Serrin-type blowup Criterion for the degenerate compressible Navier-Stokes equations

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Abstract

In this paper, we consider the Cauchy problem of the three-dimensional isentropic compressible Navier-Stokes equations with degenerate viscosities. When the shear and bulk viscosity coefficients are both given as a constant multiple of the mass density’s power (ρ^δ with δ > 1), we show that the L^∞ norms of ∇u, ∇ρ^(γ−1)/2 and ∇ρ^(δ−1)/2 control the possible breakdown of classical solutions with far-field vacuum; this criterion is analogous to Serrin’s blowup criterion for the compressible Navier–Stokes equations.

Keywords

Compressible Navier–Stokes equations / degenerate viscosity / far-field vacuum / classical solutions / Serrin-type blowup Criterion

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