Density dependent viscosity for the Poisson-Nernst-Planck-Compressible Navier-Stokes system

Abstract

This paper is dedicated to the global existence of entropy weak solutions for the Poisson-Nernst-Planck-Compressible Navier-Stokes system in a periodic domain $\Pi$ d when the shear viscosity $\mu$($\rho$) = $\mu$ $\rho$ with $\mu$ to be constant and $\lambda$($\rho$) = 0 assuming a pressure state law singular close to vacuum and a $\gamma$ power elsewhere with $\gamma$ > 1. It is important to recall that recently D. Marroquin and D. Wang (Arxiv 2024) have proved global existence of weak solutions in the spirit of P.-L. Lions and E. Feireisl when the shear and bulk viscosities are assumed to be constant without singular pressure part close to vacuum namely p($\rho$) = a$\rho$^$\gamma$ having in hand (from the energy estimate) u $\in$ L 2 (0, T\,; H 1 ($\Pi$ d )). Here the main difficulty is double first because of the possible degeneracy of the shear viscosity we have to derive a formal mathematical entropy generalizing the BD entropy equality: Up to the authors knowledge this is completely new. With this energy equality and the generalization of the BD entropy, it is then possible to prove global existence of entropy weak solutions for the system. An extra difficulty from a physical point of view remains namely if the density vanishes the compressible Navier-Stokes equations provides no information on the velocity field but the velocity itself is required in the equation related to the concentration. This explain why a singular pressure state law is needed close to vacuum in the spirit of what has been done in different papers see for instance D.