On the uniqueness and regularity of the Primitive Equations imposing additional anisotropic regularity

On the uniqueness and regularity of the Primitive Equations imposing additional anisotropic regularity. F. Guill´en-Gonz´alez♣∗† , M.A. Rodr´ıguez-Bellido♠ ♣Dpto. de Ecuaciones Diferenciales y An´alisis Num´erico, Universidad de Sevilla, Aptdo. 1160, 41080 Sevilla, Spain. e-mail: [email protected] ♠Dpto. de Matem´atica Aplicada I, E. T. S. de Arquitectura, Universidad de Sevilla, Avda. Reina Mercedes, 2, 41012 Sevilla, Spain. e-mail: [email protected] Abstract In this note, we prove that given ua weak solution of the Primitive Equations, imposing an additional condition on the vertical derivative of the velocity u(concretely ∂zu∈ L∞(0,T;L2(Ω)) ∩L2(0,T;H1(Ω))), then two different results hold; namely, uniqueness of weak solution (any weak solution associated to the same data that umust coincide with u) and global in time strong regularity for u(without “smallness assumptions” on the data). Both results are proved when either Dirichlet or Robin type conditions on the bottom are considered. In the last case, a domain with a strictly bounded from below depth has to be imposed, even for the uniqueness result. Key words: Weak-strong uniqueness, Primitive Equations, anisotropic estimates, strong solution 1 Introduction The Primitive Equations are related with a great variety of geophysical fluids [9, 10, 12]. This system can be deduced asymptotically from the Navier-Stokes equations with anisotropic (eddy) viscosity, when the aspect ratio (quotient between vertical and horizontal characteristic dimensions) tends to zero [1, 2, 3]. The 3D system can be written as follows: to find u: (0,T)×Ω→R2, the horizontal velocity field, and ps:(0,T)×S→R, a surface potential function (involving the pressure), verifying: (PE)                  ∂tu−νH∆Hu−νz∂2 zzu+αu⊥+ +(u·∇H)u+u3∂zu+∇Hps=Fin (0,T)×Ω, ∇H·�u�=0 in(0,T)×S, u|t=0 =u0in Ω, νz∂zu|Γs=Υ,u|Γl=0in (0,T), either u|Γb=0or ((νH∇Hu,ν z∂zu)·n+βu)|Γb=0in (0,T), where u3is the vertical velocity, that becomes now a diagnostic variable, depending on the horizontal velocity uas follows: u3(x,z)=�0 z∇H·u(x,s)ds. (1) We consider the domain Ω={(x, y, z)=(x,z)∈R3/x∈S, −h(x)<z<0},withS⊂R2a bounded open set (the surface) and h:S→Ra non-negative continuous function (the depth). ∗Corresponding author †The first author has been partially financed by the project BFM2000-1317. 1 2 THE MAIN RESULTS. 2 Its boundary is decomposed as ∂Ω=Γ b∪Γl∪Γswhere Γs={(x,0) : x∈S}is the surface, Γl={(x,z)∈R3:x∈∂S, −h(x)<z<0}are the side-walls and Γb={(x,z)∈R3:x∈ S, z =−h(x)}is the bottom. We have denoted by �u�(t;x)=�0 −h(x)u(t;x,z)dz the vertical integration of u. The horizontal operators ∆Hand ∇Hrepresent ∂2 xx +∂2 yy and (∂x,∂ y)trespectively. The constants νH,ν z>0 are the viscosity coefficients and nis the outward normal vector on the bottom. The external forces are data denoted by F:(0,T)×Ω→R2, and αu⊥=α(−u2,u 1)tmodels the Coriolis forces, with α∈Rdepending on the latitude. We consider either homogeneous Dirichlet or Robin type boundary conditions on