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The Dirichlet problem for the nonstationary Stokes system in a polygon

Jürgen Rossmann

TL;DR

This work analyzes the Dirichlet problem for the nonstationary Stokes system in polygonal domains by developing a weighted Sobolev framework that captures corner singularities. It reduces the time-dependent problem to a family of parameter-dependent problems in angles, establishes existence, uniqueness, and sharp a priori estimates for these problems, and then lifts the results to the polygon via a partition-of-unity argument. The time dimension is handled through Laplace transforms, giving solvability in weighted spaces with exponential time weights, and the results extend to finite time intervals through standard extension/restriction arguments. The main contribution is a comprehensive well-posedness theory for strong solutions in polygonal domains with corner singularities, including precise pressure estimates and compatibility conditions, applicable to both infinite and finite time horizons. This provides a solid analytical foundation for simulations and further study of viscous flows in domains with corners, where singular behavior and divergence constraints pose technical challenges.

Abstract

The author proves the existence of strong solutions of the Dirichlet problem for the nonstationary Stokes system in polygonal domain. Here, the solutions are elements of weighted Sobolev spaces, where the weight function is a power of the distance from the corner points.

The Dirichlet problem for the nonstationary Stokes system in a polygon

TL;DR

This work analyzes the Dirichlet problem for the nonstationary Stokes system in polygonal domains by developing a weighted Sobolev framework that captures corner singularities. It reduces the time-dependent problem to a family of parameter-dependent problems in angles, establishes existence, uniqueness, and sharp a priori estimates for these problems, and then lifts the results to the polygon via a partition-of-unity argument. The time dimension is handled through Laplace transforms, giving solvability in weighted spaces with exponential time weights, and the results extend to finite time intervals through standard extension/restriction arguments. The main contribution is a comprehensive well-posedness theory for strong solutions in polygonal domains with corner singularities, including precise pressure estimates and compatibility conditions, applicable to both infinite and finite time horizons. This provides a solid analytical foundation for simulations and further study of viscous flows in domains with corners, where singular behavior and divergence constraints pose technical challenges.

Abstract

The author proves the existence of strong solutions of the Dirichlet problem for the nonstationary Stokes system in polygonal domain. Here, the solutions are elements of weighted Sobolev spaces, where the weight function is a power of the distance from the corner points.

Paper Structure

This paper contains 10 sections, 10 theorems, 89 equations.

Table of Contents

  1. Introduction
  2. The parameter-dependent problem in an angle
  3. The parameter-dependent problem in a polygon
  4. Existence of solutions
  5. An estimate for $p$
  6. An estimate for the solution
  7. The time-dependent problem on $\Omega$
  8. Weighted Sobolev spaces in $\Omega\times (0,T)$
  9. Solvability in $\stackrel{\circ}{W}\!{}_\beta^{2,1}(\Omega\times {\Bbb R}_+,e^{-\gamma t})\times L_{2,-\gamma}(0,\infty;W_\beta^1(\Omega))$
  10. Solvability in a finite $t$-interval

Key Result

Theorem 2.1

Suppose that $s\not=0$, $\hbox{\em Re}\, s \ge 0$ and $\max(-\frac{\pi}{\alpha} \, ,1-\hbox{\em Re}\, \lambda^*(\alpha))< \beta <\min(2,\frac{\pi}{\alpha})$, $\beta\not=0$, where $\lambda^*(\alpha)$ is the solution of the equation (evequation) with smallest positive real part. Furthermore, let $f\i has a unique solution $(u,p)\in E_\beta^2(K)\times V_\beta^1(K)$ satisfying the estimate with a co

Theorems & Definitions (12)

  • Theorem 2.1
  • Remark 2.1
  • Remark 2.2
  • Lemma 3.1
  • Lemma 3.2
  • Lemma 3.3
  • Lemma 3.4
  • Lemma 3.5
  • Theorem 3.1
  • Lemma 4.1
  • ...and 2 more