  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 $v_0$.
