Initial and boundary value problems for the inviscid primitive equations and shallow water equations

This thesis contains two main objectives. The first aim of this thesis is to propose a unified way to impose suitable boundary conditions for the 3D Primitive Equations (PEs) and 2D Shallow Water Equations (SWEs) without viscosity in the Local Area Models (LAMs). The boundary conditions in LAMs are important for the numerical simulations (to avoid numerical explosion) and are also considered as a major computational issue for the geophysical fluid dynamics for the coming years since there are no physical laws giving the necessary boundary conditions. The boundary conditions that we propose do not lead to singularities and allow us to develop results of existence and uniqueness of solutions in suitable spaces for the (at least linearized) PEs and SWEs. In the nonlinear case, we have studied two special cases of 2D SWEs, where we showed local existence and uniqueness results.;The study of inviscid PEs and SWEs naturally leads us to consider more general first order hyperbolic partial differential equations (PDEs) in a rectangular domain, which is the second goal of this thesis. The hyperbolic PDEs have been well understood in the case of smooth domains compared to the domains with corners. For the general hyperbolic system, we find by simultaneous congruence diagonalization that there are only two elementary modes in the system which we call hyperbolic and elliptic modes. Therefore, the well-posedness of the fully hyperbolic system in a rectangle (or possibly curvilinear polygonal domains) has been achieved by studying these two elementary modes separately.