On blowup of classical solutions to the compressible Navier-Stokes equations

Abstract

We study the finite time blow up of smooth solutions to the Compressible Navier-Stokes system when the initial data contain vacuums. We prove that any classical solutions of viscous compressible fluids without heat conduction will blow up in finite time, as long as the initial data has an isolated mass group (see definition in the paper). The results hold regardless of either the size of the initial data or the far fields being vacuum or not. This improves the blowup results of Xin (1998) by removing the crucial assumptions that the initial density has compact support and the smooth solution has finite total energy. Furthermore, the analysis here also yields that any classical solutions of viscous compressible fluids without heat conduction in bounded domains or periodic domains will blow up in finite time, if the initial data have an isolated mass group satisfying some suitable conditions.

arXiv comment13 pages, Submitted

References in corpus (3)

• A Blow-Up Criterion for Classical Solutions to the Compressible Navier-Stokes Equations
• Global Classical and Weak Solutions to the Three-Dimensional Full Compressible Navier-Stokes System with Vacuum and Large Oscillations
• Blowup of Smooth Solutions to the Navier-Stokes Equations for Compressible Isothermal Fluids

Cited by in corpus (4)

• Global classical solutions to the two-dimensional compressible Navier-Stokes equations in ${\mathbb{R}}^2$
• Global well-posedness of the Cauchy problem of two-dimensional compressible Navier-Stokes equations in weighted spaces
• Remarks on Blow-up of Smooth Solutions to the Compressible Fluid with Constant and Degenerate Viscosities
• Global solutions to the three-dimensional full compressible Navier-Stokes equations with vacuum at infinity in some classes of large data