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Spatial decay/asymptotics in the Navier-Stokes equation

Peter Topalov

Abstract

We discuss the appearance of spatial asymptotic expansions of solutions of the Navier-Stokes equation on $\mathbb{R}^n$. In particular, we prove that the Navier-Stokes equation is locally well-posed in a class of weighted Sobolev and asymptotic spaces. The solutions depend analytically on the initial data and time and (generically) develop non-trivial asymptotic terms as $|x|\to\infty$. In addition, the solutions have a spatial smoothing property that depends on the order of the asymptotic expansion.

Spatial decay/asymptotics in the Navier-Stokes equation

Abstract

We discuss the appearance of spatial asymptotic expansions of solutions of the Navier-Stokes equation on . In particular, we prove that the Navier-Stokes equation is locally well-posed in a class of weighted Sobolev and asymptotic spaces. The solutions depend analytically on the initial data and time and (generically) develop non-trivial asymptotic terms as . In addition, the solutions have a spatial smoothing property that depends on the order of the asymptotic expansion.

Paper Structure

This paper contains 5 sections, 10 theorems, 56 equations.

Table of Contents

  1. Introduction
  2. Consequences from the conserved quantity
  3. Proof of Theorem \ref{['th:NS_W-spaces']} and Proposition \ref{['prop:asymptotics']}
  4. The spatial smoothing
  5. Basic properties of asymptotic spaces

Key Result

Theorem 1.1

Assume that $m>2+\frac{d}{p}$ and $\delta+\frac{d}{p}\ge 0$. Then, we have: The solution in (a) and (b) depends Lipschitz continuously on the initial data $u_0\in W^{m,p}_\delta$ in the sense that the data-to-solution map is Lipschitz continuous. Moreover, the solutions depend analytically on time and the initial data in the sense that the map in the case (a) and the map in the case (b), where

Theorems & Definitions (17)

  • Theorem 1.1
  • Proposition 1.1
  • Proposition 1.2
  • Proposition 1.3
  • Proposition 2.1
  • Corollary 2.1
  • proof : Proof of Corollary \ref{['coro:preservation_of_weight_A']}
  • proof : Proof of Proposition \ref{['prop:preservation_of_weight_W']}
  • proof : Proof of Theorem \ref{['th:NS_W-spaces']}
  • Remark 3.1
  • ...and 7 more