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We establish functional analytic properties of the Stokes operator withbounded measurable coefficients on $L^p_{\sigma} (\mathbb{R}^d)$, $d \geq 2$,for $\lvert 1 / p - 1 / 2 \rvert < 1 / d$. These include optimal resolventbounds and the property of maximal $L^q$-regularity. We further give regularityestimates on the gradient of the solution to the Stokes resolvent problem withbounded measurable coefficients. As a key to these results we establish thevalidity of a non-local Caccioppoli inequality to solutions of the Stokesresolvent problem.

A non-local approach to the generalized Stokes operator with bounded measurable coefficients