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

Abstract

We show the existence of global weak solutions to the three-dimensional compressible primitive equations of atmospheric dynamics with degenerate viscosities. In analogy with the case of the compressible Navier-Stokes equations, the weak solutions satisfy the basic energy inequality, the Bresh-Desjardins entropy inequality and the Mellet-Vasseur estimate. These estimates play an important role in establishing the compactness of the vertical velocity of the approximating solutions, and therefore are essential to recover the vertical velocity in the weak solutions.

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

“…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]. See also [43].…”

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

“…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. The existence of such weak solutions is recently constructed in [49,65] (see also [21,30] for the existence of global weak solutions to some variant of compressible primitive equations in two spatial dimension). In [50], we also construct local strong solutions to CPE in two cases: with gravity but no vacuum; with vacuum but no gravity.…”

“…Throughout the rest of this section, it is assumed that (ξ, ψ h ) with ψ z given by (1.10) is a solution to (1.8) which is smooth enough such that the estimates below can be established. To justify the arguments, one can employ the local wellposedness theory and the standard different quotient method to the corresponding lines below (replaced the differential operators by different quotients, for example); see, for instance, similar arguments in [49,50].…”

Global existence of weak solutions to the Navier–Stokes–Korteweg equations

“…The analysis of the long time behaviour for the isothermal quantum Navier-Stokes equations has been performed in [16]. In [39] the authors, by using a strategy similar to [35], study the existence of global-in-time finite energy weak solutions to the compressible primitive equations with degenerate viscosity.…”