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Construction of a right inverse for the divergence in non-cylindrical time dependent domains

Olli Saari, Sebastian Schwarzacher

Abstract

We construct a stable right inverse for the divergence operator in non-cylindrical domains in space-time. The domains are assumed to be Hölder regular in space and evolve continuously in time. The inverse operator is of Bogovskij type, meaning that it attains zero boundary values. We provide estimates in Sobolev spaces of positive and negative order with respect to both time and space variables. The regularity estimates on the operator depend on the assumed Hölder regularity of the domain. The results can naturally be connected to the known theory for Lipschitz domains. As an application, we prove refined pressure estimates for weak and very weak solutions to Navier--Stokes equations in time dependent domains.

Construction of a right inverse for the divergence in non-cylindrical time dependent domains

Abstract

We construct a stable right inverse for the divergence operator in non-cylindrical domains in space-time. The domains are assumed to be Hölder regular in space and evolve continuously in time. The inverse operator is of Bogovskij type, meaning that it attains zero boundary values. We provide estimates in Sobolev spaces of positive and negative order with respect to both time and space variables. The regularity estimates on the operator depend on the assumed Hölder regularity of the domain. The results can naturally be connected to the known theory for Lipschitz domains. As an application, we prove refined pressure estimates for weak and very weak solutions to Navier--Stokes equations in time dependent domains.

Paper Structure

This paper contains 19 sections, 28 theorems, 222 equations.

Table of Contents

  1. Introduction
  2. Application to the Navier Stokes equation on non-cylindrical domains
  3. Inhomogeneous boundary data for incompressible fluids
  4. Fluid-structure interactions
  5. Density estimates for compressible fluids
  6. Homogenization
  7. Preliminaries
  8. Notation
  9. Domains
  10. Function spaces
  11. Inequalities
  12. Construction of a Bogovskij operator
  13. The first auxiliary operator
  14. Decomposition of the domain
  15. Construction of the operator
  16. ...and 4 more sections

Key Result

Theorem 1.1

Let $0 < \alpha \le 1$. Assume $\Omega$ to be a $C^{\alpha,1,\theta}$ domain, $1<p< \infty$ and $\kappa, k \ge 0$ be integers. Then for all compactly supported test functions $f \in C_{smz}^{\infty}(\Omega)$ and all times $t$ with $C$ only depending on $\Omega$, $k$, $\kappa$ and $p$.

Theorems & Definitions (55)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Proposition 1.5
  • Definition 1.6
  • Theorem 1.7
  • Theorem 1.8
  • Theorem 1.9
  • Definition 2.1: $C^{\alpha,\beta,\theta}$ domain
  • ...and 45 more