the bottom (with β:S→Ra non-negative data function depending on the rugosity of the bottom) and Neumann boundary conditions on the surface, where Υ:(0,T)×S→R2is a data function depending on the wind force. The Neumann conditions on the bottom are also considered taking β= 0. Notice that Primitive Equations are variants of the Navier-Stokes equations. Now, the pressure field depends only on x(but not on z). However, the explicit form of u3given in (1) implies that the system is no longer parabolic respect to (u,u 3) and the regularity of u3and ∇H·uare comparable, hence the nonlinear term corresponding to the vertical convection u3∂zu is less regular that in the Navier-Stokes case. The existence of weak solution of (PE) were given in [10, 9]. The existence of local in time strong solution (or global for small enough data) is proved in [7] for the 2D case (where Sis a real interval) and in [6] for the 3D case, using strong regularity results for the stationary linear case given in [14]. On the other hand, some results of weak/strong uniqueness were given in [7, 6], always imposing additional regularity hypothesis over the horizontal and vertical derivatives of u. In this work, we weaken these additional hypothesis found in [7, 6] supposing only additional regularity over the vertical derivative ∂zu(avoiding the additional regularity over ∇Hu). Moreover, we will also prove that this same additional regularity implies global strong regularity when the data are more regular but without smallness assumptions. We think that the anisotropy between horizontal and vertical scales could produce anisotropic regularity for the solution. Indeed, this occurs in the 2D case; existence (and uniqueness) of weak solution ufor the 2D model such that ∂zuhas also weak regularity, i.e. ∂zu∈L∞(0,T;L2(Ω)) ∩ L2(0,T;H1(Ω)), is proved in [4] for Robin boundary conditions on the bottom and in [5] for Dirichlet conditions. In this line, the existence of weak solution for (PE) with only weak regularity for ∂zu(even local in time or global for small enough data) is an interesting open problem, that we are going to analyse in a future work. 2 The main results. Basically, uis a weak solution for (PE)in(0,T), if u∈L2(0,T;H1(Ω))2∩L∞(0,T;L2(Ω)2) and verifies the restriction ∇H·�u�= 0, the Dirichlet conditions in the trace sense and the momentum equations jointly with the Neumann and Robin conditions in a variational sense ([10, 6]). Moreover, if u∈L∞(0,T;H1(Ω)2)∩L2(0,T;H2(Ω)2) and ∂tu∈L2(0,T;L2(Ω)2), u is a strong solution for (PE)in(0,T). Theorem 2.1 (Uniqueness of solution) Let ube a weak solution of (PE)in (0,T). If there exists a weak solution ¯ uof (PE)in (0,T)verifying the additional regularity: ∂z¯ u∈L∞(0,T;L2(Ω)2)∩L2(0,T;H1(Ω)2),(2) then both solution coincided in [0,T). When Robin conditions are considered on the bottom, the assumption h≥hmin >0in Shas to be imposed. Remark 2.1 Notice that we have reduced the hypotheses on ¯ uimposed in [6] for getting uniqueness of weak/strong solution. Concretely, in [6] we considered ∇H¯ u∈L2(0,T;L∞ zL2 x)and ∂z¯ u∈L∞(0,T;L2(Ω)2)∩L2(0,T;H1(Ω)2) 3 SOME AUXILIARY ANISOTROPIC ESTIMATES. 