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On regularity of solutions to the Navier--Stokes equation with initial data in $\mathrm{BMO}^{-1}$

Hedong Hou

Abstract

We prove that any mild solution in the Koch--Tataru space to the incompressible Navier--Stokes equation with initial data in $\mathrm{BMO}^{-1}$ is weak*-continuous in time, valued in $\mathrm{BMO}^{-1}$. We also show that the global mild solution vanishes in $\mathrm{BMO}^{-1}$ at infinity in time.

On regularity of solutions to the Navier--Stokes equation with initial data in $\mathrm{BMO}^{-1}$

Abstract

We prove that any mild solution in the Koch--Tataru space to the incompressible Navier--Stokes equation with initial data in is weak*-continuous in time, valued in . We also show that the global mild solution vanishes in at infinity in time.

Paper Structure

This paper contains 13 sections, 6 theorems, 75 equations.

Table of Contents

  1. Introduction
  2. Function spaces
  3. Outline of the proofs
  4. Proof of Proposition \ref{['prop:L0']}
  5. Continuity at t=0
  6. Estimates of $\Phi_1$
  7. Estimates of $R_1$
  8. Continuity at t>0
  9. Long-time limit
  10. Proof of Proposition \ref{['prop:L1']}
  11. Continuity at t=0
  12. Continuity at t>0
  13. Long-time limit

Key Result

theorem 1

Let $0<T \le \infty$. Let $u_0 \in \bmo^{-1}$ be divergence free and $u \in X_T$ be a mild solution to the Navier--Stokes equation e:NS with initial data $u_0$. Then $u$ belongs to $C([0,T);\bmo^{-1})$ with $u(0)=u_0$ and

Theorems & Definitions (14)

  • theorem 1
  • theorem 2
  • remark 1
  • proof : Proof of Theorems \ref{['thm:cont-bmo-1']} and \ref{['thm:long-time']}, admitting Proposition \ref{['prop:L']}
  • proof : Proof of Proposition \ref{['prop:L']}
  • lemma 1
  • proof : Proof of Lemma \ref{['lemma:R']}
  • lemma 2
  • lemma 3
  • proof
  • ...and 4 more