Abstract

The compressible Navier-Stokes system with degenerate viscosities was little known in many physical and mathematical considerations. Such a degeneracy is a highly challenging obstacle for development of well-posedness theory. In this paper, We establish the blow-up criterion of classical solutions to the Cauchy problem for the compressible Navier-Stokes equations, as well as Shallow water equations, with degenerate viscosities. Moreover, we prove that the maximum norm of the gradients of velocity controls the possible breakdown of regular solutions for compressible Navier-Stokes equations with degenerate viscosities.