A review on the improved regularity for the primitive equations

A review on the improved regularity for the Primitive Equations∗ F. GUILL´ EN-GONZ´ ALEZ, & M. A. RODR´ IGUEZ-BELLIDO Dpto. de Ecuaciones Diferenciales y An´alisis Num´erico, Universidad de Sevilla, Aptdo. 1160, 41080 Sevilla, Spain. [email protected], [email protected] Abstract In this work we will study, some types of regularity properties of solutions for the geophysical model of hydrostatic Navier-Stokes equations, so-called the Primitive Equations (PE). Also, we will present some results about uniqueness and asymptotic behavior in time. Keywords: Regularity, uniqueness, time asymptotic behavior. AMS Subject Classification: 35Q30, 35B40, 76D05 1 Introduction. The knowledge of seas and oceans has always been a human interest. We cannot forget that at least two thirds of the Earth surface are covered by oceans, and it is surrounded by the atmosphere. From the beginning of XIXth century, some scientists such as Pierre Simon de Laplace thought that the physical laws that govern atmosphere and ocean could serve to predict the future weather and climate. Nevertheless, it was not until the XXth century that people started to treat this prediction by solving differential problems in mathematical physics. The dynamics of geophysical fluids is a subject born in the fifties, including the Oceanography and the Meteorology, and that study large scale fluids (in space and, sometimes, in time). What Meteorology tries to describe are the weather changes, the coast winds, the influence of topography in the local or regional weather, the general circulation, the climate variation,... On the other hand, Oceanography studies “upwelling” phenomena (circulation of deep water), oceanic streams (as the Mexico Gulf Stream) and large scale general circulation (meso-scale and climate scale). According to J. L. Lions, R. Temam and S. Wang [17], in order to understand the turbulent behavior of both the atmosphere and the ocean, and to predict the climate, the following requirements are needed: (a) to establish the equations and mathematical models that govern the movement and the atmosphere and ocean states, and the interactions appearing among them; (b) to know the mathematical basis of these equations and models; ∗Research of the authors has been partially supported by the project BFM2003-06446-CO-01. 1 (c) to design and compute numerical approximates to these equations. The atmosphere is a compressible fluid described mathematically by the Hydrodynamic and Thermodynamic equations, where centripetal and Coriolis forces are also acting. Such equations describe the big scale movements, and small scales are considered as “noises” in the numerical treatment. However, because of the vertical scale is much smaller than the horizontal scales, we can use the hydrostatic approximations in order to obtain the Primitive Equations of the atmosphere and the ocean. The interaction between atmosphere and ocean can be observed when the wind force moves the ocean or when the ocean interfere the behavior of the atmosphere. From a physical point of view, the water dynamics, the distribution of temperature and salinity, and the chemical and biological components of water are interesting in Oceanography. In Meteorology, air dynamics, temperature, humidity and pressure are interesting. Although it seems that the incompressible (or slightly incompressible) Navier-Stokes equations, with variable density and free surface, is one of the most realistic models to simulate the hydrodynamic behavior in Oceanography, the high