(Ω)∩ V). Then, we can conclude the existence of a strong solution u∈L∞(0,+∞;V)∩ L2(0,+∞;H2(Ω)∩V)and∂tu∈L2(0,+∞;H). Now, let us see the asymptotic behaviour of u. Adding in both parts of (32) d dt�e(t)�2 V+ 1 4C�e(t)�2 V, taking into account that d dt�e(t)�2 V≤2�e(t)�V�∂te(t)�V, we obtain for z(t)=�wm(t)�2 V+�e(t)�2 Vthe inequality: z�(t)+�1 4C−a(t)�z(t)≤b(t)+ 1 2C�e(t)�2 V+4C�∂te(t)�2 V. Multiplying by exp �1 4Ct−�t 0a(s)ds�and integrating on (0,t), z(t)≤exp �−1 4Ct+�t 0a(s)ds��z(0) +�t 0exp �1 4Cs−�s 0a(σ)dσ��b(s)+ 1 2C�e(s)�2 V+4C�∂te(s)�2 V�ds�. (34) Now, using that �um(t)�2 V≤2z(t), �um(t)�2 V≤exp �−1 4Ct�K1�z(0) +2 K1�t 0exp �1 4Cs��b(s)+ 1 2C�e(s)�2 V+4C�∂te(s)�2 V�ds�, 17 where K1=2exp��a�L1(0,+∞)�.Sincez(0) ≤2�u0�2 V+C2�τ(0)�2 H−1/2(Γs), bounding in a convenient way b,eand ∂te(in function of τ,∂tτand F), we can deduce the asymptotic behaviour (5) whenever the hypothesis (4) holds. 6 Local strong solution for any data (proof of Theorem 1.4) We want to apply now a fixed point argument to obtain strong solution of (EP2), local in time, but for any data. Now, we study problem (Q), which is similar to (P)butwhose solution is (w=u−v, ˜πs=ps−qs), where (v,qs) is the solution of (L)withv0=0 anf f= 0. With this purpose, we rewrite (Q)asafixedpointequationbymeansofa linearisation. We define, for each T>0: Y(T)=�¯w;¯w∈L2(0,T;D(A)) ∩L∞(0,T;V),∂ t¯w∈L2(0,T;H), ¯w(0) = u0,�¯w�2 L∞(V)+�¯w�2 L2(D(A)) +�∂t¯w�2 L2(H)≤R2�. Given vthe strong solution of (L)in(0,T)and ¯w∈Y(T), we consider the linear problem: (Ql)                      ∂tw−νh∂2 xxw−νv∂2 zzw+∂xπs=G(¯w,v)in(0,T)×Ω, ∂x�w�=0 in(0,T)×ω, w|t=0 =u0in Ω, νv∂zw=0 on(0,T)×Γs, w=0 on(0,T)×(Γb∪Γl), where G(¯w,v)=F−(¯w+v)∂x(¯w+v)−(¯w3+v3)∂z(¯w+v). Problem (Ql) is similar to problem (R), which has already been studied in Section 4. Therefore, since u0∈Vand G∈L2((0,T)×Ω), then w∈L2(0,T;D(A)) ∩L∞(0,T;V)and∂tw∈L2(0,T;H). First, we are going to prove that, there exists R2large enough such that Y(T)�=∅, ∀T>0. Indeed, let w∗be the unique solution of the hydrostatic Stokes problem:                      ∂tw∗−νh∂2 xxw∗−νv∂2 zzw∗+∂xπs=0 in(0,T)×Ω, ∂x�w∗�=0 in(0,T)×ω, w∗|t=0 =u0in Ω, νv∂zw∗=0 on(0,T)×Γs, w∗=0 on(0,T)×(Γb∪Γl). Following the reasoning of the problem (R), see (21) and (22), we know that: �w∗�2 L∞(V)+�w∗�2 L2(D(A)) +�∂tw∗�2 L2(H)≤�u0�2 V,(35) therefore, taking R2≥�u0�2 V,thenw∗∈Y(T), ∀T>0. 