Finite time blow-up of regular solutions for compressible flows

Abstract

The development of finite time singularity of smooth solutions to the compressible Navier-Stokes system as well as its blowup mechanism is discussed in the presence of vacuum. It is shown that any smooth solutions to the compressible Navier-Stokes equations for polytropic fluids in the absence of heat conduction will blow up in finite time as long as the initial densities have compact support or isolated mass group. Besides, unified Serrin-type regularity criteria are established for the barotropic and full compressible Navier-Stokes equations with or without heat conduction. As an immediate corollary, it gives an affirmative answer to a problem proposed by J. Nash in the 1950s which asserts that the finite time blowup must be due to the concentration of either the density or the temperature.