complexity of this model and the high dimensions of the domain of study motivate some simplifications (see [19]). We describe here one of them. This work is organized as follows. In section 2, we present the physical derivation of the model and some mathematical simplifications. In section 3 the (PE) problem is considered: fist, we present some the functional spaces and definitions; secondly, we show the main steps in the proof of the strong regularity for the solution of (PE), global in time for small data and local in time for any data; and then, we deal with the asymptotic behavior in time. Section 4 is devoted to the uniqueness of solution, proved for weak solutions when some additional hypothesis over the derivative with respect to the z-variable are made. In the fiftieth section, we show the existence of a very weak solution for the linear problem that will help us to weaken the regularity hypothesis over the data in order to prove strong regularity for (PE). Finally, in section 6 we prove anisotropic global regularity and uniqueness of solution global in time for a 2D (PE) model provided with friction boundary conditions on the bottom. 2 Derivation of the Primitive Equations of the ocean model. The ocean can be considered as a slightly compressible fluid, with the influence of centripetal and Coriolis forces. The set of equations that form the so called “large scale ocean model” are: the momentum equation, the continuity equation, the thermodynamic equation (with temperature θ), the diffusion equation for the salinity Sand the state equation for the density: (1)                                      ρDU Dt +2ρW×U+ρW×(W×r)+∇P+ρg=D Dρ Dt +ρ∇·V=0 Dθ Dt =Qθ DS Dt =QS ρ=ρ(θ,S) 2 where Vis the 3D velocity field, Pis the pressure, g=(0,0,g) is the gravity, 2ρW×Vis the Coriolis term and ρW×(W×r) the centripetal forces (W=f(0,cos λ,sin λ) is the Earth rotation vector, f its module, λ=λ(y) is the latitude and ris the Earth ratio). On the other hand, Dis the molecular dissipation, Qθand QSare the temperature and salinity diffusions, respectively. We will use the following operators: ∇=(∂x,∂ y,∂ z) the 3D gradient, with ∇·the divergent operator and D Dt the material derivative, i.e. D Dt =∂t+U·∇ In what follows, we will do a β-plane approximation, that means to suppose that the earth surface can be approached locally by the tangent plane to a central point of this neighborhood, where βis the deformation angle from the sphere over the plane. In this case, the domain of ocean Ω, can be described in cartesian coordinates as: Ω={(x, y, z)=(x,z)∈IR 3,x∈S, −H(x)<z<0}. Its boundary is ∂Ω=Γb∪Γl∪Γswhere the bottom Γb, the sidewalls Γland the surface Γsare defined by: Γb={(x,z)∈IR 3:x∈S, z =−H(x)}, Γl={(x,z)∈IR 3:x∈∂S, −H(x)<z<0}, Γs={(x,0) : x∈S}, where the horizontal section Sis an open set in IR2and the depth His a non-negative continuous function over S. In order to avoid theoretical and computational difficulties, two main simplifications are considered in (1): a) Boussinesq approximation, that neglect the differences of density in all the equations of the system except the gravity term and the state equation. In this way, once a medium density ρ0 is fixed, then ρ=ρ0+ρ�with ρ�<< ρ0. The continuity equation is then the incompressibility equation for the velocity U. The inclusion of the centripetal forces in the gradient of a potential function p (along with the pressure), they allow to consider the following model of Navier-Stokes with anisotropic viscosities: (BEs)                  D DtU−∇·(Dν(U)) + 2W×U+∇p=−ρ� ρge3 ρ=ρ(θ,S),∇·U=0 D Dtθ−∇·(Dνθ(θ)) = 0,D DtS−∇·(DνS(S)) = 0 Here, D Dt =∂t+U·∇is the material derivative, ν,νθ,ν S>0 are anisotropic (eddy) diffusion coefficients (with different order in horizontal and vertical) of (U,θ,S) respectively, where Dν(U)= ∇νU+∇νUtand ∇ν=(νx∇x,ν z∂z)t,with∇x=(∂x,∂ y)tthe horizontal gradient operator. 