Unbounded Yudovich Solutions of the Euler Equations

TL;DR

This paper establishes the global existence and uniqueness of unbounded solutions to the 2D incompressible Euler equations with specific growth conditions, expanding the understanding of solutions beyond traditional finite energy frameworks.

Contribution

It introduces a novel approach to prove existence and uniqueness of solutions with unbounded initial data, using pressure decomposition and local energy estimates.

Findings

Proved global existence for initial data with square-root growth.

Established uniqueness via adapted Yudovich's argument.

Demonstrated continuity of the initial data to solution map.

Abstract

In this article, we will study unbounded solutions of the 2D incompressible Euler equations. One of the motivating factors for this is that the usual functional framework for the Euler equations (e.g. based on finite energy conditions, such as $L^2$) does not respect some of the symmetries of the problem, such as Galileo invariance. Our main result, global existence and uniqueness of solutions for initial data with square-root growth $O(|x|^{\frac{1}{2} - \epsilon})$ and bounded vorticity, is based on two key ingredients. Firstly an integral decomposition of the pressure, and secondly examining local energy balance leading to solution estimates in local Morrey type spaces. We also prove continuity of the initial data to solution map by a substantial adaptation of Yudovich's uniqueness argument.