Vanishing viscosity non-unique solutions to the forced 2D Euler Equations

TL;DR

This paper explores how the inviscid limit of the forced 2D Navier-Stokes equations can select unique solutions from a non-unique set of solutions to the 2D Euler equations, revealing a threshold dependent on viscosity and initial perturbation size.

Contribution

It identifies a critical threshold for initial perturbations relative to viscosity that determines whether the inviscid limit yields a unique or non-unique solution in the forced 2D Euler equations.

Findings

Below the threshold, solutions are unique and radial.

At the threshold, viscous solutions converge to non-unique, non-radial solutions.

The inviscid limit acts as a selection principle depending on initial perturbation size.

Abstract

The forced 2D Euler equations exhibit non-unique solutions with vorticity in $L^p$, $p > 1$, whereas the corresponding Navier-Stokes solutions are unique. We investigate whether the inviscid limit $\nu \to 0^+$ from the forced 2D Navier-Stokes to Euler equations is a selection principle capable of "resolving" the non-uniqueness. We focus on solutions in a neighborhood of the non-uniqueness scenario discovered by Vishik; specifically, we incorporate viscosity $\nu$ and consider $O(\varepsilon)$ size perturbations of the initial datum. We discover a uniqueness threshold $\varepsilon \sim \nu^{\kappa_{\rm c}}$, below which the vanishing viscosity solution is unique and radial, and at which there are viscous solutions converging to non-unique, non-radial solutions.