Local well-posedness of the incompressible Euler equations in $B^1_{\infty,1}$ and the inviscid limit of the Navier-Stokes equations

TL;DR

This paper establishes local well-posedness of the Euler equations in a critical Besov space and proves the inviscid limit of Navier-Stokes in the same topology, addressing open problems in fluid dynamics.

Contribution

It demonstrates the Euler equations are locally well-posed in the borderline Besov space and proves the inviscid limit of Navier-Stokes in this setting, using a Bona-Smith type method.

Findings

Euler equations are locally well-posed in $B^{rac dp+1}_{p,1}$ spaces.

Inviscid limit of Navier-Stokes holds in the same Besov topology.

Addresses open problems by Bourgain, Li, Misio{ extl}ek, and Yoneda.

Abstract

We prove the inviscid limit of the incompressible Navier-Stokes equations in the same topology of Besov spaces as the initial data. The proof is based on proving the continuous dependence of the Navier-Stokes equations uniformly with respect to the viscosity. To show the latter, we rely on some Bona-Smith type method in the $L^p$ setting. Our obtained result implies a new result that the Cauchy problem of the Euler equations is locally well-posed in the borderline Besov space $B^{\frac dp+1}_{p,1}(\mathbb{R}^d)$, $1\leq p\leq \infty$, $d\geq 2$, in the sense of Hadmard, which is an open problem left in recent works by Bourgain and Li in \cite{BL,BL1} and by Misio{\l}ek and Yoneda in \cite{MY,MY2, MY3}.