Uniqueness of solution for the 2D Primitive Equations with friction condition on the bottom

Uniqueness of solution for the 2D Primitive Equations with friction condition on the bottom∗ D. Bresch† , F. Guill´en-Gonz´alez‡ , N. Masmoudi§ , M. A. Rodr´ıguez-Bellido¶ Monograf´ıas del Semin. Matem. Garc´ıa de Galdeano. 27: 135–143, (2003). Abstract Uniqueness of solution for the Primitive Equations with Dirichlet conditions on the bottom is an open problem even in 2D domains. In this work we prove a result of additional regularity for a weak solution vfor the Primitive Equations when we replace Dirichlet boundary conditions by friction conditions. This allows to obtain uniqueness of weak solution global in time, for such a system [3]. Indeed, we show weak regularity for the vertical derivative of the solution, ∂zvfor all time. This is because this derivative verifies a linear pde of convection-diffusion type with convection velocity v, and the pressure belongs to a L2-space in time with values in a weighted space. Keywords: Boundary conditions of type Navier, 2D Primitive Equations, uniqueness AMS Classification: 35Q30, 35B40, 76D05 1 Introduction and motivation. Primitive Equations are one of the models used to forecast the fluid velocity and pressure in the ocean. Such equations are obtained from the dimensionless Navier-Stokes equations, letting the aspect ratio (quotient between vertical dimension and horizontal dimensions) go to zero. The first results about existence of solution (weak, in the sense of the NavierStokes equations) are proved for boundary conditions of Dirichlet type on the bottom of the domain and with wind traction on the surface, in the works by Lions-Temam-Wang, ∗The second and fourth authors have been financed by the C.I.C.Y.T project MAR98-0486. †Laboratoire de Math´ematiques Appliqu´ees CNRS 6620 Univ. Blaise Pascal, 63177 Aubi`ere (FRANCE), bresc[email protected]clermont.fr ‡Dpto. Ecuaciones Diferenciales y An´alisis Num´erico, Fac. Matem´aticas, Univ. Sevilla, C/ Tarfia, s/n - 41012 Sevilla (SPAIN), [email protected] §Courant Institute of Mathematical Science, New York University, [email protected]yu.edu ¶Dpto. Matem´atica Aplicada I, E. T. S. de Arquitectura, Univ. Sevilla, Avda. Reina Mercedes, s/n - 41012 Sevilla (SPAIN), [email protected] 135 [5, 6], for domains with vertical walls and in the work of Az´erad-Guill´en, [1], for domains without vertical walls. However, uniqueness of solution remained as an open problem due to the necessity of a more regular solution. In the case of vertical sidewalls, the authors proved in [4] the existence of a more regular solution, global in time for small data or local in time for any data. In these cases, uniqueness of solution is guaranteed. But, from a physical point of view, homogeneous Dirichlet boundary conditions (on the bottom) are only justified when considering a molecular viscosity fluid. In many geophysical fluids, the role of this viscosity is negligible, being more relevant the viscosity due to turbulent effects. It seems then logical to use Navier boundary conditions for the Primitive Equations. Moreover, they prevent the appearance of a boundary layer phenomena on the bottom. The authors obtained the Primitive Equations model with Navier type boundary conditions from the Navier-Stokes equations in [2]. Here, we will focus on the uniqueness problem in the 2D case, see also [3]. We will present what we consider is the first result of uniqueness of weak solution for the 2D Primitive Equations. 