On the asymptotic behavior of solutions of the 2d Euler equation

TL;DR

This paper proves global well-posedness of the 2d Euler equation for vector fields with specified asymptotic expansions at infinity, detailing the evolution and properties of these asymptotic terms over time.

Contribution

It establishes the global well-posedness of the 2d Euler equation in spaces with prescribed asymptotic expansions, including the holomorphic evolution of asymptotic coefficients.

Findings

Solutions have holomorphic asymptotic coefficients in time.

Asymptotic terms evolve without involving logarithmic factors.

Results hold even for initial data with rapid decay at infinity.

Abstract

We prove that the 2d Euler equation is globally well-posed in a space of vector fields having spatial asymptotic expansion at infinity of any a priori given order. The asymptotic coefficients of the solutions are holomorphic functions of $t$, do not involve (spacial) logarithmic terms, and develop even when the initial data has fast decay at infinity. We discuss the evolution in time of the asymptotic terms and their approximation properties.