18 Now, we introduce the Banach space XT=L2(0,T;V) and the mapping Φ:Y(T)−→ XT,given by Φ( ¯w)=w, where wis the unique solution of (Ql). Obviously, a fixed point of Φsolves problem (Q). Arguing as in problem (R), we have: �w�2 L∞(V)+�w�2 L2(D(A)) +�∂tw�2 L2(H)≤�u0�2 V+C�G(¯w,v)�2 L2(L2(Ω)) (36) On the other hand, vverifies problem (L), with inicial data zero and homogeneous second member (i.e. v0=0andf= 0) but a nonhomogeneous Neumann boundary condition (τ)onthesurface.Then,vsatisfies the estimate (see (16)): �v�2 L∞(V)+�v�2 L2(H2(Ω)) +�∂tv�2 L2(H)≤B(τ)2,(37) where B(τ)2=C��τ(0)�2 H−1/2(Γs)+�τ�2 L2(H1/2+ε 0(Γs)) +�∂tτ�2 L2(H−1/2(Γs))�. Now, we want to find conditions to apply Schauder’s Theorem. 1) ∃T∗∈(0,T]such that Φ(Y(T∗)) ⊂Y(T∗): Let ¯w∈Y(T)andw=Φ(¯w). Then: �G(¯w,v)�2 L2(L2(Ω)) ≤9��F�2 L2(L2(Ω)) +�(¯w+v)∂x(¯w+v)�2 L2(L2(Ω)) +�(¯w3+v3)∂z(¯w+v)�2 L2(L2(Ω))�≡ 3 � i=1 Ii (38) We bound each term Ii(constant Gwill come from the Gagliardo-Nirenberg’s inequalities, see Lemmas 2.2and2.3, whereas Cwe will denote different constants independent of R,B(τ), Dmax and T). First, we estimate �(¯w+v)∂x(¯w+v)�2 L2(Ω)≤�¯w+v�2 L4(Ω)�∂x(¯w+v)�2 L4(Ω) ≤4��¯w�2 L4(Ω)+�v�2 L4(Ω)���∂x¯w�2 L4(Ω)+�∂xv�2 L4(Ω)� ≤4G2��¯w�V�¯w�L2(Ω)+�v�V�v�L2(Ω)���¯w�H2(Ω)�¯w�V+�v�H2(Ω)�v�V�. Integrating in (0,T), taking into account definition of Y(T) and (37), I2≤4G2T1/2��¯w�L∞(V)�¯w�L∞(H)+�v�L∞(V)�v�L∞(H)� ×��¯w�L∞(V)�¯w�L2(H2(Ω)) +�v�L∞(V)�v�L2(H2(Ω))� ≤CT1/2(B(τ)2+R2)2. In a similar way, we bound the vertical velocity terms as follows: �(¯w3+v3)∂z(¯w+v)�2 L2(Ω)≤�¯w3+v3�2 L4(Ω)�∂z(¯w+v)�2 L4(Ω) ≤4��¯w3�2 L4(Ω)+�v3�2 L4(Ω)���∂z¯w�2 L4(Ω)+�∂zv�2 L4(Ω)� ≤4D2 max ��∂x¯w�2 L4(Ω)+�∂xv�2 L4(Ω)���∂z¯w�2 L4(Ω)+�∂zv�2 L4(Ω)� ≤4G2D2 max ��¯w�H2(Ω)�¯w�V+�v�H2(Ω)�v�V�2. 19 Therefore, integrating in (0,T), I3≤4CD 2 max ��¯w�L2(H2(Ω))�¯w�L∞(V)+�v�L2(H2(Ω))�v�L∞(V)�2 ≤CD 2 max (B(τ)2+R2)2. In the last estimates, we could not obtain any power of T, and this fact is the main difficulty in our argument. Indeed, inserting all the above bound in (38), �G(¯w,v)�2 L2(L2(Ω)) ≤C��F�2 L2(L2(Ω)) +(B(τ)2+R2)2�D2 max +T1/2�� (39) Then, from (36) and (39), �w�2 L∞(V)+�w�2 L2(D(A)) +�∂tw�2 L2(H)≤�u0�2 V +C��F�2 L2(L2(Ω)) +(B(τ)2+R2)2�D2 max +T1/2�� (40) The above inequality can be written as �w�2 L∞(V)+�w�2 L2(H2(Ω)) +�∂tw�2 L2(H)≤aR4+bR2+c, where, for some C=C(Ω,ν h,ν v)>0, a=C�D2 max +T1/2�, b=2CB(τ)2�D2 max +T1/2�, c=�u0�2 V+C��F�2 L2(0,T;L2(Ω)) +B(τ)4�D2 max +T1/2��. Taking R2≥�u0�2 V(hence Y(T)�=∅,∀T>0), one has w∈Y(T)whenever aR4+bR2+c≤R2.