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

In a previous work (arXiv:2306.05948), we constructed by convex integration examples of energy dissipating solutions to the 2D Euler equations on R² with vorticity in the real Hardy space Hᵖ(R²). In the present paper, we develop tools that significantly improve that result in two ways: Firstly, we achieve vorticities in Hᵖ(R²) in the optimal range p∈ (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.