Premium
Blowup of smooth solutions to the compressible Navier‐Stokes equation with compact density
Author(s) -
Xin Zhouping
Publication year - 1998
Publication title -
communications on pure and applied mathematics
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 3.12
H-Index - 115
eISSN - 1097-0312
pISSN - 0010-3640
DOI - 10.1002/(sici)1097-0312(199803)51:3<229::aid-cpa1>3.0.co;2-c
Subject(s) - polytropic process , mathematics , euler equations , compressibility , mathematical analysis , bounded function , constant (computer programming) , perturbation (astronomy) , compact space , euler system , initial value problem , physics , classical mechanics , mechanics , quantum mechanics , computer science , programming language
We present a sufficient condition on the blowup of smooth solutions to the compressible Navier‐Stokes equations in arbitrary space dimensions with initial density of compact support. As an immediate application, 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, and an upper bound, which depends only on the initial data, on the blowup time follows from our elementary analysis immediately. Another implication is that there is no global small (decay in time) or even bounded (in the case that all the viscosity coefficients are positive) smooth solutions to the compressible Navier‐Stokes equations for polytropic fluids, no matter how small the initial data are, as long as the initial density is of compact support. This is in contrast to the classical theory of global existence of small solutions to the same system with initial data being a small perturbation of a constant state that is not a vacuum. The blowup of smooth solutions to the compressible Euler system with initial density and velocity of compact support is a simple consequence of our argument. © 1998 John Wiley & Sons, Inc.