2 The model. The domain considered is defined by: Ω = {(x, z)∈R2/ x ∈S, −h(x)<z<0}, where S(ocean surface) is an open interval and h:¯ S→R+is a nonnegative continuous function defined on ¯ Sthat vanishes on ∂S. The boundary of the domain is ∂Ω = ¯ Γb∪Γs, where the bottom is Γb={(x, z)∈R2:x∈S, z =−h(x)}and the surface Γs={(x, 0) : x∈S}. Therefore, the fluid velocity (v, w) and the pressure psatisfy the following equations: (PE)                                    ∂tv+v ∂xv+w ∂zv−νh∂2 xxv−νv∂2 zzv+∂xp=fin (0, T)×Ω, ∂zp= 0, w(t, x, z) = Z0 z ∂xv(t, x, s)ds, hvi= 0 in (0, T)×S, νv∂zv=α|vair|(vair −v) on (0, T)×Γs, νv∂zv=β(x)von (0, T)×Γb, v|t=0 =v0in Ω, where hvi(t;x) = Z0 −h(x) v(t;x, z)dz,vair is the horizontal velocity of the wind at the surface, v0the horizontal initial velocity, (νh, νv) the anisotropic turbulent viscosity, α∈R 136 a positive constant and β=β(x) a positive function defined on S. Remark 2.1 The model for Primitive Equations with Navier conditions deduced in [2] was formed by (P E)1,∂zp= 0 and ∂xv+∂zw= 0 in (0, T)×Ω,νv∂zv=α(vair −v)and w= 0 on (0, T)×Γs,νv∂zv=βv and (v, w)·n= 0 on (0, T)×Γband v|t=0 =v0in Ω. The equation ∂xv+∂zw= 0 and boundary conditions for wimply that w(t;x, z) = R0 z∂xv(t;x, s)ds and ∂xhvi= 0. Finally, as hviis a 1-dimensional function, the hypothesis hvi= 0 on (0, T )×∂S implies that hvi= 0 on (0, T )×S. 3 Definitions and previous results. For the velocity v, we introduce the following spaces: V={ϕ∈C∞ s(Ω) : hϕi= 0 in S,} where C∞ s(Ω) is the space of C∞-functions that vanish in a neighbourghood of ∂Γs. We will denote by Hand Vits closures in the L2(Ω) and H1(Ω)−norms respectively. Definition 3.1 (Weak solution) We say that vis a weak solution for (PE)in (0, T ) if: v∈L∞(0, T;H)∩L2(0, T;V), satisfies the variational formulation: ∀ϕ∈C1([0, T ]; V)with ϕ(T) = 0,                        −ZT 0ZΩ (∂tϕ+v∂xϕ+w∂zϕ)v+ZT 0ZΩ (νh∂xv∂xϕ+νv∂zv∂zϕ) +ZT 0ZS δ(x)v|Γbϕ|Γb+ZT 0ZS α|vair|(v|Γs−vair)ϕ|Γs =ZΩ v0ϕ(0) + ZT 0ZΩ f ϕ +νhZT 0ZS v|Γb∂x[ϕ|Γbh0(x)], with w=R0 z∂xvand satisfies the following energy inequality            1 2kv(t)k2 L2(Ω) +νhZt 0k∂xv(s)k2 L2(Ω) +νvZt 0k∂zv(s)k2 L2(Ω) +Zt 0ZS γ(x)|v|Γb|2+1 2Zt 0ZS α|vair||v|Γs|2≤1 2kv0k2 L2(Ω) +1 2Zt 0ZS α|vair|3 with δ(x) = β(x)1 + νh νv|h0(x)|2and γ(x) = δ(x)−νh 2h00(x). Remark 3.1 In order to ensure that the system is dissipative (necessary property from a physical point of view), we assume that γ(x)≥0. 137 Remark 3.2 Notice that the boundary condition on the bottom is not standard because ∂zvis not the Neumann condition respect to the laplacian operator. This fact produces the term νhRT 0RSv|Γb∂x[ϕ|Γbh0(x)] in the variational formulation. In other words, giving a weak solution v, we can get an associate pressure pthrough the De Rham Lemma (as a Lagrange multiplier) in such a way that (v, w, p)verify the differential problem (PE) in the distribution sense (see [3] for more details). In particular, the following mixed variational formulation can be obtained: ∀ϕ∈C1([0, T]; C∞ s(Ω)), with ϕ(T) = 0, there exists a function ψsmooth enough, satisfying (ϕ, ψ)·n|∂Ω= 0 such that: −ZT 0ZΩ (∂tϕ+v∂xϕ+w∂zϕ)v+ZT 0ZΩ (νh∂xv∂xϕ+νv∂zv∂zϕ) +ZT 0ZS δ(x)v|Γbϕ|Γb+ZT 0ZS α|vair|(v|Γs−vair)ϕ|Γs =ZΩ v0ϕ(0) + ZT 0ZΩ fϕ +νhZT 0ZS v|Γs∂x(ϕ|Γbh0) + ZT 0ZΩ p∇·(ϕ, ψ). (1) Theorem 3.2 (See [2] for a proof of this result.) Suppose that h∈H2(S)with |h0|>0 on ∂S,β∈L∞(S),f∈L2(0, T ;L2(Ω)),vair ∈L3(0, T ;L3(S)),v0∈Hand γ(x)≥0on S. Then, there exists a weak solution vfor (P E)in (0, T). Definition 3.3 (Weak-vorticity solution) We will say that vis a