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Resolvent Estimates in $L^\infty$ for the Stokes Operator in Nonsmooth Domains

Jun Geng, Zhongwei Shen

TL;DR

This work resolves $L^ olinebreak∞$ resolvent estimates for the Stokes operator in nonsmooth domains by blending localization on Lipschitz graph domains with new scale-based pressure-velocity estimates. The authors establish that, for bounded domains with $C^1$ (or $C^{1, olinebreak α}$) boundaries in $d ext{≥}3$ (and Lipschitz in $d=2$), the resolvent satisfies $| ext{λ}| ig\| u ig\ig olinebreak_{L^ olinebreak∞} ≤ C ig\| F ig\ig olinebreak_{L^ olinebreak∞}$ for $ ext{λ} olinebreak∈ oldsymbol{eta}$, and $| ext{λ}|^{1/2} ig\| abla u ig\ig olinebreak_{L^ olinebreak∞}$ is bounded when the domain is smoother. A key innovation is novel $L^q$-based pressure estimates that relate $p$ to the velocity gradient on scales above $| ext{λ}|^{-1/2}$, enabling control without relying on boundary-bounded gradient estimates. The paper also extends results to exterior domains for large $| ext{λ}|$ and develops a perturbation argument to carry the approach from graph Lipschitz domains to general Lipschitz domains via interpolation, thereby obtaining a comprehensive theory of resolvent bounds and analytic semigroup generation in bounded and exterior non-smooth domains. These contributions advance the understanding of Stokes dynamics in rough geometries and have implications for the analysis of Navier–Stokes equations in such domains.

Abstract

We establish resolvent estimates in spaces of bounded solenoidal functions for the Stokes operator in a bounded domain $Ω$ in $R^d$ under the assumptions that $Ω$ is $C^1$ for $d\ge 3$ and Lipschitz for $d=2$. As a corollary, it follows that the Stokes operator generates a uniformly bounded analytic semigroup in the spaces of bounded solenoidal functions in $Ω$. The smoothness conditions on $Ω$ are sharp. The case of exterior domains with nonsmooth boundaries is also studied.The key step in the proof involves new estimates which connect the pressure to the velocity in the $L^q$ average, but only on scales above certain level.

Resolvent Estimates in $L^\infty$ for the Stokes Operator in Nonsmooth Domains

TL;DR

This work resolves resolvent estimates for the Stokes operator in nonsmooth domains by blending localization on Lipschitz graph domains with new scale-based pressure-velocity estimates. The authors establish that, for bounded domains with (or ) boundaries in (and Lipschitz in ), the resolvent satisfies for , and is bounded when the domain is smoother. A key innovation is novel -based pressure estimates that relate to the velocity gradient on scales above , enabling control without relying on boundary-bounded gradient estimates. The paper also extends results to exterior domains for large and develops a perturbation argument to carry the approach from graph Lipschitz domains to general Lipschitz domains via interpolation, thereby obtaining a comprehensive theory of resolvent bounds and analytic semigroup generation in bounded and exterior non-smooth domains. These contributions advance the understanding of Stokes dynamics in rough geometries and have implications for the analysis of Navier–Stokes equations in such domains.

Abstract

We establish resolvent estimates in spaces of bounded solenoidal functions for the Stokes operator in a bounded domain in under the assumptions that is for and Lipschitz for . As a corollary, it follows that the Stokes operator generates a uniformly bounded analytic semigroup in the spaces of bounded solenoidal functions in . The smoothness conditions on are sharp. The case of exterior domains with nonsmooth boundaries is also studied.The key step in the proof involves new estimates which connect the pressure to the velocity in the average, but only on scales above certain level.
Paper Structure (11 sections, 35 theorems, 233 equations)

This paper contains 11 sections, 35 theorems, 233 equations.

Table of Contents

  1. Introduction
  2. Localization
  3. Estimates for the velocity
  4. A Neumann problem in a bounded Lipschitz domain
  5. A Neumann problem in an exterior Lipschitz domain
  6. Estimates for the pressure
  7. Proof of Theorem \ref{['main-1']} for $d\ge 3$
  8. Proof of Theorem \ref{['main-2']} for $d\ge 3$
  9. A perturbation argument for Lipschitz domains
  10. Regularity for the Stokes equations in Lipschitz and $C^1$ domains
  11. A divergence problem for a Lipschitz graph domain

Key Result

Theorem 1.1

Let $\Omega$ be a bounded $C^{1}$ domain in $\mathbb{R}^{d}$, $d\ge 3$, or a bounded Lipschitz domain in $\mathbb{R}^2$. Let $\lambda \in \Sigma_\theta$, where $\theta \in (0, \pi/2)$. Then for any $F\in L_\sigma^\infty(\Omega)$, the solution of the Dirichlet problem eq-0 in $W_0^{1, 2}(\Omega; \mat Moreover, if $\Omega$ is a bounded $C^{1, \alpha}$ domain for some $\alpha>0$, then The constants

Theorems & Definitions (78)

  • Theorem 1.1
  • Remark 1.2
  • Remark 1.3
  • Remark 1.4
  • Corollary 1.5
  • Theorem 1.6
  • Corollary 1.7
  • Theorem 2.1
  • Theorem 2.2
  • proof
  • ...and 68 more