Control and analysis for partial differential equations in fluid mechanics and geoscience.

摘要

This dissertation is divided into three parts.; In the first part, we consider the robust boundary control of the Kuramoto-Sivashinsky equation. A robust boundary control problem is formulated, and the existence and uniqueness of the solutions to the control problem are proved. A data assimilation problem to the Kuramoto-Sivashinsky equation is considered.; In the second part, we study the regularity of solutions of the GFD-Stokes problem and of some second order elliptic differential equations related to the Primitive equations of the ocean. We take into consideration the non-homogeneous boundary conditions and the dependence of solutions on the thickness &egr; of the domain occupied by the ocean and its varying bottom topography. We derive various Sobolev-type inequalities.; In the third part, we study the global existence of strong solutions of the Primitive Equations for the lame scale ocean under small depth hypothesis. We establish the global existence of strong solutions for initial data and boundary ‘forces’ which belong to large sets in their respective phase spaces, provided &egr; is sufficiently small.