Inviscid limit of vorticity distributions in Yudovich class

TL;DR

This paper proves that solutions of the Navier-Stokes equations with bounded initial vorticity converge to Euler solutions in the inviscid limit, ensuring vorticity distributions also converge, and establishes continuity of the solution map in vorticity topology.

Contribution

It demonstrates strong convergence of Navier-Stokes solutions to Euler solutions for Yudovich class data and establishes continuity of the solution map in vorticity topology.

Findings

Vorticity distribution functions converge to inviscid solutions.

Strong convergence in $L^ abla(0,T;W^{1,p})$ for Navier-Stokes to Euler.

Continuity of Euler solution map in $L^p$ vorticity topology.

Abstract

We prove that given initial data $\omega_0\in L^\infty(\mathbb{T}^2)$, forcing $g\in L^\infty(0,T; L^\infty(\mathbb{T}^2))$, and any $T>0$, the solutions $u^\nu$ of Navier-Stokes converge strongly in $L^\infty(0,T;W^{1,p}(\mathbb{T}^2))$ for any $p\in [1,\infty)$ to the unique Yudovich weak solution $u$ of the Euler equations. A consequence is that vorticity distribution functions converge to their inviscid counterparts. As a byproduct of the proof, we establish continuity of the Euler solution map for Yudovich solutions in the $L^p$ vorticity topology. The main tool in these proofs is a uniformly controlled loss of regularity property of the linear transport by Yudovich solutions. Our results provide a partial foundation for the Miller--Robert statistical equilibrium theory of vortices as it applies to slightly viscous fluids.