weak-vorticity solution of (P E)in (0, T)if it is a weak solution that also satisfies the additional regularity: ∂zv∈L∞(0, T;L2(Ω)) ∩L2(0, T;H1(Ω)). Remark 3.3 ∂zvcan be seen as the vorticity associated to the Primitive Equations. Indeed, if we consider the vorticity for the 2D Navier-Stokes equations, ωNS =∂zvNS − ∂xwNS, letting the aspect ratio go to zero we arrive at ∂zv. 4 Main result. Theorem 4.1 (Uniqueness of weak solution) Under the hypothesis of Theorem 3.2, if we also consider that β∈H1 0(S),vair ∈L∞(0, T;H1 0(S)),∂tvair ∈L2(0, T;L1(S)),∂zf∈ L2(0, T;H−1(Ω)),∂zv0∈L2(Ω) and the depth function hverifies |h0|/h ≤c/dist(x, ∂S), then there exists a unique weak solution for (P E). Moreover, this solution is a weakvorticity solution. Outline of the proof: Here, we will explain the main ideas that we have followed to prove Theorem 4.1. For a complete proof of this result see [3]. Following the method of P. L. Lions, [7], to prove uniqueness of weak solution for the Navier-Stokes equations we observed that additional regularity is necessary for one 138 of the two solutions compared. Applying the argument to (PE), we observed that this regularity should be ∂zv∈L4(0, T;L4(Ω)). In order to obtain more regularity for ∂zv, we search for the problem verified by ∂zv. First, we formally derive (PE)1respect to z, obtaining that ∂zvsatisfies in D0((0, T)×Ω): ∂t(∂zv) + v ∂x(∂zv) + w ∂z(∂zv)−νh∂2 xx(∂zv)−νv∂2 zz(∂zv) = ∂zf. Knowing vand w, the previous equation is linear and parabolic, because the pressure p has disappeared, so we could expect weak regularity for ∂zv. To this end, we need to study a homogeneous system, so we consider the auxiliary function ψ=νv∂zv−φ v −e with φ(t;x, z) = −α1 + z h(x)|vair(t;x)|− z h(x)β(x) and e(t;x, z) = α|vair(t;x)|vair(t;x)1 + z h(x) auxiliary functions such that ψ|∂Ω= 0. Then, ψverifies the problem: (P)       ∂tψ+v ∂xψ+w ∂zψ−νv∂2 xxψ−νv∂2 zzψ=Fin (0, T)×Ω, ψ= 0 on (0, T)×∂Ω, ψ|t=0 =νv∂zv0−φ|t=0v0−e|t=0 in Ω, where F=G(φ, v, w, e, f) + φ ∂xp. At this point, we have 2 problems: getting an additional regularity for the pressure pto obtain weak regularity for ψ, and identifying ψ+φ v +ewith νv∂zv. Once these problems are solved, then ∂zv∈L2(0, T ;H1(Ω)) ∩L∞(0, T ;L2(Ω)) and in particular belongs to L4(0, T;L4(Ω)), so we will be able to conclude weak uniqueness for (PE). 5 Additional regularity for the pressure. Thanks to ∂zp= 0, we can identify pwith a function psonly defined on S,ps(x) = p(x, z), through the relation: ZΩ p(x, z)ϕ(x, z)dx dz =ZS ps(x)hϕi(x)dx ∀ϕ∈L2(Ω). Theorem 5.1 Assume the hypothesis for the data of Theorem 4.1. If (v, p)is a weak solution of (P E), we have: √h ∂xps∈L2(0, T;H−1(S)). Outline of the proof: For the equations of Navier-Stokes type, the pressure regularity is normally obtained from the regularity of the remaining terms of the equation. The term 139 ∂tvprevents a L2-regularity in time for the pressure. The fact that hvi= 0 on (0, T )×S implies that ∂thvi= 0 on (0, T )×S, so integrating (PE)1in zwe try to improve the regularity for the pressure. In a rigorous form, this vertical integration corresponds to take test functions independent from zin the mixed variational formulation (1). On the other hand, as the pressure pis independent from z, its integration on zonly adds a factor h(x) multiplying p. Moreover, for (ϕ, ψ) any test functions in (1), ZΩ p∇·(ϕ, ψ)dΩ = ZS ps∂xhϕidx. Then, we choose ϕ=ζ/√hwith ζ∈C1 0([0, T]; C∞ 0(S)) as a test function (in particular, this space is dense in L2(0, T ;H1 0(S))). Concretely, we have to give sense to the term ZT 0ZS ps(t;x)∂x(√h ζ)(t;x)dx dt. To this aim, we prove that the others terms from the mixed variational formulation are well-defined and bounded in function of the L2(0, T ;H1 0(S))-norm of ζ. Additional regularity required for the data, hypothesis |h0|/h ≤c/dist(x, ∂S) jointly with Hardy inequalities and the fact that ∂thvi= 0 let finish the proof. 