(41) In the following, we will see that for any data F,τ,u0,(41)isverified. Anecessary condition to (41) is b<1. But, it can also find some sufficient conditions. Indeed, one possibility is to impose the following three conditions: Condition 1: Dmax and Tare small enough sucht that b≤1 2. For instance, 2 CB(τ)2D2 max ≤1/4and2CB(τ)2T1/2≤1/4. Condition 2: R2big enough such that c≤1 4R2. Condition 3: asmall enough (i.e. Dmax and Tsmall enough) such that aR 2≤1 4. 20 In conclusion, there exists T∗∈(0,T]andDmax >0 small enough, such that for some Rbig enough, one has Φ(Y(T∗)) ⊂Y(T∗). 2) Y(T∗)is relatively compact in XT∗. Let WT∗={¯w;¯w∈L2(0,T ∗;D(A)) and ∂t¯w∈L2(0,T ∗;H)}.Y(T∗)isaboundedset of WT∗and WT∗is embedded in a compact way in XT∗.Therefore,Y(T∗) is relatively compact in XT∗. 3) Y(T∗)is closed in XT∗. Let {¯wn}n≥1⊆Y(T∗)suchthat ¯wn−→ ¯wstrongly in XT∗(i.e. in the L2(0,T;V)- norm). Let us see that ¯w∈Y(T∗).As {¯wn}n≥1is bounded in WT∗, in particular, there exists a subsequence {¯wk}of {¯wn}such that: ¯wk�¯win L2(0,T ∗;D(A)∩V), ∂t¯wk�∂ t¯win L2(0,T ∗;H).(42) Then, applying a compactness result of Aubin-Lions type [13]: ¯wk−→ ¯win C([0,T ∗]; H). Therefore, since ¯wk(0) = u0,∀k≥1, then ¯w(0) = u0.Bylowersemi-continuityofthe norm, �¯w�2 L∞(V)+�¯w�2 L2(D(A)) +�∂t¯w�2 L2(H) ≤lim infk→+∞��¯wk�2 L∞(V)+�¯wk�2 L2(D(A)) +�∂t¯wk�2 L2(H)�≤R2, then ¯w∈Y(T∗), hence Y(T∗)isclosedinXT∗.Thisone,jointlywith2), imply that Y(T∗) is compact in XT∗. 4) Φ:Y(T∗)−→ Y(T∗)is continuous respect to XT∗topology. Let {¯wn}n≥1⊆Y(T∗) such that ¯wn→¯wstrongly in XT∗.Letusprovethat: Φ( ¯wn)=wn−→ Φ( ¯w)=wstrongly in XT∗. As also {wn}n≥1⊆Y(T∗), there are subsequences {¯wk}of {¯wn}and {wk}of {wn}such that ¯wk�¯w, ¯w∈WT∗ wk�˜w, ˜w∈WT∗ (where the above convergences are as in (42)). If we consider the system verified by wkand we pass to the limit as k→+∞,weobtain that ˜wis a solution of the problem (Ql)withsecondmemberG(¯w,v). By uniqueness ˜w=Φ(¯w)=w.Therefore,wk−→ wweakly in WT∗and, by compactness, wk−→ win XT∗. Finally, all the sequence converges. 5) Existence of a fixed point. As Y(T∗)isaconvexcompactsetofXT∗and Φis continuous respect to XT∗topology, applying the Schauder’s Theorem we deduce the existence of a fixed point wof Φin Y(T∗). Therefore, wis a strong solution of (Q)in(0,T ∗) (if T∗verifies jointly with Dmax the conditions 1 and 3). 21 Remark 6.1 Again, in the 3D case we cannot bound the nonlinear vertical convection �¯w3∂z¯w�2 L2(Ω)in function of the strong regularity. Concretely, we obtain a bound of the form �¯w�3 D(A)�¯w�V which cannot be bounded using the definition of Y(T). Therefore, we cannot continue with the Fixed Point reasoning. 