Maximal $L_p$-$L_q$ regularity for the Stokes equations with various boundary conditions in the half space

TL;DR

This paper establishes resolvent and maximal $L_p$-$L_q$ regularity estimates for the Stokes equations with various boundary conditions in the half space, using Fourier multipliers and kernel operator bounds.

Contribution

It introduces a new simple method to achieve maximal regularity for the Stokes equations with Dirichlet, Neumann, and Robin boundary conditions in the half space.

Findings

Proves resolvent $L_p$ estimates for the Stokes equations.

Establishes maximal $L_p$-$L_q$ regularity estimates.

Provides a new approach based on Fourier multipliers and kernel operator bounds.

Abstract

We prove resolvent $L_p$ estimates and maximal $L_p$-$L_q$ regularity estimates for the Stokes equations with Dirichlet, Neumann and Robin boundary conditions in the half space. Each solution is constructed by a Fourier multiplier of $x'$-direction and an integral of $x_N$-direction. We decompose the solution such that the symbols of the Fourier multipliers are bounded and holomorphic. We see that the operator norms are dominated by a homogeneous function of order $-1$ for $x_N$-direction. The basis are Weis's operator-valued Fourier multiplier theorem and a boundedness of a kernel operator. We give a new simple approach to get maximal regularity in the half space.