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

TL;DR

This paper proves local existence of strong solutions for 3D incompressible Euler equations in bounded domains within a critical Besov space, utilizing vanishing viscosity and commutator estimates.

Contribution

It introduces a framework for analyzing Euler equations in critical Besov spaces on bounded domains, extending previous work with new energy bounds.

Findings

Established unique local existence of solutions in critical Besov space

Developed energy bounds uniform in viscosity using commutator estimates

Extended analysis of Euler equations to bounded domains with smooth boundary

Abstract

We consider the 3D incompressible Euler equations in bounded domains $\Omega$ with smooth boundary $\partial\Omega$. 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 $\partial\Omega$, 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.