6 Identification of ψ+φ v +ewith νv∂zv. Using a Galerkin method, the additional regularity for plet us obtain weak regularity for ψ, so ψ∈L2(0, T ;H1(Ω)) ∩L∞(0, T ;L2(Ω)). To get ∂zv∈L2(0, T;H1(Ω)) ∩ L∞(0, T;L2(Ω)), we prove that ψ+φ v +e=νv∂zv. The first idea to get this result was to use the uniqueness of weak solution for problem (P), but the problem was that we could not assure the weak regularity for ∂zv(only ∂zv∈L2(0, T;L2(Ω))). Consequently, we looked for a new method to our purpose: We call a=ψ+φ v +eand define ev∈L2(0, T;H1 0(Ω)) ∩L∞(0, T;L2(Ω)) such that νv∂zev=a in Ω and hevi= 0 on S. In fact, we can choose: ev(x, z) = −1 νvZ0 z a(x, s)ds +1 νv 1 h(x)Z0 −h(x)Z0 z a(x, s)dsdz. The idea is to obtain uniqueness for both velocities vand ev, and then ∂zv=∂zev∈ L2(0, T;H1(Ω)) ∩L∞(0, T;L2(Ω)). Starting from the variational formulation for ψ, taking χ=Z0 z η(x, s)ds as test functions, where η∈ D(Ω) with hηi= 0 and taking into account that νv∂zev=α|vair|(vair −v) on Γsand νv∂zev=βv on Γb, 140 we can easily deduce that evverifies the following variational formulation (g FV ): ∀η∈ C1([0, T]; V),                                    Zt 0h∂tev, ηiΩ+Zt 0ZΩ (v ∂xev+w ∂zev)η +Zt 0ZΩ (νh∂xev ∂xη+νv∂zev ∂zη) + Zt 0ZS α|vair|(v|Γs−vair)η|Γs +Zt 0ZS δ(x)v|Γbη|Γb=Zt 0ZΩ f η +Zt 0ZΩv ∂xev+Z0 z ∂x(v∂zev)(x, s)dsη+νvZt 0ZS v|Γb∂x[η|Γbh0(x)] . On the other hand, we know that vsatisfies the following variational formulation (FV ): ∀ϕ∈C1([0, T]; V),                                    hv(t), ϕ(t)iΩ−Zt 0ZΩ (∂tϕ+v∂xϕ+w∂zϕ)v +Zt 0ZΩ (νh∂xv∂xϕ+νv∂zv∂zϕ) +Zt 0ZS α|vair|(v|Γs−vair)ϕ|Γs+Zt 0ZS δ(x)v|Γbϕ|Γb =ZΩ v0ϕ(0) + νhZt 0ZS v|Γb∂x[ϕ|Γbh0(x)] + Zt 0ZΩ fϕ, Taking into account the weak regularity for evand ∂zevand arguing by density, we can take evas a test function in (F V ) and vas a test function in (g FV ). Subtracting both expressions to the energy equality of evand the energy inequality of v, we arrive at ([3]): a. e. t∈(0, T ), 1 2kv(t)−ev(t)k2 L2(Ω) +Zt 0νhk∂x(v−ev) (s)k2 L2(Ω) +νvk∂z(v−ev) (s)k2 L2(Ω)ds ≤Zt 0ZΩv ∂xev+Z0 z ∂x(v ∂zev) (x, s)ds(ev−v)dΩds +νh 2Zt 0ZS|ev|Γb−v|Γb|2h00(x)dxds ≡I+J. (2) Notice that if ev=v, then I= 0 and J= 0. Integrating by parts respect to z, we 141 rewrite Ias: I=Zt 0ZΩ{∂zev ∂x(v−ev)−∂xev ∂z(v−ev)}Z0 z (v−ev)(x, s)dsdΩds ≤min{νh, νv} 4Zt 0kv−evk2 H1(Ω)ds +C(νh, νv)Zt 0k∂xevk2 L2(Ω) +k∂x(∂zev)k4/3 L2(Ω)kv−evk2 L2(Ω)ds. We bound Jusing the Trace and Interpolation Theory in Hs(Ω)-spaces with s∈R: J≤CZt 0kh00kL2(S)k(v−ev)|Γbk2 L4(S)ds ≤CZt 0kh00kL2(S)kv−evk2 H3/4(Ω)ds ≤CZt 0kh00kL2(S)kv−evk1/2 L2(Ω)kv−evk3/2 H1(Ω)ds ≤min{νh, νv} 4Zt 0kv−evk2 H1(Ω)ds +C(νh, νv)Zt 0kh00k4 L2(S)kv−evk2 L2(Ω)ds Then, (2) becomes: kv(t)−ev(t)k2 L2(Ω) +Zt 0νhk∂x(v−ev) (s)k2 L2(Ω) +νvk∂z(v−ev) (s)k2 L2(Ω)ds ≤C(νh, νv)Zt 0k∂zevkL2(Ω)k∂zevkH1(Ω) +k∂xevk2 L2(Ω) +k∂x(∂zev)k4/3 L2(Ω) +kh00k4 L2(S)kv−evk2 L2(Ω)ds. Since ∂zevhas weak regularity, we can use the Gronwall Lemma and deduce that ev=v. References [1] P. Az´erad & F. Guill´en-Gonz´alez. Mathematical justification of the hydrostatic approximation in the Primitive Equations of Geophysical fluid dynamics. To appear in Siam J. Math. 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