Blow-up criterion for the 3D viscous polytropic fluids with degenerate viscosities

Abstract

<p style="text-indent:20px;">In this paper, the Cauchy problem of the <inline-formula><tex-math id="M1">$ 3 $</tex-math></inline-formula>D compressible Navier-Stokes equations with degenerate viscosities and far field vacuum is considered. We prove that the <inline-formula><tex-math id="M2">$ L^\infty $</tex-math></inline-formula> norm of the deformation tensor <inline-formula><tex-math id="M3">$ D(u) $</tex-math></inline-formula> (<inline-formula><tex-math id="M4">$ u $</tex-math></inline-formula>: the velocity of fluids) and the <inline-formula><tex-math id="M5">$ L^6 $</tex-math></inline-formula> norm of <inline-formula><tex-math id="M6">$ \nabla \log \rho $</tex-math></inline-formula> (<inline-formula><tex-math id="M7">$ \rho $</tex-math></inline-formula>: the mass density) control the possible blow-up of regular solutions. This conclusion means that if a solution with far field vacuum to the Cauchy problem of the compressible Navier-Stokes equations with degenerate viscosities is initially regular and loses its regularity at some later time, then the formation of singularity must be caused by losing the bound of <inline-formula><tex-math id="M8">$ D(u) $</tex-math></inline-formula> or <inline-formula><tex-math id="M9">$ \nabla \log \rho $</tex-math></inline-formula> as the critical time approaches; equivalently, if both <inline-formula><tex-math id="M10">$ D(u) $</tex-math></inline-formula> and <inline-formula><tex-math id="M11">$ \nabla \log \rho $</tex-math></inline-formula> remain bounded, a regular solution persists.</p>