3 b) Hydrostatic approximation. An analysis of spatial scales says that the aspect quotient (reason between the vertical Zand horizontal Lcharacteristic lengths) is small, namely: δ=Z L≈10−3. It is also possible to observe that the vertical water velocity is much smaller than the horizontal ones, which is modelled approximating the third momentum equation by the so-called hydrostatic equation: ∂p ∂z=−ρg, which relates the ocean pressure and density with the gravity, and that has become a fundamental equation in Oceanography. This analysis also shows that for the viscosities in each direction to be of the same order (respect to δ), we have to suppose: (2) νz=δ2νv,ν x=νh,with νv=O(1) and νh=O(1). By simplicity, we only treat the (nonlinear) system for velocity U=(u,v) (where u=(u1,u 2) and vare the horizontal and vertical velocities respectively) and pressure p, because of the system coupled with temperature and salinity (of convection-diffusion type) do not introduce any new mathematical difficulties. This system is called Hydrostatic Navier-Stokes equations, which can be described as follows: (HNS)       ∂tu+u·∇xu+v∂zu−νh∆xu−νv∂2 zzu+αu⊥+1 ρ0∇xp=0 ρ=ρ0+ρ�(θ,S),∂ zp=−ρg, ∇x·u+∂zv=0, where α=2fsin(λ). The surface Γsis the same as before, where the new Ω,Γland Γb(with h=H Z) are described as follows: (3) Ω={(x,z)∈IR 3,x∈S, −h(x)<z<0}, Γl={(x,z)∈IR 3:x∈∂S, −h(x)<z<0}, Γb={(x,z)∈IR 3:x∈S, z =−h(x)}. A derivation of (HNS) of the ocean from the hydrostatic approximation hypothesis is obtained in the works of J. L. Lions, R. Temam and S. Wang, [15, 16]. Such hypothesis can be justified as the limit of the weak solution of the Navier-Stokes equations or (BEs)whenδ→0 imposing (2) (see the work of O. Besson and M. R. Laydi, [3], for the stationary case, and the work of P. Az´erad and F. Guill´en- Gonz´alez, [2], for the evolutionary case). 2.1 The boundary conditions. The exchange between atmosphere and ocean determine the interface conditions, called surface boundary conditions when the isolated model of the ocean is considered. A simplifying hypothesis is the “rigid lid” hypothesis; namely, the interface atmosphere-ocean is assumed flat, thanks to two facts: 4 (a) the water density is much greater than the air density; ρa/ρ≈10−3,whereρaand ρare the air and oceanic water density respectively. Then, the atmosphere-ocean interface is very stable considering great spatial scales, due to the intensity of the gravitational force. (b) in the oceanic scale, the vertical displacement of the tides waves usually is neglected in most of Global Circulation models. Denoting with the upper-index athe variables of the atmosphere, the surface boundary conditions are: v|Γs=0,u|Γs=ua|Γs. Nevertheless, due to the difference of density between both states, a thin boundary layer appears in the atmosphere (of 1 km of thickness) and very fine in the ocean (between 10 and 100 m.). A possible modelling of this boundary layer is given by: v=0 −ρ0νv∂zu=ρaCa D(ua−u)|ua−u|αon Γs, where Ca Dis a momentum transfer coefficient. Following the references [16, 15], we consider the simplification: v=0,ν v∂zu=Υon Γs, where Υis the wind stress tensor on the surface of the ocean, which is given as a datum or as a linear function of u: for instance Υ=−C|ua|α(ua−u). With respect to the bottom and