Regularity Results for a Navier-Stokes Type Problem Related to Oceanography
Résumé
The problem studied in this paper derives from the modelization of an oceanic phenomenon occurring in tropical zones: the presence of zonal waves superimposed on the mean currents. Our assumptions are justified by the physical features: we have studied the perturbation of a given mean flow that verifies Navier-Stokes type equations and mixed boundary conditions of Dirichlet, Neumann or periodic types. The existence and uniqueness of the solution are studied for the problem linearized around the mean flow and for the complete nonlinear problem. In the second case, we obtain sufficient conditions, which are physically sensible: the space gradient of the mean velocity, the perturbation of the windstress and its time derivative cannot be too large. The value of the pressure pon the surface of the ocean is of great interest for physical interpretation. To define that quantity, it is necessary to have the regularity p \in H1. We have proved that the perturbation (u,p) of the mean circulation is such that u\in L2(0,T,H2)), p \in L2(0,T,H1)), provided the perturbation of the windstress is sufficiently regular and satisfies compatibility relations. This result is valid for the linearized and the nonlinear cases, in a domain with corners. It is proved by means of an extension method, with even-odd reflection.