Existence of Global Weak Solutions for 3D Degenerate Compressible Navier-Stokes Equations

摘要

In this paper, we prove the existence of global weak solutions for 3Dcompressible Navier-Stokes equations with degenerate viscosity. The method isbased on the Bresch and Desjardins entropy conservation. The main contributionof this paper is to derive the Mellet-Vasseur type inequality for the weaksolutions, even if it is not verified by the first level of approximation. Thisprovides existence of global solutions in time, for the compressibleNavier-Stokes equations, for any $\gamma>1$, in three dimensional space, withlarge initial data possibly vanishing on the vacuum. This solves an openproblem proposed by Lions.