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Navier--Stokes flow in the exterior of a moving obstacle with a Lipschitz boundary

Tomoki Takahashi, Keiichi Watanabe

Abstract

Consider the three-dimensional Navier--Stokes flow past a moving rigid body $\mathscr{O} \subset \mathbb{R}^3$ with prescribed translational and angular velocities, where $\mathscr{O}$ stands for a bounded Lipschitz domain. We prove that the solution to the linearized problem is governed by a $C_0$-semigroup on solenoidal $L^q$-vector spaces with the $L^q$-$L^r$ estimates provided that $|1/q-1/2|<1/6+\varepsilon$ with some $\varepsilon>0$, where $r \ge q$ may be taken arbitrary large. As an application, we prove the existence and uniqueness of global mild solutions to the Navier--Stokes problem if the translational and angular velocities as well as the initial are sufficiently small.

Navier--Stokes flow in the exterior of a moving obstacle with a Lipschitz boundary

Abstract

Consider the three-dimensional Navier--Stokes flow past a moving rigid body with prescribed translational and angular velocities, where stands for a bounded Lipschitz domain. We prove that the solution to the linearized problem is governed by a -semigroup on solenoidal -vector spaces with the - estimates provided that with some , where may be taken arbitrary large. As an application, we prove the existence and uniqueness of global mild solutions to the Navier--Stokes problem if the translational and angular velocities as well as the initial are sufficiently small.
Paper Structure (12 sections, 27 theorems, 237 equations)

This paper contains 12 sections, 27 theorems, 237 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Notation
  4. Whole space problem
  5. Bogovskiı̆ operator
  6. Interior problem
  7. Generation of a $C_0$-semigroup
  8. $L^q$-$L^r$ estimate with additional restriction for $r$
  9. Proofs of Theorems \ref{['thmlqlr']} and \ref{['thmlqlr2']}
  10. Local energy decay estimates
  11. Proofs of Theorems \ref{['thmlqlr']} and \ref{['thmlqlr2']}
  12. Nonlinear problem

Key Result

Theorem 1.1

Let $\Omega \subset \mathbb{R}^3$ be an exterior Lipschitz domain. Let $\eta, \omega \in \mathbb{R}^3$. Then the following assertions are valid.

Theorems & Definitions (51)

  • Theorem 1.1
  • Remark 1.2
  • Remark 1.3
  • Theorem 1.4
  • Theorem 1.5
  • Remark 1.6
  • Lemma 2.1
  • Lemma 2.2: H20
  • Proposition 2.3
  • Proposition 2.4
  • ...and 41 more