Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Asymptotic properties of the Stokes flow in an exterior domain with slowly decaying initial data and its application to the Navier-Stokes equations

Tongkeun Chang, Bum Ja Jin

Abstract

In this paper, we study the decay rate of the Stokes flow in an exterior domain with a slowly decaying initial data ${\bf u}_0(x)=O(|x|^{-\al}), 0<\al\leq n$. %which is not $L^1$ integrable. As an application we find the unique strong solution of the Navier-Stokes equations corresponding to a slowly decaying initial data. We also derive the pointwise decay estimate of the Navier-Stokes flow. Our decay rates will be optimal compared with the decay rates of the heat flow.

Asymptotic properties of the Stokes flow in an exterior domain with slowly decaying initial data and its application to the Navier-Stokes equations

Abstract

In this paper, we study the decay rate of the Stokes flow in an exterior domain with a slowly decaying initial data . %which is not integrable. As an application we find the unique strong solution of the Navier-Stokes equations corresponding to a slowly decaying initial data. We also derive the pointwise decay estimate of the Navier-Stokes flow. Our decay rates will be optimal compared with the decay rates of the heat flow.
Paper Structure (9 sections, 13 theorems, 156 equations)

This paper contains 9 sections, 13 theorems, 156 equations.

Table of Contents

  1. Introduction
  2. Notations and Preliminaries
  3. Proof of Theorem \ref{['theorem_stokes1']}
  4. Proof of Theorem \ref{['theorem1']}
  5. Uniform boundedness.
  6. Convergence in ${\mathcal{X}}(\alpha,q)$
  7. Uniqueness in ${\mathcal{X}}(\alpha,q)$
  8. Proof of Corollary \ref{['navier_wieght_space']}
  9. Proof of Lemma \ref{['lemma1']}

Key Result

Theorem 1.1

Let $\Omega\subset{\mathbb R}^n$, $n\geq 3$ be an exterior domain of smooth boundary with $B_{\frac{1}{2}}\subseteq \Omega^c\subseteq B_1$. Let $0 <\alpha \leq n$ and $\frac{n}{\alpha} < q \leq \infty$. Assume that ${\bf u}_0$ satisfies the conditions H1-H2 and ${\bf u}_0=O(|x|^{-\alpha})$ for some Then it holds that Moreover, it holds that

Theorems & Definitions (20)

  • Theorem 1.1
  • Corollary 1.2
  • Theorem 1.3
  • Corollary 1.4
  • Remark 1.5
  • Proposition 2.1
  • Theorem 2.2: galdikozono
  • Lemma 2.3
  • proof
  • Remark 2.4
  • ...and 10 more