Global regularity of weak solutions for steady motions of electrorheological fluids in 3D smooth domain

In this paper, we first show higher integrability of the gradient of weak solutions for steady motions of electrorheological fluids under a further assumption on regularity of external force f in 3D smooth domain. Based on the higher integrability result, we then prove that any weak solution to the problem becomes global strong one under such an assumption on f. Moreover we improve the lower bound on p(x) for global regularity of solutions to the problem and their Stokes setting even if p(x) is a constant.