The 2D Euler equations are well-posed for generic initial data in $L^2$

TL;DR

This paper proves that for a generic set of initial data in L^2, the 2D Euler equations are well-posed, with solutions exhibiting energy conservation and stability properties.

Contribution

It establishes the generic well-posedness of 2D Euler equations in L^2, including energy conservation, vanishing viscosity limits, and well-posed transport equations.

Findings

Existence of a residual set of initial data with unique solutions

Solutions satisfy energy balance and are limits of Navier-Stokes solutions

Associated flow and transport equations are well-posed

Abstract

In this note we show the existence of a residual set (in the sense of Baire) of divergence free initial data $u_0\in L^2(D)$, $D=\mathbb{R}^2$ or $\mathbb{T}^2$, for which global existence and uniqueness of weak solutions to the incompressible 2D Euler equations holds. The associated solutions $u$ satisfy the energy balance and are recovered in the vanishing viscosity limit from solutions to 2D Navier-Stokes, which as a consequence cannot display anomalous dissipation of energy. Additionally, there exists a unique regular Lagrangian flow associated to such $u$, and the associated transport equation is well-posed. Finally, when $D=\mathbb{T}^2$, the solution $u$ is recovered as the limit of Galerkin approximations. The proof relies on global existence of smooth solutions and weak-strong uniqueness arguments.