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Neumann Problems for the Stokes Equations in Convex Domains

Jun Geng, Zhongwei Shen

Abstract

This paper studies the Neumann boundary value problems for the Stokes equations in a convex domain in $\mathbb{R}^d$. We obtain nontangential-maximal-function estimates in $L^p$ and $W^{1, p}$ estimates for $p$ in certain ranges depending on $d$. These ranges are larger than the known ranges for Lipschitz domains. The proof relies on a $W^{2, 2}$ estimate for the Stokes equations in convex domains.

Neumann Problems for the Stokes Equations in Convex Domains

Abstract

This paper studies the Neumann boundary value problems for the Stokes equations in a convex domain in . We obtain nontangential-maximal-function estimates in and estimates for in certain ranges depending on . These ranges are larger than the known ranges for Lipschitz domains. The proof relies on a estimate for the Stokes equations in convex domains.

Paper Structure

This paper contains 8 sections, 16 theorems, 123 equations.

Table of Contents

  1. Introduction
  2. $W^{2,2}$ estimates
  3. Nontangential-maximal-function estimates
  4. W1p estimates
  5. The case $F=0$ and $g=0$
  6. The case of $f=0$ and $F=0$
  7. The case of $f=0$ and $g=0$
  8. The proof of Theorem \ref{['W1p']}

Key Result

Theorem 1.1

Let $\Omega$ be a bounded convex domain in $\mathbb{R}^{d}$, $d\geq 2$. Then the $L^p$ Neumann problem stokesSystem is uniquely solvable if where $\varepsilon=\varepsilon(\Omega)>0$.

Theorems & Definitions (33)

  • Theorem 1.1
  • Remark 1.2
  • Theorem 1.3
  • Remark 1.4
  • Theorem 2.1
  • Lemma 2.2
  • proof
  • proof : Proof of Theorem \ref{['W22']}
  • Corollary 2.3
  • proof
  • ...and 23 more