Inviscid limit of vorticity distributions in Yudovich class

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.

16 pgs, accepted version (3/9/2020)