sidewalls, we will always impose the slip condition (u,v)·n= 0 on Γl∪Γb.OnΓ lthis condition yields u|Γl= 0 (allowing vertical sliding on the sidewalls). On the bottom, two additional conditions should be imposed, that could be of adherence or friction type: u=0on Γb(or (∇νu)n+βu=0on Γb) where β=β(x)>0 is a coefficient depending of the bottom roughness. From a physical point of view, the election of homogeneous Dirichlet boundary conditions for the velocity on the bottom is only justify when the molecular viscosity of the fluid is important. Nevertheless, in many geophysical models eddy viscosity is considered, neglecting the molecular viscosity. On the other hand, one knows that the use of the friction boundary condition in the Navier-Stokes equations prevents the appearance of boundary layers. 2.2 The reduced model. The unknowns (u,v,p) of the (HNS) system have different roles: the horizontal velocity uverifies an evolution problem and therefore needs initial data (prognostic variable). The vertical velocity vcan be determined from u(diagnostic variable). Indeed, integrating the incompressibility equation in (z,0) and using the rigid lid hypothesis v|Γs= 0, one has: (4) v(t;x,z)=�0 z∇x·u(t;x,s)ds. With regard to the pressure, integrating the hydrostatic equation in (z,0), one has: p(t;x,z)=ps(t;x)+�0 z (ρg)(t;x,s)ds =ps(t;x)−ρ0gz +g�0 z ρ�(θ,S)(t;x,s)ds, 5 where ps(t;x)=p(t;x,0) is a potential function only defined in the surface of the ocean, namely the atmospheric pressure plus the surface lid pressure (this latter is the pressure exerted by undulations of a free surface), and −ρ0gz +g�0 zρ�(θ,S)(t;x,s)ds is the baroclinic pressure (where −ρ0gz is an average of the pressure exerted by the water column between zand 0). Then, horizontal gradient of pressure is rewritten as: 1 ρ0∇xp=1 ρ0∇xps+F(θ,S). On the other hand, using in (4) the slip boundary condition on the bottom (u,v)·n|Γb= 0, one arrives to the constraint (see [16, 14]) (5) ∇x·�u�=0in(0,T)×S,where�u�(t;x)=�0 −h(x) u(t;x,z)dz. Then, we arrive at the following reduced system, that it will called Primitive Equations : (PE)                                  ∂tu+u·∇xu+v∂zu−νh∆xu−νv∂2 zzu+αu⊥ +1 ρ0∇xps=Fin (0,T)×Ω, ∇x·�u�=0 in(0,T)×S, u|t=0 =u0in Ω, νv∂zu|Γs=Υ,u|Γb∪Γl=0in (0,T), where vdepends on uas in (4). This system displays some advantages, from the computational point of view, eliminating the unknown vand reducing the unknown pressure to a surface function. Nevertheless, due to the dependency of vwith respect to ∇x·u, an anisotropy in the regularity of the derivatives of vis produced. For example, in the weak solution framework, u∈H1(Ω), hence using the incompressibility equation, ∂zv=−∇x·u∈L2(Ω) but ∇xv/∈L2(Ω) in general. This anisotropy implies that the nonlinear terms of the momentum equations are less regular than in the Navier-Stokes case. Another fact to consider, is that whereas (BEs) is a differential model, (PE) it is a integral-differential one. Remark 1 When in the Navier-Stokes model, free surface is considered (as a new unknown), the rigid lid condition (v=0on Γs) must be changed by the free surface equation, arriving at the so-called 3D Shallow Water model. 3 Regularity for the Primitive Equations model (PE). 