On the L^p-L^q estimates of the gradient of solutions to the Stokes problem

The paper is concerned with estimates of the gradient of the solutions to the Stokes IBVP both in a bounded and in an exterior domain. More precisely, we look for estimates of the kind $\|\nabla v(t)\|_q\leq g(t)\|\nabla v_0\|_p, q\geq p>1$, for all $t>0$, where function $g$ is independent of $v_0$.