Abstract. In this article, an L p -approach to the primitive equations is developed. In particular, it is shown that the three dimensional primitive equations admit a unique, global strong solution for all initial data a ∈ [Xp, D(Ap)] 1/p provided p ∈ [6/5, ∞). To this end, the hydrostatic Stokes operator Ap defined on Xp, the subspace of L p associated with the hydrostatic Helmholtz projection, is introduced and investigated. Choosing p large, one obtains global well-posedness of the primitive equations for strong solutions for initial data a having less differentiability properties than H 1 , hereby generalizing in particular a result by Cao and Titi [8] to the case of non-smooth initial data.

In this paper, we propose the mathematical and finite element analysis of a second order Partial Differential Equation endowed with a generalized Robin boundary condition which involves the Laplace-Beltrami operator, by introducing a function space H 1 (Ω; Γ) of H 1 (Ω)-functions with H 1 (Γ)-traces, where Γ ⊆ ∂Ω. Based on a variational method, we prove that the solution of the generalized Robin boundary value problem possesses a better regularity property on the boundary than in the case of the standard Robin problem. We numerically solve generalized Robin problems by means of the finite element method with the aim of validating the theoretical rates of convergence of the error in the norms associated to the space H 1 (Ω; Γ).

Abstract. We consider the P1/P1 or P1b/P1 finite element approximations to the Stokes equations in a bounded smooth domain subject to the slip boundary condition. A penalty method is applied to address the essential boundary condition u · n = g on ∂Ω, which avoids a variational crime and simultaneously facilitates the numerical implementation. We give O(h 1/2 + 1/2 +h/ 1/2 )-error estimate for velocity and pressure in the energy norm, where h and denote the discretization parameter and the penalty parameter, respectively. In the two-dimensional case, it is improved to O(h + 1/2 + h 2 / 1/2 ) by applying reduced-order numerical integration to the penalty term. The theoretical results are confirmed by numerical experiments.

It is shown that the hydrostatic Stokes operator on L σ ¯ p ( Ω ) L^p_{\overline {\sigma }}(\Omega ) , where Ω ⊂ R 3 \Omega \subset \mathbb {R}^3 is a cylindrical domain subject to mixed periodic, Dirichlet and Neumann boundary conditions, admits a bounded H ∞ H^\infty -calculus on L σ ¯ p ( Ω ) L^p_{\overline {\sigma }}(\Omega ) for p ∈ ( 1 , ∞ ) p\in (1,\infty ) of H ∞ H^\infty -angle 0 0 . In particular, maximal L q − L p L^q-L^p -regularity estimates for the linearized primitive equations are obtained.

Consider the full primitive equations, i.e. the three dimensional primitive equations coupled to the equation for temperature and salinity, and subject to outer forces. It is shown that this set of equations is globally strongly well-posed for arbitrary large initial data lying in certain interpolation spaces, which are explicitly characterized as subspaces of H 2/p,p , 1 < p < ∞, satisfying certain boundary conditions. In particular, global well-posedeness of the full primitive equations is obtained for initial data having less differentiability properties than H 1 , hereby generalizing by result by Cao and Titi [6] to the case of non-smooth data. In addition, it is shown that the solutions are exponentially decaying provided the outer forces possess this property.2010 Mathematics Subject Classification. Primary: 35Q35; Secondary: 76D03, 47D06, 86A05.The system is complemented by the boundary conditionsand α > 0. The rigorous analysis of the primitive equations started with the pioneering work of Lions, Temam and Wang [25][26][27], who proved the existence of a global weak solution for this set of equations for initial data a ∈ L 2 and b τ ∈ L 2 , b σ ∈ L 2 . For recent results on the uniqueness problem for weak solutions, we refer to the work of Li and Titi [28] and Kukavica, Pei, Rusin and Ziane [21]. The existence of a local, strong solution for the decoupled velocity equation with data a ∈ H 1 was proved by Guillén-González, Masmoudi and Rodiguez-Bellido in [15].In 2007, Cao and Titi [6] proved a breakthrough result for this set of equation which says, roughly speaking, that there exists a unique, global strong solution to the primitive equations for arbitrary initial data a ∈ H 1 and b τ ∈ H 1 neglecting salinity. Their proof is based on a priori H 1 -bounds for the solution, which in turn are obtained by L ∞ (L 6 ) energy estimates. Note that the boundary conditions on Γ b ∪ Γ l considered there are different from the ones we are imposing in (1.2). Kukavica and Ziane considered in [23,24] the primitive equations subject to the boundary conditions on Γ u ∪ Γ b as in (1.2) and they proved global strong well-posedness of the primitive equations with respect to arbitrary H 1 -data. For a different approach see also Kobelkov [20].Modifications of the primitive equations dealing with either only horizontal viscosity and diffusion or with horizontal or vertical eddy diffusivity were recently investigated by Cao and Titi in [7], by Cao, Li and Titi in [8][9][10]. Here, global well-posedness results are established for initial data in H 2 . For recent results concerning the presence of vapor, we refer to the work of Coti-Zelati, Huang, Kukavica, Teman and Ziane [37].The existence of a global attractor for the primitive equations was proved by Ju [19] and its properties were investigated by Chueshov [11].For local well-posedness results concerning the inviscid primitive equations, we refer to Brenier [4], Masmoudi and Wong [30], Kukavica, Temam, Vicol and Ziane [22] as well as Hamouda, Jung and Temam [16].Recentl...