We prove the existence of global strong solutions of the primitive equations of the ocean in the case of the Dirichlet boundary conditions on the side and the bottom boundaries including the varying bottom topography. Previously, the existence of global strong solutions was known in the case of the Neumann boundary conditions in a cylindrical domain(Cao and Titi, [CT]).

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The Primitive Equations are a basic model in the study of large scale Oceanic and Atmospheric dynamics. These systems form the analytical core of the most advanced General Circulation Models. For this reason and due to their challenging nonlinear and anisotropic structure the Primitive Equations have recently received considerable attention from the mathematical community.On the other hand, in view of the complex multi-scale nature of the earth's climate system, many uncertainties appear that should be accounted for in the basic dynamical models of atmospheric and oceanic processes. In the climate community stochastic methods have come into extensive use in this connection. For this reason there has appeared a need to further develop the foundations of nonlinear stochastic partial differential equations in connection with the Primitive Equations and more generally.In this work we study a stochastic version of the Primitive Equations. We establish the global existence and uniqueness of strong, pathwise solutions for these equations in dimension 3 for the case of a nonlinear multiplicative noise. The proof makes use of anisotropic estimates, L p t L q x estimates on the pressure and stopping time arguments.MSC2010: 35Q86, 60H15, 35Q35 * To appear in Nonlinearity.1 Physical uncertainties on the boundary of the oceanic or coupled oceanic-atmospheric system are also present and are responsible for another source of stochasticity in these models. The boundary conditions (1.4) on Γ i (which is physically the sea surface-air interface) are a simplified version of the more realistic boundary conditions(1.1)In the form (1.1) the boundary condition expresses two fundamental aspects of oceanic-atmospheric interaction namely the driving force of the wind (the term gv) and the heating or cooling of the air by the ocean (the term g T ). These functions gv and g T are not well known and are estimated by modelers through very rough averages. Thus the uncertainties of these functions lead to stochastic PDEs with white noise on the boundary a subject which we intend to pursue in future work.

We consider sufficient conditions for the regularity of Leray-Hopf solutions of the Navier-Stokes equations. We prove that if the third derivative of the velocity ∂u∕∂x3 belongs to the space Lts0Lxr0, where 2∕s0+3∕r0⩽2 and 9∕4⩽r0⩽3, then the solution is regular. This extends a result of Beirão da Veiga [Chin. Ann. Math., Ser. B 16, 407–412 (1995); C. R. Acad. Sci, Ser. I: Math. 321, 405–408 (1995)] by making a requirement only on one direction of the velocity instead of on the full gradient. The derivative ∂u∕∂x3 can be substituted with any directional derivative of u.

We establish sufficient conditions for the regularity of solutions of the Navier-Stokes system based on conditions on one component of the velocity. The first result states that if ∇u 3 ∈ L s t L r x , where 2 s + 3 r 11 6 and 54/23 r 18/5, then the solution is regular. The second result is that if u 3 ∈ L s t L r x , where 2 s + 3 r 5 8 and 24/5 r ∞, then the solution is regular. These statements improve earlier results on one component regularity.

We address the question of well-posedness in spaces of analytic functions for the Cauchy problem for the hydrostatic incompressible Euler equations (inviscid primitive equations) on domains with boundary. By a suitable extension of the Cauchy-Kowalewski theorem we construct a locally in time, unique, real-analytic solution and give an explicit rate of decay of the radius of real-analyticity.

Abstract. A modification of the classical primitive equations of the atmosphere is considered in order to take into account important phase transition phenomena due to air saturation and condensation. We provide a mathematical formulation of the problem that appears to be new in this setting, by making use of differential inclusions and variational inequalities, and which allows to develop a rather complete theory for the solutions to what turns out to be a nonlinearly coupled system of non-smooth partial differential equations. Specifically we prove global existence of quasi-strong and strong solutions, along with uniqueness results and maximum principles of physical interest.