Compactly supported anomalous weak solutions for 2D Euler equations with vorticity in Hardy spaces

TL;DR

This paper advances the construction of compactly supported weak solutions to the 2D Euler equations with vorticity in Hardy spaces, extending results to the optimal range of p and preserving key physical quantities.

Contribution

It improves previous convex integration methods to produce compactly supported solutions with vorticity in the optimal Hardy space range, maintaining linear and angular momentum.

Findings

Vorticities in H^p for p in (0,1) achieved.

Solutions have compact support and preserve physical quantities.

Enhanced tools for constructing weak solutions to 2D Euler equations.

Abstract

In a previous work (arXiv:2306.05948), we constructed by convex integration examples of energy dissipating solutions to the 2D Euler equations on $\mathbb{R}^2$ with vorticity in the real Hardy space $H^p(\mathbb{R}^2)$. In the present paper, we develop tools that significantly improve that result in two ways: Firstly, we achieve vorticities in $H^p(\mathbb{R}^2)$ in the optimal range $p\in (0,1)$ compared to $(2/3,1)$ in our previous work. Secondly, the solutions constructed here possess compact support and in particular preserve linear and angular momenta.