In this talk, we will introduce the global existence and incompressible limit of weak solutions to compressible primitive equations (CPEs) with density-dependent viscosity for large initial data, when physical number gamma equals 1 in the pressure of atmosphere.

The primitive equations (PEs) of atmosphere are the fundamental equations in geophysical fluid dynamics. They are based on the so-called hydrostatic approximation, in which the conservation of momentum in the vertical direction is replaced by the hydrostatic equation. The CPE model can be derived from the 3D compressible and anisotropic Navier–Stokes equations by hydrostatic approximation.

Firstly, we can obtain the global existence of weak solutions to the nondimensional CPE model with degenerate viscosity, which need three steps in the proof. First of all, by using the method of work which is obtained by Wang, Dou and Jiu in 2020 and Faedo-Galerkin method, we firstly obtain the global existence of the approximate solutions to CPE model. Then, we get  the lower bound of the density, which is the key estimates. And we arrive at the Bresch–Desjardins entropy of solutions to the approximate system. Finally, we apply compactness arguments to vanish the parameters in the approximate system that we construct to obtain global existence of weak solutions to nondimensional CPE model. 

Secondly, we can obtain the incompressible limit to CPE model as epsilon  goes to zero (where epsilon means the Mach number), and the primitive equations with the incompressibility condition are identified as the limiting equations. The convergence with well-prepared initial data (that is, initial data has no acoustic oscillations) is rigorously justified. In the proof, it is crucial that we obtain the estimates of the solutions to the compressible primitive equations are uniform about the Mach number. So, we can let epsilon tends to zero and obtain the incompressible limit. And, the density is a function on the vertical variable, rather than a constant in the incompressbile system.

compressible primitive equations,density-dependent viscosity,weak solution,incompressible limit