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Incompressible Euler equations in 3D bounded domains in a critical space

Tsukasa Iwabuchi, Hideo Kozono

Abstract

We consider the 3D incompressible Euler equations in bounded domains $Ω$ with smooth boundary $\partialΩ$. Based on the paper by Iwabuchi, Matsuyama and Taniguchi (2019), we define the Besov space $B^s_{p, q}(A)$ by means of the Stokes operator $A$ with the Neumann boundary condition on $\partialΩ$, and prove unique local existence theorem of strong solution for the initial data in the critical Besov space $B^{\frac52}_{2, 1}(A)$. Our proof relies on the method of vanishing viscosity. The commutator estimate plays an essential role for derivation of energy bounds which hold uniformly with respect to viscosity constants.

Incompressible Euler equations in 3D bounded domains in a critical space

Abstract

We consider the 3D incompressible Euler equations in bounded domains with smooth boundary . Based on the paper by Iwabuchi, Matsuyama and Taniguchi (2019), we define the Besov space by means of the Stokes operator with the Neumann boundary condition on , and prove unique local existence theorem of strong solution for the initial data in the critical Besov space . Our proof relies on the method of vanishing viscosity. The commutator estimate plays an essential role for derivation of energy bounds which hold uniformly with respect to viscosity constants.

Paper Structure

This paper contains 8 sections, 22 theorems, 181 equations.

Table of Contents

  1. Introduction
  2. Preliminary
  3. The Stokes operator and Besov spaces
  4. Analytical tools
  5. Solutions with $\varepsilon \Delta$ and Regularity
  6. Uniform Boundedness with respect to $\varepsilon$
  7. Proof of Theorem \ref{['thm:1']}
  8. Compact Embedding and Time Continuity

Key Result

Theorem 1.1

For every $u_0 \in B^{\frac{5}{2}}_{2,1}(A)$ there exist a time $T > 0$ and a unique solution $u \in C([0,T], B^{\frac{5}{2}}_{2,1}(A))$ of (eq:1). Such a solution $u$ satisfies $u \in C^1 ([0,T], W^{1,p}(\Omega))$ for all $1 \leq p \leq 3$.

Theorems & Definitions (25)

  • Theorem 1.1
  • Remark 1.2
  • Lemma 2.1
  • proof
  • Proposition 2.2
  • Lemma 2.3
  • Lemma 2.4
  • Lemma 2.5
  • proof
  • Lemma 2.6
  • ...and 15 more