3.1 Functional spaces and definitions. Before making a mathematical study of problem (PE), we describe the functional spaces and the definitions of very weak, weak and strong solution: C∞ b,l(Ω) = {ϕ∈C∞(Ω)2;supp(ϕ) is a compact set ⊆Ω\(Γb∪Γl)}, H1 b,l(Ω) = C∞ b,l(Ω)H1 ={v∈H1(Ω)2;v= 0 on Γb∪Γl}, 6 H−1 b,l (Ω) = dual space of H1 b,l(Ω), V={ϕ∈C∞ b,l(Ω)2;∇x·�ϕ�=0 inS}, H=VL2 ={v∈L2(Ω); ∇x·�v�=0 inS, �v�·n|∂S=0}, V=VH1 ={v∈H1(Ω); ∇x·�v�=0 inS, v|Γb∪Γl=0}. Taking regular test functions in (PE) and integrating by parts, we obtain: Definition 1 (Weak solution) Let u0∈H,F∈L2(0,T;H−1 b,l (Ω)) and Υ∈L2(0,T;H−1/2(Γs)) be given functions. We say u:(0,T)×Ω→IR 2is a weak solution of (PE)in (0,T)if u∈L∞(0,T;H)∩L2(0,T;V), verifying the variational formulation: ∀ϕ∈C1([0,T]; V)such that ϕ(T)=0, �T 0�Ω�−u·�∂tϕ+(u·∇x)ϕ+u3∂zϕ�+αu⊥·ϕ�dΩdt +�T 0�Ω (νh∇xu:∇xϕ+νv∂zu·∂zϕ)dΩdt =�Ω u0·ϕ(0) dΩ+�T 0�F,ϕ�Ωdt +�T 0�Υ,ϕ�Γsdt, and, moreover, usatisfying the energy inequality: (6) 1 2�u�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�Υ,u�Γsds c.p.d. t ∈(0,T). In the case T=+∞,wesaythatuis a weak solution of (PE)in (0,+∞)if uis a weak solution of (PE)in (0,T),∀T<+∞. Observe that �·,·�Ωdenotes the duality between H−1 b,l (Ω)and H1 b,l(Ω),and�·,·�Γsdenotes the duality between H−1/2(Γs)and H1/2(Γs). In this section, u3will denote the vertical velocity associated to u. Finally, we denote the V-norm by �ϕ�2 V=νh�∇xϕ�2 L2(Ω)+νv�∂zϕ�2 L2(Ω),andtheH1 b,l(Ω)-norm by �ϕ�2 H1(Ω)=�∇xϕ�2 L2(Ω)+�∂zϕ�2 L2(Ω). Definition 2 (Strong solution) Let u0∈V,F∈L2(0,T;L2(Ω)), Υ∈L2(0,T;H1/2(Γs)) and ∂tΥ∈L2(0,T;H−1/2(Γs)) be given functions. If uis a weak solution of (PE)in (0,T),wesaythatuis a strong solution if it verifies the following additional regularity: u∈L∞(0,T;V)∩L2(0,T;H2(Ω)∩V),∂ tu∈L2(0,T;H). The existence of weak solution of (PE) is well-known from the works of Lewandovski [14] and Lions- Teman-Wang [16] in domain whose depth is strictly bounded from below (i.e., h≥hmin >0inS). They use a Galerkin method in order to obtain the velocity uin a space with the restriction ∇·�u�= 0. The pressure will be recovered later thanks to a De Rham Lemma, specific for this kind of spaces. In domain 7 without this restriction the existence of weak solution is obtained as a consequence of a limit process applied to the Navier-Stokes equations with anisotropic viscosity, when the aspect quotient tends to zero (see Besson-Laydi [3] for the stationary case and Azerad-Guill´en [2] for the evolutionary case. Other proofs by internal approximations can be seen in [6] for the stationary case and [9] for the evolutionary case. The novelty of the results of authors is the proof of existence of strong solution for the nonlinear system (PE) and the uniqueness. The linear stationary case has been studied by M. Ziane, [21]. One of the main difficulty for this study is the treatment of the boundary conditions: Neumann non homogeneous on the surface and Dirichlet homogeneous on the bottom and sidewalls. Uniqueness of weak solution is still an open problem, but the regularity hypothesis for it have been weakened. 3.2 Strong regularity for the Primitive Equations. We start our study by the linear evolutionary system associated to the Primitive Equations (for simplicity in the exposition, we will omit the Coriolis term): (S)                        ∂tv−νh∆xv−νv∂2 zzv+∇xqs=Fin (0,T)×Ω, ∇x·�v�=0 in(0,T)×S, v|t=0 =u0in Ω, νv∂zv|Γs=Υ,v|Γb∪Γl=0in (0,T). The stationary problem associated will be called (Sst). Theorem 1 (Weak solution of (Sst))Let S⊆IR d(d=1or 2)andΩ⊆IR d+1 be Lipstchitz-continuous domains defined by (3).IfF∈H−1 b,l (Ω)and Υ∈H−1/2(Γs), then the problem (Sst)has a unique solution v∈H1(Ω). Moreover, there exists a constant C=C(Ω)>0such that if ν=min{νh,ν v}, we obtain: (7) �v�2 H1(Ω)≤C ν2��Υ�2 H−1/2(Γs)+�F�2 H−1 b,l (Ω)�. In [3], [6] and [14], there are different proofs of this result. Theorem 2 (Strong solution for (Sst))([21]) Let S⊆IR d(d=1or 2)beaC3domain and h∈ C3(S)the