7 Uniqueness of weak/strong solution (proof of Theorem 1.6) We start from a weak solution uof the system (EP) (see definition 1.1), in particular u verifies the energy inequality: 1 2�u(t)�2 L2(Ω)+�t 0�νh�∂xu�2 L2(Ω)+νv�∂zu�2 L2(Ω)�ds ≤1 2�u0�2 L2(Ω)+�t 0�F,u�Ωds +�t 0�τ,u�Γsds, a.e. t ∈(0,T). (43) Suppose that there exists another weak solution ¯umore regular (associated to the same data u0and F). The idea is to find under what regularity conditions, only on ¯u,wehave that u≡¯u. Observe that, starting from the weak variational formulation of u(definition 1.1), it is easy to verify that ∂tu∈L4/3(0,T;W�), where W={ψ∈V;∂zψ∈L4(Ω)}. In fact, if we want to take ϕ=¯uas test function in the weak variational formulation of u, the unique term that presents problems is �Ωu3∂z¯uudΩ. Then, with the additional regularity of the Theorem 1.6for¯u(recall ∂z¯u∈L4(0,T;L4(Ω))) this term has a sense, hence one verifies the following equality: a.e. t∈(0,T), �u(t),¯u(t)�Ω−�t 0�∂t¯u, u�Ωds +�t 0�Ω(νh∂xu∂x¯u+νv∂zu∂z¯u)dΩds =�u0�2 L2(Ω)+�t 0�F, ¯u�Ωds +�t 0�Ω(u∂x¯u+u3∂z¯u)udΩds +�t 0�τ,¯u�Γsds. (44) Now, we write the differential system for (¯u, ¯ps)as: ∂t¯u+u∂x¯u+u3∂z¯u−νh∂2 xx ¯u−νv∂2 zz ¯u+∂x¯ps =F+(u−¯u)∂x¯u+(u3−¯u3)∂z¯u. (45) Thanks to the additional regularity of ¯u,wecanmultiply(45)byuand integrate on Ω×(0,t): �t 0�∂t¯u, u�Ωds +�t 0�Ω�(u∂x¯u+u3∂z¯u)u+νh∂x¯u∂xu+νv∂z¯u∂zu�dΩds =�t 0�F,u�Ωds +�t 0�Ω�(u−¯u)∂x¯u+(u3−¯u3)∂z¯u�udΩds +�t 0�τ,u�Γsds (46) 22 Adding (44) and (46), the terms �t 0�∂t¯u, u�Ωds and �t 0�Ω(u∂x¯u+u3∂z¯u)udΩds are cancelled, obtaining: �u(t),¯u(t)�Ω+�t 0�Ω2(νh∂xu∂x¯u+νv∂zu∂z¯u)dΩds =�u0�2 L2(Ω)+�t 0�F,u +¯u�Ωds +�t 0�τ,u +¯u�Γsds +�t 0�Ω�(u−¯u)∂x¯u+(u3−¯u3)∂z¯u�udΩds a.e. t∈(0,T). (47) Finally, we multiply (45) by ¯uand integrate on Ω×(0,t), obtaining the energy equality: 1 2�¯u(t)�2 L2(Ω)+�t 0�νh�∂x¯u�2 L2(Ω)+νv�∂z¯u�2 L2(Ω)�ds =1 2�u0�2 L2(Ω)+�t 0�F, ¯u�Ωds +�t 0�τ,¯u�Γsds +�t 0�Ω�(u−¯u)∂x¯u+(u3−¯u3)∂z¯u�¯udΩds, (48) where the last term on the right hand of (48) vanishes (by the free divergence condition). Then, doing (43) + (48) −(47), we obtain: a.e. t ∈(0,T), 1 2�u(t)−¯u(t)�2 L2(Ω)+�t 0�u(s)−¯u(s)�2 Vds ≤−�t 0�Ω�(u−¯u)∂x¯u+(u3−¯u3)∂z¯u�udΩds =−�t 0�Ω�(u−¯u)∂x¯u+(u3−¯u3)∂z¯u�(u−¯u)dΩds =−�t 0�Ω|u−¯u|2∂x¯udΩds −�t 0�Ω(u3−¯u3)∂z¯u(u−¯u)dΩds ≡I1+I2 (49) We estimate the Ii-terms (using lemmas of Section 2): I1≤�t 