3 (see the next section for the definition of the anisotropic space L∞ zL2 x). Therefore, we have removed the hypothesis for ∇H¯ u. Theorem 2.2 (Global strong regularity) Let S⊆R2with ∂S∈C3and h∈C3(¯ S)with h≥hmin >0in ¯ S.Supposethatu0∈H1(Ω)with ∇H·�u0�=0(and u0|Γb=0in the case of Dirichlet conditions on the bottom), F∈L2(0,T;L2(Ω)2)and Υ∈L2(0,T;H1/2+ε 0(Γs)2)∩ L∞(0,T;H−1/2(Γs)2)for some ε>0such that ∂tΥ∈L2(0,T;H−3/2(Γs)2)with Υ(0) ∈ H−1/2(Γs)2.If∂zuverifies the additional regularity of (2), then uis a strong solution of (PE) in (0,T). 3 Some auxiliary anisotropic estimates. Let us to introduce the anisotropic Lp,q spaces for any exponents p, q ∈[1,+∞]. It will said say that a function vbelongs to Lq zLp x(Ω) if: v(·,z)∈Lp(Sz) and �v(·,z)�Lp(Sz)∈Lq(−hmax,0), where hmax = max S hand Sz={x∈S:(x,z)∈Ω}for each z∈(−hmax,0). We will use the following three anisotropic results, the first one has already been considered and proved in [6], and the other ones are new in this work (see Appendix for the proofs). Lemma 3.1 a) Let v∈H1(Ω). Then v∈L2 zL4 x(Ω)and verifies: �v�L2 zL4 x≤C�v�1/2 L2(Ω)�v�1/2 H1(Ω)(3) b) Let v∈L2(Ω)2such that ∇H·v∈L2(Ω),andv3defined as in (1). Then, v3∈L∞ zL2 x(Ω) and �v3�L∞ zL2 x≤h1/2 max�∇H·v�L2(Ω)(4) Lemma 3.2 Let v∈H1(Ω)such that ∂zv∈H1(Ω)and v|Γb=0. Then v∈L∞ zL4 x(Ω)and �v�L∞ zL4 x≤C�v�1/4 L2(Ω)�v�1/4 H1(Ω)�∂zv�1/4 L2(Ω)�∂zv�1/4 H1(Ω)(5) Lemma 3.3 Assume h≥hmin >0in S. a) Let v∈L2(Ω)such that ∂zv∈L2(Ω). Then v∈L∞ zL2 x(Ω)and hmin�v�2 L∞ zL2 x≤�v�2 L2(Ω)+2�v�L2(Ω)�∂zv�L2(Ω).(6) b) Let v∈H1(Ω)such that ∂zv∈H1(Ω). Then v∈L∞ zL4 x(Ω)and h1/2 min�v�L∞ zL4 x≤C�v�1/4 L2(Ω)�v�1/4 H1(Ω)��v�1/4 L2(Ω)�v�1/4 H1(Ω)+�∂zv�1/4 L2(Ω)�∂zv�1/4 H1(Ω)�(7) 4 Proof of the main results in the Dirichlet case. Proof of Theorem 2.1:We follow the direct method to prove uniqueness, used for instance in [11] for the 3D Navier-Stokes equations. Denoting v=u−¯ uand v3=u3−¯u3, one has [6] (see [13] for more details): 1 2�v(t)�2 L2(Ω)+ν�t 0�v(s)�2 H1(Ω)ds ≤− �t 0�Ω [v·∇H¯ u+v3∂z¯ u]·vdΩds := I1+I2, (8) 4 PROOF OF THE MAIN RESULTS IN THE DIRICHLET CASE. 4 where ν=min{νH,ν z}. With respect to the proof of uniqueness done in [6], we will change the treatment of the term I1. Now, integrating by parts and applying (3) and (5) one has (here, Dirichlet condition on Γbis used) I1=�t 0�Ω [(v·∇H)v·¯ u+(∇H·v)v·¯ u]dΩds ≤C�t 0�Ω|v||∇Hv||¯ u|dΩds ≤C�t 0�v�L2 zL4 x�∇Hv�L2(Ω)�¯ u�L∞ zL4 xds ≤C�t 0�v�1/2 L2(Ω)�∇Hv�3/2 L2(Ω)�¯ u�1/4 L2(Ω)�¯ u�1/4 H1(Ω)�∂z¯ u�1/4 L2(Ω)�∂z¯ u�1/4 H1(Ω)ds. Using the Young inequality for the indexes (4/3,4), we have: I1≤ε�t 0�v�2 H1(Ω)ds +Cε�t 0�¯ u�L2(Ω)�¯ u�H1(Ω)�∂z¯ u�L2(Ω)�∂z¯ u�H1(Ω)�v�2 L2(Ω)ds. We bound I2as in [6] (using (3) and (4)), obtaining I2=�t 0�Ω v3∂z¯ u·vdΩds ≤�t 0�v3�L∞ zL2 x�∂z¯ u�L2 zL4 x�v�L2 zL4 xds ≤ε�t 0�v�2 H1(Ω)ds +Cε�t 0�∂z¯ u�2 L2(Ω)�∂z¯ u�2 H1(Ω)�v�2 L2(Ω)ds Using the previous bounds in (8), one has: �v(t)�2 L2(Ω)+ν�t 0�v(s)�2 H1(Ω)ds ≤C�t 0 a(s)�v(s)�2 L2(Ω)ds (9) where a=�¯ u�L2(Ω)�¯ u�H1(Ω)�∂z¯ u�L2(Ω)�∂z¯ u�H1(Ω)+�∂z¯ u�2 L2(Ω)�∂z¯ u�2 H1(Ω).Sincea∈L1(0,T) (thanks to the regularity hypothesis