depth verifying h≥hmin >0in S.IfF∈L2(Ω)and Υ∈H1/2+ε 0(Γs)(for some ε>0), then there exists a (unique) strong solution vof (Sst)(i.e., v∈H2(Ω)∩V). Moreover, there exists a constant C=C(Ω)>0such that: (8) �v�2 H2(Ω)≤C ν2��F�2 L2(Ω)+�Υ�2 H1/2+ε 0(Γs)�. This result of strong regularity ([10, 20]), must be extended to the linear evolutionary case (S). First of all, we get a lift of the boundary conditions: In this way, we define the operator B:a∈H−1/2(Γs)→ u=Ba∈V,whereuis the weak solution of the hydrostatic Stokes problem (Sst)withF=0and Υ=a. Then, we consider e(t)=B(Υ(t)) which has strong regularity, and we prove that ∂te(t) coincides with B(∂tΥ(t)) which has weak regularity, and therefore e∈C0([0,T]; V). Secondly, we consider the homogeneous problem verified by y=v−e. The estimates of energy deduced for eand ∂tethanks to Theorem 1 and Theorem 2 let us conclude the following result: 8 Theorem 3 (Strong solution of (S))Let S⊆IR d(d=1or 2) be a C3-domain and h∈C3(S)the depth with h≥hmin >0in S.IfF∈L2((0,T)×Ω),u0∈V,Υ∈L2(0,T;H1/2+ε 0(Γs)),foranyε>0, with ∂tΥ∈L2(0,T;H−1/2(Γs)), then there exists a unique strong solution vof (S)in (0,T). Moreover, there exists a constant C>0such that: (9) �v�2 L∞(V)+�v�2 L2(H2(Ω)) +�∂tv�2 L2(H)≤C��u0�2 V+�Υ(0)�2 H−1/2(Γs) +�F�2 L2(L2(Ω)) +�Υ�2 L2(H1/2+ε 0(Γs)) +�∂tΥ�2 L2(H−1/2(Γs))� Once the linear problem has been studied, we deal with the strong regularity for the nonlinear problem (PE). We use the previous theorem to lift the boundary conditions of (PE) problem and thus to study the homogeneous boundary problem verified by (w,π s), where w=u−v,πs=ps−qs, for (v,q s)the solution of (S): (NL)                    ∂tw−νh∆xw−νv∂2 zzw+(w+v)∂x(w+v) +(w3+v3)∂z(w+v)+∂xπs=0in (0,T)×Ω, ∇x·�w�=0 in(0,T)×S, w|t=0 =0 inΩ, νv∂zw= 0 on (0,T)×Γs,w= 0 on (0,T)×(Γb∪Γl), with w3=�0 z∇x·wds and in a similar manner for v3. In the spirit of Galerkin method, we approach wfunctions by wm. They are the Galerkin approximates in the m-dimensional space Vm, composed by an orthogonal and unitary base in Vof eigenfunctions of the hydrostatic operator A:V→V�such that: (10) �Au,v�V�,V =�Ω�νh∇xu:∇xv+νv∂zu·∂zv�dΩ∀u,v∈V, is associated to the homogeneous boundary conditions (Neumann on the surface and Dirichlet on the bottom and sidewalls). In order to obtain estimates in the H2(Ω)-norm we take Awm(t)∈Vmas test functions, obtaining: (11) 1 2 d dt�wm�2 V+�Awm�2 L2(Ω)=G(wm,v), for a certain function G. Using the estimates in the strong norm for v(depending on data) and controlling the terms in wmwith the term appearing on the left side of (11), we try to bound G. The big difficulty appears in the terms: I1=−�Ω (wm·∇x)wm·AwmdΩand I2=−�Ω (wm)3∂zwm·AwmdΩ respective to the nonlinear term of (PE). Observe that I2is less regular than I1due to the anisotropy regularity of the vertical velocity. In order to bound the I2-term, the following Lemma ([10]) will be basic: Lemma 4 Let Ω⊆IR N(N=2or 3) be the domain considered in the (PE)problem. Then, for all function v∈W1,p(Ω)N−1(p>1), if we define v3as v3(x,z)=−�z −h(x)∇x·v(x,s)ds, we have: �v3�Lp(Ω)≤hmax�∇x·v�Lp(Ω) 9 5.2 The very weak regularity. Definition 4 Apair(u,p)is called a very weak solution of (26) iffthe following conditions are verified: (29)                    u∈L2(Ω),p∈(H1(S))�/IR , �Ω u·gdΩ+�p, ϕ�S=l(g,ϕ), ∀g∈L2(Ω),∀ϕ∈H1(S)such that �S ϕdx=0, where �·,·�Sdenote the duality between (H1(S))�and H1(S), where l:L2(Ω)×H→IR is defined by:        l(g,ϕ)=�Ω F·φdΩ+�Υ,φ�Γssi F∈L2(Ω), l(g,ϕ)=�Fφ�Ω+�Υ,φ�Γssi F∈(H2(Ω)∩H1 b,l(Ω))�, where (φ,π)is the solution of the dual problem (27) and �·,·�Γsthe duality