  We consider Stokes systems with measurable coefficients and Lions-type
boundary conditions. We show that, in contrast to the Dirichlet boundary
conditions, local boundary mixed-norm $L_{s,q}$-estimates hold for the spatial
second-order derivatives of solutions, assuming the smallness of the mean
oscillations of the coefficients with respect to the spatial variables in small
cylinders. In the un-mixed norm case with $s=q=2$, the result is still new and
provides local boundary Caccioppoli-type estimates. The main challenges in the
work arise from the lack of regularity of the pressure and time derivatives of
the solutions and from interaction of the boundary with the nonlocal structure
of the system. To overcome these difficulties, our approach relies heavily on
several newly developed regularity estimates for both divergence and
non-divergence form parabolic equations with coefficients that are only
measurable in the time variable and in one of the spatial variables.

Analysis of PDEs