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

TL;DR

This paper investigates $L^p$-$L^q$ estimates for the gradient of solutions to the Stokes initial boundary value problem in bounded and exterior domains, establishing bounds that are independent of initial data.

Contribution

It provides new $L^p$-$L^q$ gradient estimates for the Stokes problem applicable in various domain types, extending previous results with domain-independent bounds.

Findings

Derived $L^p$-$L^q$ gradient estimates for the Stokes problem

Established bounds valid in both bounded and exterior domains

Results are independent of initial data norms

Abstract

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 $v_0$.