On the Stability of Weak Solution for Compressible Primitive Equations

Abstract

In this paper, we study a compressible Primitive Equations (CPEs) of the ocean in the two dimensional space, with horizontal periodic and vertical mixed boundary conditions. Thanks to an effective change of variables, we obtain a new CPEs model, which is similar as viscous shallow water equation. Using a new entropy estimate, we prove the stability of weak solutions for this new two dimensional CPEs model.

Local Well-Posedness of Strong Solutions to the Three-Dimensional Compressible Primitive Equations

“…Recently, Gatapov, Kazhikhov, Ersoy, Ngom construct a global weak solution to some variant of two-dimentional compressible primitive equations in [16,19]. Meanwhile, Ersoy, Ngom, Sy, Tang, Gao study the stability of weak solutions to the CPE in [17,41] in the sense that a sequence of weak solutions satisfying some entropy conditions contains a subsequence converging to another weak solution. In recent work, we show the existence of such weak solutions in [31].…”

Zero Mach Number Limit of the Compressible Primitive Equations: Well-Prepared Initial Data

“…On the other hand, as mentioned before, the authors in [22] introduce these equations with a formal deviation, and a rigorous justification is still an open question for now. In [22], the stability of weak solutions is also investigated (see also [63]). The stability is meant in the sense that a sequence of weak solutions, satisfying some entropy conditions, contains a subsequence converging to another weak solution, i.e., a very weak sense of stability.…”

Global Existence of Weak Solutions to the Compressible Primitive Equations of Atmospheric Dynamics with Degenerate Viscosities

“…Recently, Gatapov, Kazhikhov, Ersoy, Ngom construct a global weak solution to some two dimensional compressible primitive equations in [15,19]. Meanwhile, Ersoy, Ngom, Sy, Tang, Gao study the stability of weak solutions to CPE in [16,50], in the sense that a sequence of weak solutions satisfying some entropy conditions contains a subsequence converging to another weak solution. In this work, we address the problem of existence of such global weak solutions to (CPE').…”

“…Furthermore, the vertical velocity is less regular than the horizontal velocity in general. To overcome this difficulty, in [16,50], the authors make use of the Bresch-Desjardins entropy inequality by taking the viscosity coefficients as in (CPE'). This will give some extra regularity to the density and the vertical velocity.…”