for ¯ uand ∂z¯ u), we are in the hypothesis of Gronwall Lemma, hence the uniqueness is deduced. Proof of Theorem 2.2:First, we lift the boundary data Υusing an adequate (strong) solution (e,q s) of a stationary hydrostatic Stokes system. Observe that hypothesis ∂tΥ∈ L2(0,T;H−3/2(Γs)2) implies that ∂te∈L2(0,T;L2(Ω)2) (see [8]). Then, we reason over the homogeneous variables (v,v 3,π s)=(u−e,u 3−e3,p s−qs), verifying:        ∂tv−νH∆Hv−νz∂2 zzv+u·∇Hv+v3∂zu+∇Hπs=Gin (0,T)×Ω, ∇H·�v�=0in (0,T)×S, v|t=0 =v0in Ω, νz∂zv|Γs=0,v|Γb=0,v|Γl=0in (0,T), (10) where v0=u0−e(0) and G=F−∂te−u·∇He+e3∂zu.Thanks to the additional regularity of ∂zuand the strong regularity of e, one has that G∈L2(0,T;L2(Ω)2). Indeed, in the convective terms, we have products of u(and e3) belonging to L4 tL∞ zL4 x,by∂zu(and ∇He) belonging to L4 tL2 zL4 x(accordingly Lemmas 3.1 and 3.2). Using the same argument than in [6], we apply a Galerkin method, using Avmas test functions, where Ais the hydrostatic Stokes operator and vmits eigenfunctions. In order to bound the convection terms, we use the inequalities (3) and (5) as follows (for simplicity, we drop the m-indexes): �Ω u·∇Hv·AvdΩ≤�u�L∞ zL4 x�∇Hv�L2 zL4 x�Av�L2(Ω) ≤C�u�1/4 L2(Ω)�u�1/4 H1(Ω)�∂zu�1/4 L2(Ω)�∂zu�1/4 H1(Ω)�∇Hv�1/2 L2(Ω)�Av�3/2 L2(Ω) ≤ε�Av�2 L2(Ω)+a(t)�∇Hv�2 L2(Ω), (11) 5 PROOFS FOR ROBIN CONDITIONS ON THE BOTTOM 5 where a(t)=C�u�L∞(0,T;L2(Ω))�u(t)�H1(Ω)�∂zu�L∞(0,T;L2(Ω))�∂zu(t)�H1(Ω), and �Ω v3∂zu·AvdΩ≤�v3�L∞ zL4 x�∂zu�L2 zL4 x�Av�L2(Ω) ≤C�∇H·v�1/2 L2(Ω)�∂zu�1/2 L2(Ω)�∂zu�1/2 H1(Ω)�Av�3/2 L2(Ω)≤ε�Av�2 L2(Ω)+b(t)�∇Hv�2 L2(Ω), (12) where b(t)=C�∂zu�2 L∞(0,T;L2(Ω))�∂zu(t)�2 H1(Ω). The additional regularity for ∂zuguarantees that a,b∈L1(0,T). Then, 1 2 d dt�∇v�2 L2(Ω)+�Av�2 L2(Ω)≤(a(t)+b(t))�∇v�2 L2(Ω)+�G(t)�2 L2(Ω). Gronwall’s Lemma allows us to conclude that v∈L∞(0,T;H1(Ω)2)∩L2(0,T;H2(Ω)2) and, thanks to the strong regularity of e, one has the same regularity for u. Regularity for ∂tuis followed by a standard way. 5 Proofs for Robin conditions on the bottom Proof of Theorem 2.1:In this case, one arrives at (8) with the supplementary non-negative term �t 0�Γbβ|v|2dσds in the left hand-side. Now, it is necessary to change the bound for the term I1in (8). Indeed, using directly (8) (without by parts integration), we get: I1≤�t 0�v�2 L2 zL4 x�∇H¯ u�L∞ zL2 x≤�t 0�v�L2(Ω)�v�H1(Ω)�∇H¯ u�L∞ zL2 x(13) In order to bound ∇H¯ u, we cannot use the following inequality (proved in [6]) �∇H¯ u�L∞ zL2 x≤C�∇H¯ u�1/2 L2(Ω)�∇H¯ u�1/2 H1(Ω) because ∇H¯ u�∈ H1(Ω). Instead of this, we will use Lemma 3.3 a). Indeed, applying (6) for v=∇H¯ uin (13), we arrive at I1≤ε�t 0�v�2 H1(Ω)ds +Cε hmin �t 0��∇H¯ u�2 L2(Ω)+�∇H¯ u�L2(Ω)�∂z(∇H¯ u)�L2(Ω)��v�2 L2(Ω)ds Adding this expression to the estimate for I2, one can also prove uniqueness of solution. Proof of Theorem 2.2:The main difference in the proof is the use of inequality (7) instead of (5) of Lemma 3.2 in order to bound the convective terms. A Appendix Proof of Lemma 3.2.We will use the following inequality, proved in [6]; for any p, q ∈[1,+∞] with q>p, �v�Lq xLp z≤�v�Lp zLq x(14) With the same arguments one can change the integration order, i.e., �v�Lq zLp x≤C�v�Lp xLq z(15) Let vin the hypothesis of Lemma 3.2. Since v|Γb= 0, we have v(x,z)4=�2�z −h(x) v(x,s)∂zv(x,s)ds�2 ≤4�v(x,·)�2 L2 z�∂zv(x,·)�2 L2 z REFERENCES 6 Consequently, �v(x,·)�L∞ z≤√2�v(x,·)�1/2 L2 z�∂zv(x,·)�1/2 L2 z.