Poisson kernel and blow-up of the second derivatives near the boundary for Stokes equations with Navier boundary condition

TL;DR

This paper derives the Poisson kernel for Stokes equations with Navier boundary conditions in a half space, demonstrating that second derivatives of velocity can blow up near the boundary despite bounded velocity and first derivatives.

Contribution

It provides an explicit Poisson kernel for Stokes equations with Navier boundary conditions and shows the unboundedness of second derivatives near the boundary, contrasting with first derivatives.

Findings

Explicit Poisson kernel derived for Navier boundary conditions.

Second derivatives of velocity can be unbounded near the boundary.

First derivatives remain bounded despite second derivative blow-up.

Abstract

We derive the explicit Poisson kernel of Stokes equations in the half space with nonhomogeneous Navier boundary condition (BC) for both infinite and finite slip length. By using this kernel, for any $q>1$, we construct a finite energy solution of Stokes equations with Navier BC in the half space, with bounded velocity and velocity gradient, but having unbounded second derivatives in $L^q$ locally near the boundary. While the Caccioppoli type inequality of Stokes equations with Navier BC is true for the first derivatives of velocity, which is proved by us in [CPAA 2023], this example shows that the corresponding inequality for the second derivatives of the velocity is not true. Moreover, we give an alternative proof of the blow-up using a shear flow example, which is simple and is the solution of both Stokes and Navier--Stokes equations.