Global existence and time behaviour of strong solution to a compressible non-Newtonian fluid

Abstract

This paper studies the initial–boundary problem of a one-dimensional compressible non-Newtonian fluid with vacuum, whose power-law exponent is r∈(1,2). The global existence of strong solution is obtained by combining the local existence and a priori estimates, provided that H1-norm of the initial density is small enough and the initial data satisfies a compatibility condition. Moreover, the time behaviour of ‖ρut‖L2 and ‖ux‖Lr are obtained on time interval of existence. As a byproduct, a blow-up criterion in term of ‖uxx‖L2 for strong solution to the initial–boundary value problem of the compressible non-Newtonian fluid. Our result is an extension of the work of Yuan and Xu (2008), where the local strong solution in one-dimensional space is obtained. Our analysis is based on some new mathematical techniques.