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.