0�∂x¯u�L2(Ω)�(u−¯u)�2 L4(Ω)ds ≤�t 0�∂x¯u�L2(Ω)�(u−¯u)�L2(Ω)�∇(u−¯u)�L2(Ω)ds ≤1 4�t 0�(u−¯u)�2 Vds +C�t 0�(u−¯u)�2 L2(Ω)�∂x¯u�2 L2(Ω)ds 23 I2≤�t 0�u−¯u�L4(Ω)�∂z¯u�L4(Ω)�u3−¯u3�L2(Ω)ds ≤Dmax �t 0�∇(u−¯u)�1/2 L2(Ω)�u(s)−¯u�1/2 L2(Ω)�∂z¯u�L4(Ω)�∂x(u3−¯u3)�L2(Ω)ds ≤CD max �t 0�u−¯u�3/2 V�u−¯u�1/2 L2(Ω)�∂z¯u�L4(Ω)ds ≤1 4�t 0�(u−¯u)�2 L2(Ω)ds +CD 4 max �t 0�∂z¯u�4 L4(Ω)�u−¯u�2 L2(Ω)ds Hence, the inequality (49) becomes: a.e. t∈(0,T), �u(t)−¯u(t)�2 L2(Ω)+�t 0�u(s)−¯u(s)�2 Vds ≤C�t 0��∂x¯u(s)�2 L2(Ω)+D4 max�∂z¯u(s)�4 L4(Ω)��u(s)−¯u(s)�2 L2(Ω)ds (50) Then, from Gronwall lemma, we can conclude the uniqueness. Remark 7.1 In the 3D case, the bound obtaining for I2is 1 4�t 0��u(s)−� ¯u(s)�2 Vds +C�t 0�∂z� ¯u(s)�8 L4(Ω)��u(s)−� ¯u(s)�2 L2(Ω)ds. Now, to obtain uniqueness, we have to impose in � ¯uthe following additional regularity ∂z� ¯u∈L8(0,T;L4(Ω)2), which it is not a consequence of the strong regularity. References [1] J. L. Lions, R. Teman, S. Wang, New formulation of the primitive equations of the atmosphere and applications. Nonlinearity, 5, 1992, 237-288. [2] J. L. Lions, R. Teman, S. Wang, On the equations of the large scale Ocean. Nonlinearity, 5, 1992, 1007-1053. [3] R. Lewandowski, Analyse Math´ematique et Oc´eanographie, Masson, 1997. [4] J. Pedlosky, Geophysical fluid dynamics, Springer-Verlag, 1987. [5] O. Besson & M. R. Laydi, Some Estimates for the Anisotropic Navier-Stokes Equations and for the Hydrostatic Approximation, M2AN-Mod. Math. Ana. Nume., Vol. 7, 1992, 855-865. [6] P. Az´erad & F. Guill´en, ´ Equations de Navier-Stokes en bassin peu profond: l’approximation hydrostatique, C. R. Acad. Sci. Paris, t.329, S´erie I, 1999, 961-966. [7] T. Chac´on & F. Guill´en, An intrinsic analysis of existence of solutions for the hydrostatic approximation of Navier-Stokes equations, submitted. [8] F. Guill´en & M. V. Redondo, Convergencia de algunos esquemas num´ericos hacia el modelo evolutivo de Ecuaciones Primitivas, Actas XVI CEDYA, VI CMA, University of Las Palmas de Gran Canaria 1999, 1165-1172. 24 [9] F. Guill´en & M. V. Redondo, work in preparation. [10] M. Ziane, Regularity Results for Stokes Type Systems. Applicable Analysis, Vol. 58, 1995, 263-292. [11] L. Nirenberg, On Elliptic Partial Differential Equations, Ann. Scuo. Norm. Sup. Pisa, 13 (3), 1959, 115-162. [12] R. A. Adams Sobolev spaces, Academic Press, New York, 1975. [13] J. L. Lions, Quelques M´ethodes de R´esolution des Probl`emes aux Limites Non Lin´eaires, Dunod, Paris, 1969. 25