The Stokes resolvent problem: Optimal pressure estimates and remarks on resolvent estimates in convex domains

TL;DR

This paper investigates the Stokes resolvent problem with different boundary conditions, establishing decay rates of the operator norm and providing optimal resolvent and gradient estimates in L^p spaces for convex domains.

Contribution

It provides the first L^p resolvent estimates for the Stokes problem with Neumann-type boundary conditions on convex domains, and analyzes decay rates under Dirichlet and Neumann boundary conditions.

Findings

Operator norm decay rate under Neumann conditions matches the equation's scaling.

Decay rate under Dirichlet conditions cannot be improved beyond 1/4.

Established optimal L^p resolvent and gradient estimates for convex domains.

Abstract

The Stokes resolvent problem $\lambda u - \Delta u + \nabla \phi = f$ with $\mathrm{div}(u) = 0$ subject to homogeneous Dirichlet or homogeneous Neumann-type boundary conditions is investigated. In the first part of the paper we show that for Neumann-type boundary conditions the operator norm of $\mathrm{L}^2_{\sigma} (\Omega) \ni f \mapsto \pi \in \mathrm{L}^2 (\Omega)$ decays like $\lvert \lambda \rvert^{- 1 / 2}$ which agrees exactly with the scaling of the equation. In comparison to that, we show that the operator norm of this mapping under Dirichlet boundary conditions decays like $\lvert \lambda \rvert^{- \alpha}$ for $0 \leq \alpha < 1 / 4$ and we show that this decay rate cannot be improved to any exponent $\alpha > 1 / 4$, thereby, violating the natural scaling of the equation. In the second part of this article, we investigate the Stokes resolvent problem subject to homogeneous Neumann-type boundary conditions if the underlying domain $\Omega$ is convex. We establish optimal resolvent estimates and gradient estimates in $\mathrm{L}^p (\Omega ; \mathbb{C}^d)$ for $2d / (d + 2) < p < 2d / (d - 2)$ (with $1 < p < \infty$ if $d = 2$). This interval is larger than the known interval for resolvent estimates subject to Dirichlet boundary conditions on general Lipschitz domains and is to the best knowledge of the author the first result that provides $\mathrm{L}^p$-estimates for the Stokes resolvent subject to Neumann-type boundary conditions on general convex domains.