Global weak solution to the viscous two-phase model with finite energy

Alexis Vasseur, Huanyao Wen, Cheng Yu

Published 2017-04-24Version 1

In this paper, we prove the existence of global weak solutions to the compressible Navier-Stokes equations when the pressure law is in two variables.The method is based on the Lions argument and the Feireisl-Novotny-Petzeltova method. The main contribution of this paper is to develop a new argument for handling a nonlinear pressure law $P(\rho,n)=\rho^{\gamma}+n^{\alpha}$ where $\rho,\,n$ satisfy the mass equations. This yields the strong convergence of the densities, and provides the existence of global solutions in time, for the compressible barotropic Navier-Stokes equations, with large data. The result holds in three space dimensions on condition that the adiabatic constants $\gamma>\frac{9}{5}$ and $\alpha\geq 1$. Our result is the first global existence theorem on the viscous compressible two-phase model with pressure law in two variables, for large initial data, in the multidimensional space.