On the problem of the compressible primitive equations

In the talk we will describe the rigorous derivation of the compressible primitive equations. Moreover, we will focus on the problem of the energy conservation for the weak solutions to the compressible Primitive Equations (CPE) system with degenerate viscosity. The sufficient conditions on the regularity of weak solutions for the energy equality are obtained even for the case when the solutions may include vacuum. We will discuss two theorems, the first one gives regularity in the classical isotropic Sobolev and Besov spaces. The second one  states regularity in the anisotropic spaces. We obtain new regularity results in the second theorem due to the special structure of CPE system, which are in  contrast to compressible Navier-Stokes equations. The talk is based on thee joint collaboration with  M. A. Rodriguez-Bellido,  H. J. Gao and T. Tang.