Serrin-type blowup Criterion for the degenerate compressible Navier-Stokes equations - Zhigang Wang (Communications in Analysis and Mechanics)
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 $ \nabla u $, $ \nablaρ^{\frac{γ-1}{2}} $ and $ \nablaρ^{\frac{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.