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Theory Seminar
Boundary gradient estimates and second derivative blow-up for Stokes equations with Navier boundary condition

Abstract

For the non-stationary Stokes system, it is well-known that one can improve spatial regularity in the interior, but not near the boundary if it is coupled with the no-slip boundary condition (BC). We show that, under the Navier BC with either infinite or finite slip length, we have Caccioppoli type gradient estimates near a flat boundary, contrary to the no-slip BC case. However, for every finite slip length and q>1, we construct a finite energy solution of Stokes equations with nonhomogeneous Navier BC in the half space with bounded, compactly supported boundary data, bounded velocity and velocity gradient, but unbounded second derivatives in L^q locally near the boundary. To show this, we first derive the explicit Poisson kernel of Stokes equations in the half space with nonhomogeneous Navier BC for both infinite and finite slip length. Moreover, we give an alternative proof of the blow-up using a shear flow example, which solves both Stokes and Navier-Stokes equations, but has no spatial decay. This talk is based on joint work with Hui Chen and Su Liang, arXiv:2306.16480 and arXiv:2406.15995