On classical solutions to 2D Shallow water equations with degenerate viscosities

Abstract

In this paper, the $2$-D isentropic Navier-Stokes systems for compressible fluids with density-dependent viscosity coefficients are considered. In particular, we assume that the viscosity coefficients are proportional to density. These equations, including several models in $2$-D shallow water theory, are degenerate when vacuum appears. We introduce the notion of regular solutions and prove the local existence of solutions in this class allowing the initial vacuum in the far field. This solution is further shown to be stable with respect to initial data in $H^2$ sense. A Beal-Kato-Majda type blow-up criterion is also established.

arXiv comment43pages. arXiv admin note: substantial text overlap with arXiv:1407.7828

References in corpus (3)

• Vanishing of vacuum states and blow-up phenomena of the compressible Navier-Stokes equations
• Global Well-Posedness of Classical Solutions to the Cauchy problem of Two-Dimensional Baratropic Compressible Navier-Stokes System with Vacuum and Large Initial Data
• On regular solutions of the 3-D compressible isentropic Euler-Boltzmann equations with vacuum

Cited by in corpus (1)

• On classical solutions for viscous polytropic fluids with degenerate viscosities and vacuum