## Young Researchers Workshop: Multiscale phenomena: modeling, analysis and computation

### Existence of global weak solutions to the compressible Navier-Stokes equations with density dependent viscosity

Cheng Yu

University of Texas at Austin

Abstract:

In this talk, we will discuss the existence of global weak solutions of the barotropic compressible Navier-Stokes equations with density dependent coefficients vanishing on vacuum. In particular,we will focus on the existence of globally defined weak solutions in three dimensional space for any $\gamma>1$ with large initial data possibly vanishing on vacuum for a special relationship of viscosity coefficients first proposed by Bresch and Desjardins. In two dimensional space with $\gamma=2$, the model recover the physical viscous shallow water equations. Thus, we resolved Lions' open problem on the weak solutions of viscous shallow water equations. This is a joint work with Alexis Vasseur.