Local-in-time existence of strong solutions to a class of compressible Power-Law flows

TL;DR

This paper proves the local-in-time existence of strong solutions for a class of compressible power-law non-Newtonian fluids in three dimensions, with conditions on the power-law exponent and an improved blow-up criterion.

Contribution

It establishes the local existence of strong solutions for variable exponent power-law fluids and provides an improved blow-up criterion based on velocity gradient norms.

Findings

Existence of strong solutions for 7/5 < p(t,x) ≤ 2.

Blow-up criterion involving the L-infinity in time and L^3 in space norm of velocity gradient.

Results applicable to periodic domains in three-dimensional space.

Abstract

We consider a model of the compressible non-Newtonian fluids for power-law flow fulfilling a periodic domain in ${\mathbb R}^3,$ in which the extra stress tensor is induced by a potential with $p(t,x)$-structure. The local-in-time existence of strong solution is proved for all $\frac{7}{5} < \inf p(t,x) \leqslant \sup p(t,x) \leqslant 2.$ Further, an improved blow-up criterion for strong solutions is given in terms of the $L^\infty(0,T;L^3(\Omega))$-norm of the gradient of the velocity.