Spatial asymptotic expansions in the incompressible Euler equation

TL;DR

This paper establishes the well-posedness of the incompressible Euler equation for solutions with specified spatial asymptotic expansions, including log terms, and explores resulting conservation laws.

Contribution

It introduces a framework for analyzing Euler solutions with prescribed asymptotic expansions at infinity, extending classical well-posedness results.

Findings

Solutions develop non-trivial asymptotic expansions

Conservation laws associated with asymptotic terms

Framework accommodates log terms in expansions

Abstract

In this paper we prove that the Euler equation describing the motion of an ideal fluid in $\R^d$ is well-posed in a class of functions allowing spatial asymptotic expansions as $|x|\to\infty$ of any a priori given order. These asymptotic expansions can involve log terms and lead to a family of conservation laws. Typically, the solutions of the Euler equation with initial data in the Schwartz class develop non-trivial spatial asymptotic expansions of the type considered here.