Global Existence of Weak Solutions for Compresssible Navier--Stokes--Fourier Equations with the Truncated Virial Pressure Law

Abstract: This paper concerns the existence of global weak solutions {\it `a la Leray} for compressible Navier--Stokes--Fourier systems with periodic boundary conditions and the truncated virial pressure law which is assumed to be thermodynamically unstable. More precisely, the main novelty is that the pressure law is not assumed to be monotone with respect to the density. This provides the first global weak solutions result for the compressible Navier-Stokes-Fourier system with such kind of pressure law which is strongly used as a generalization of the perfect gas law. The paper is based on a new construction of approximate solutions through an iterative scheme and fixed point procedure which could be very helpful to design efficient numerical schemes. Note that our method involves the paper by the authors published in Nonlinearity (2021) for the compactness of the density when the temperature is given.