Let $\Omega$ be abounded domain of $\mathbb{R}^n$. We consider the stationary Stokes problem \begin{cases} \Delta u + \nabla p = f \qquad in \ \ \ \ \Omega \\ \nabla . u =0 \qquad in \qquad \Omega \\ u =0 \quad in \qquad \partial \Omega \end{cases} If we hit the problem with a suitable test function $v$ and integrate by parts, we get our bilinear form $$\int_{\Omega} \nabla u \nabla v dx = \int_{\Omega} f v dx $$ After applying the Lax-Milgram theorem we get the existence and uniqueness, and further, we have $$\|u\|_{H} \leq C \|f\|_{H'}$$
$H$ hilbert space, $H'$ dual of the hilbert space. The above scenario is all good well known. Now my question are there any a priori estimates in the literature that relates $f$ and $u $ in a way
$$\|f\|_{\text{such space}} \leq \|u\|_{\text{such space, preferly $\dot{H}^1(\Omega)$ }}$$ Best regards.