Abstract
The linearized Primitive Equations with vanishing viscosity are considered. Some new boundary conditions (of transparent type) are introduced in the context of a modal expansion of the solution which consist of an infinite sequence of integral equations. Applying the linear semi-group theory, existence and uniqueness of solutions is established. The case with nonhomogeneous boundary values, encountered in numerical simulations in limited domains, is also discussed.
Highlights
The Primitive Equations of the ocean and the atmosphere are fundamental equations of geophysical fluid mechanics ([14],[20],[24])
It is generally accepted that the viscosity terms do not affect numerical simulations in a limited domain, over a period of a few days, and these viscosities are generally not used, see [25]
For the PEs without viscosity, and to the best of our knowledge, no result of well-posedness has ever been proven, since the negative result of Oliger and Sundstrom [12] showing that these equations are ill-posed for any set of local boundary conditions
Summary
The Primitive Equations of the ocean and the atmosphere are fundamental equations of geophysical fluid mechanics ([14],[20],[24]). In this article the full 2D-PEs, without viscosity, and linearized around a stratified state with constant velocity are considered. Concerning well-posedness, we are faced with boundary value problems for nonlinear hyperbolic systems of equations in a limited domain, a subject not yet extensively studied (see the important results of [4, 11, 27]). [3, 6]), making them appropriate for computations This initial boundary value problem is set as an abstract linear evolution equation in a suitable Hilbert space (Section 2.1). We multiply (1.22), (1.23) and (1.26) by Un, (1.24) and (1.25) by Wn and integrate on (−L3, 0), and we find:
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