Global $\dot H^1 \cap \dot H^{-1}$ solutions to a logarithmically regularized 2D Euler equation

TL;DR

This paper constructs global solutions in specific Sobolev spaces for a logarithmically regularized 2D Euler equation, introducing a new interpolation inequality that leverages vorticity conservation.

Contribution

The paper presents a novel approach to solving a modified 2D Euler equation by developing a new logarithm interpolation inequality and establishing global solutions in $ ext{dot} H^1 igcap ext{dot} H^{-1}$ spaces.

Findings

Established global solutions in $ ext{dot} H^1 igcap ext{dot} H^{-1}$ spaces.

Developed a new logarithm interpolation inequality.

Utilized vorticity conservation to achieve results.

Abstract

We construct global $\dot H^1\cap \dot H^{-1}$ solutions to a logarithmically modified 2D Euler vorticity equation. Our main tool is a new logarithm interpolation inequality which exploits the $L^{\infty-}$-conservation of the vorticity.