between H−3/2(Γs)and H3/2 0(Γs)(and �·,·�Ωthe duality between (H2(Ω)∩H1 b,l(Ω))�and H2(Ω)∩H1 b,l(Ω)). It is easy to see that lis a continuous linear operator from L2(Ω)×Hinto IR . Using (27), we rewrite the previous definition as: (30)                u∈L2(Ω),p∈H �, �Ω u·�−ν∆φ−ν3∂2 zzφ+∇π�dΩ−�p, ∇·�φ��S=�F,φ�Ω+�Υ,φ�Γs, ∀φ∈H2(Ω)∩H1 b,l(Ω),∇·�φ�∈H1(S) and ∂zφ|Γs=0,∀π∈H1(S). Therefore, we give the following result: Theorem 16 Given F∈(H2(Ω)∩H1 b,l(Ω))�and Υ∈H−3/2(Γs)there exists a unique very weak solution (u,p)of (29) in L2(Ω)×(H1(S))�/IR (p∈(H1(S))�unique up to additive constant). Moreover, (31) �u�L2(Ω)+�p�(H1(S))�/IR ≤C��F�(H2(Ω)∩H1 b,l(Ω)2)�+�Υ�H−3/2(Γs)�. As in [7], the proof of Theorem 16 needs the result: Proposition 17 The space (H1(S))�/IR is isomorphic to H�,thedualspaceofH. Proof. (scheme of the proof of Theorem 16) Since l:L2(Ω)×H→IR is a linear continuous operator, there exists a unique pair (u,�p)∈L2(Ω)×H�(H�the dual space of H) such that: �Ω u·gdΩ+��p, ϕ�H�,H=l(g,ϕ)∀g∈L2(Ω),∀ϕ∈H. From Proposition 17 we can identify �pwith a distribution pin (H1(S))�/IR such that ��p, ϕ�H�,H= �p, ϕ�(H1(S))�,H1(S),∀ϕ∈H. Therefore, we conclude that (u,p) is a solution of (29), and this proves the existence of solution. The uniqueness follows from the method used in the construction of the solution. The continuous dependence of the solution with respect to the data the estimate (28) is used. Once the regularity of problem (26) is obtained, we proved: 16 Proposition 18 Let (u,p)∈L2(Ω)×H�the unique solution of (29). Then, (u,p)satisfy (26)1−2in the sense of distributions in Ωand Srespectively. Finally, it is possible to give a sense to the boundary conditions in certain dual spaces, defining that we call “generalized traces” and which coincides with the standard trace operator for regular functions (see [5] for the details). As we explained before, the final version of the regularity result for Primitive Equations (S)is: Theorem 19 Let S⊆IR d(d=1or 2)aC3-domain and h∈C3(¯ S)the depth function verifying h≥hmin >0in ¯ S.IfF∈L2(0,T)×Ω),u0∈V,Υ∈L2(0,T;H1/2+ε 0(Γs)) ∩L∞(0,T;H−1/2(Γs)), for any ε>0with ∂tΥ∈L2(0,T;H−3/2(Γs)) and Υ(0) ∈H−1/2(Γs), then there exists a unique strong solution vof (S)in (0,T). Moreover, there exists a constant C>0such that: (32) �v�2 L∞(V)+�v�2 L2(H2(Ω)) +�∂tv�2 L2(H)≤C��u0�2 V+�F�2 L2(L2(Ω)) +�Υ(0)�2 H−1/2(Γs)+�Υ�2 L∞(H−1/2(Γs)) +�Υ�2 L2(H1/2+ε 0(Γs)) +�∂tΥ�2 L2(H−3/2(Γs))� Remark 4 In the case of S⊆IR 2of C∞-class, the hypothesis Υ∈L∞(0,T;H−1/2(Γs)) and Υ(0) ∈ H−1/2(Γs)are not needed. Indeed, from Υ∈L2(0,T;H1/2+ε 0(Γs)) and ∂tΥ∈L2(0,T;H−3/2(Γs)) we can obtain Υ∈C([0,T]; H−1/2(Γs)) with continuous dependence (see [13]). Remark 5 (Application to the nonlinear evolutionary Primitive Equations) The extension of Theorem 19 to the nonlinear case is identical to the extension obtained in [10, 11], replacing Theorem 3 by Theorem 19. 6 Regularity and uniqueness for the 2D model. The main object is to obtain existence of weak solution uwith additional weak regularity for ∂zufor case of friction on the bottom ∂zu|Γb=βu|Γb. This model was obtained, in the 2D case, from (BEs) with friction boundary condition on the bottom as the aspect quotient δtends to zero ([5]) (note that the ”usual” model is obtained in the same way when homogeneous Dirichlet boundary conditions on the bottom are considered). In particular, this solution is unique. The problem is: To find velocity (u, v) and pressure psuch that: (PE)2D                    ∂tu+u∂xu+v∂zu−νh∂2 xu−νv∂2 zu+∂xps=fin (0,T)×Ω, v(t;x, z)=�0 z∂xu(t;x, s)ds in (0,T)×Ω,�u�=0 in(0,T)×S, νv∂zu|Γs=α|ua|(ua−u),u |Γl=0,ν v∂zu|Γb=β(x)uin (0,T), u|t=0 =u0in Ω. Remark 6 In 2Ddomains, the constraint derive to �u�≡0that is deduced from ∂x�u�=0in (0,T)×S and �u�=0on (0,T)×∂S. Remark 7 To assure that the model is dissipative, one must impose γ(x)≥0,with γ(x)=�β(x)�1+νh νv|D�(x)|2�−νh 2D��(x)�, 17 which is derived from the limit of dissipative hypothesis of 2D(BEs) as δ→0. Definition 5 (Weak-vorticity solution) We say that uis a weak-vorticity solution of (PE)in (0,T) if it is a weak solution (u∈L∞(0,T;H)∩L2(0,T;V)), that satisfies the additional regularity: ∂zu∈L∞(0,T;L2(Ω)) ∩L2(0,T;H1(Ω)). Remark 8 The function ∂zucan be called the vorticity of (PE)2D,becauseisthelimitofthe2D(BEs) vorticity as δ→0([4]). We state the main result: Theorem 20 Let h∈H2(S)with |h�|>0on ∂S,β∈H1 0(S),f∈L2(0,T;L2(Ω)),∂zf∈L2(0,T;H−1(Ω)), ua∈L∞(0,T;H1 0(S)),∂tua∈L2(0,T;L1(S)),u0∈H,∂zu0∈L2(Ω). Assuming dissipative hypothesis γ(x)≥0in Sand that depth function hsatisfies |h�(x)|/D(x)≤c/dist(x, ∂S)in S, then there exists a unique weak-vorticity solution of (PE)2Din (0,T). In the proof of this result, a problem verified for ∂zuis used. Indeed, differentiating (PE) with respect to z, one has that w=∂zusatisfies the initial-boundary problem:              ∂tw+u∂xw+v∂zw−νh∂2 xw−νv∂2 zw=∂zf, w|Γs=α|ua|(ua−u)/νv,w|Γl=0,w|Γb=β(x)u/νv, w|t=0 =∂zu0 that we will called the vorticity problem. Notice that, given uand v, this problem is linear and parabolic, and in addition the pressure “has disappeared”. Therefore, we can expect weak regularity for ∂zu. But, boundary conditions for ∂zuon Γsand Γbdepend on uand uand pare coupled by the (PE) problem. Consequently, the problem verified by ∂zudepends also on the pressure, because a lifting of the nonhomogeneous boundary conditions must be done. The main problems to solve are two: First, we need to improve the regularity of the pressure, in order to obtain a weak solution wof the vorticity problem. Then, we need to identify wwith ∂zu, where the difficulty is that ∂zuin principle only has L2(0,T;L2(Ω)) regularity. The following lemma guarantees a certain weight regularity for the pressure that a posteriori will be sufficient to obtain a weak solution of the vorticity problem. Lemma 21 Under hypothesis of Theorem 20,if(u, v, p)is a weak solution of (PE)2D, then one has: √h∂ xps∈L2(0,T;H−1(S)). In the context of weak solution of Navier-Stokes, the regularity of the pressure is obtained from the regularity of the rest of the terms of the momentum equations. In particular, the term ∂tuimplies time regularity of H−1dual type. In this case, using that �u�= 0, in particular ∂t�u�= 0, we can improve the time regularity for the pressure, integrating previously the equation in vertical. We have then to identify the solution vorticity wwith ∂zu. As we already said, the main difficulty is that ∂zuonly belongs to L2(0,T;L2(Ω)), what causes that the well-known results of uniqueness cannot 18 be applied. Then, an alternative is to compare uwith a suitable function �usuch that ��u�= 0 on Sand ∂z�u=w(that can be directly obtained). One has that �u(jointly with a potential function �psdefined in S) verifies the problem: (33) ∂t�u+u∂ x�u+v∂ z�u−νh∂2 xx�u−νv∂2 zz�u+∂x�ps=G, with the same initial and boundary conditions that u,where G=u∂ x�u+�0 z ∂x(u∂z�u)(x, s)ds +f. It is important to notice that G=fwhereas �u=u. Making an uniqueness argument for problems verified by uand ˜u, and using the additional regularity for ∂z�u(since �uis a weak-vorticity solution), one can conclude that u=�u(see [5]). References [1] C. 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