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

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 $\frac{7}{5}<\inf p(t,x)⩽\sup p(t,x)⩽2.$ Further, an improved blow-up criterion for strong solutions is given in terms of the $L^{\mathrm{\infty }}(0,T;L^3(\mathrm{\Omega }))$-norm of the gradient of the velocity.