(16) Taking L4 x-norm, �v�L4 xL∞ z≤√2�v�1/2 L4 xL2 z�∂zv�1/2 L4 xL2 z. Now using (14) and (3), we obtain: �v�L4 xL∞ z≤√2�v�1/2 L2 zL4 x�∂zv�1/2 L2 zL4 x≤C�u�1/4 L2(Ω)�v�1/4 H1(Ω)�∂zv�1/4 L2(Ω)�∂zv�1/4 H1(Ω). To finish, it suffices to consider (15) in the left hand side, getting (5). Proof of Lemma 3.3.For any function g=g(z)definedinz∈(−h(x),0) with x∈S, we write g2(z)=g2(z�)+2�z z�g(s)∂zg(s)ds. Integrating in z�∈(−h(x),0), h(x)g2(z)≤ �g�2 L2 z+2�g�L2 z�∂zg�L2 z,thush(x)1/2�g�L∞ z≤�g�L2 z+√2�g�1/2 L2 z�∂zg�1/2 L2 z. Applying the previous inequality to g=v(x,·) and bounding from below h(x)≥hmin, we get h1/2 min�v(x,·)�L∞ z≤�v(x,·)�L2 z+√2�v(x,·)�1/2 L2 z�∂zv(x,·)�1/2 L2 z.(17) In order to prove estimate (7), we follow the same argument that in the proof of Lemma 3.2, replacing (16) by (17) and adapting the calculus therein. Finally, (6) follows directly taking L2 x-norm in (17). References [1] P. Az´erad & F. Guill´en-Gonz´alez, Mathematical justification of the hydrostatic approximation in the Primitive Equations of Geophysical fluid dynamics. Siam J. Math. Anal. ,Vol. 33, No. 4, 847-859 (2001). [2] 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, 855-865 (1992). [3] D. Bresch, F. Guill´en-Gonz´alez, N. Masmoudi & M. A. Rodr´ıguez-Bellido, Asymptotic derivation of a Navier condition for the Primitive Equations, to appear in Asymptotic Analysis. [4] D. Bresch, F. Guill´en-Gonz´alez, N. Masmoudi & M. A. Rodr´ıguez-Bellido, On the uniqueness of weak solutions of the two-dimensional primitive equations. Diff. and Integral Eqs. ,Volume 16, Number 1, 77-94 (2003). [5] D. Bresch, A. Kazhikhov & J. Lemoine, On the two-dimensional hydrostatic equations. Submitted to Arch. Rat. Mech. Anal., (2001). [6] F. Guill´en-Gonz´alez, N. Masmoudi & M. A. Rodr´ıguez-Bellido, Anisotropic estimates and strong solutions of the Primitive Equations. Diff. and Integral Eqs. ,Volume 14, Number 11, 1381-1408 (2001). [7] F. Guill´en-Gonz´alez & M.A. Rodr´ıguez-Bellido. On the strong solutions of the Primitive Equations in 2D domains, Nonlinear Analysis 50, 621-646 (2002). [8] F. Guill´en-Gonz´alez, M. A. Rodr´ıguez-Bellido & M. A. Rojas-Medar, Hydrostatic Stokes equations with non-smooth Neumann data. Submitted. [9] R. Lewandowski, Analyse Math´ematique et Oc´eanographie, Masson, (1997). [10] J. L. Lions, R. Teman & S. Wang,On the equations of the large scale Ocean. Nonlinearity, 5, 10071053 (1992). [11] P. L. Lions, Mathematical Topics in Fluids Mechanics, Vol.1, Incompressible Models, Univ. ParisDauphine & ´ Ecole Polytechnique, Oxford Univ. Press. Inc., NY, (1996). [12] J. Pedlosky, Geophysical fluid dynamics, Springer-Verlag, (1987). [13] M. A. Rodr´ıguez Bellido, Ph. Thesis, University of Seville (2001). [14] M. Ziane, Regularity Results for Stokes Type Systems. Applicable Analysis, Vol. 58, 263-292 (1995).