Perfect fluid flows on $\R^d$ with growth/decay conditions at infinity

TL;DR

This paper investigates the well-posedness and asymptotic behavior of perfect fluid flows on , demonstrating that solutions develop analytic asymptotic expansions at infinity influenced by initial conditions.

Contribution

It introduces a framework for analyzing Euler equations with initial data in weighted Sobolev spaces allowing growth/decay at infinity and characterizes the asymptotic expansions of solutions.

Findings

Solutions develop non-vanishing asymptotic terms at infinity.

Asymptotic terms depend analytically on time and initial data.

Identifies evolution space for initial data in Schwartz class.

Abstract

We study the well-posedness and the spatial behavior at infinity of perfect fluid flows on $\R^d$ with initial data in a scale of weighted Sobolev spaces that allow spatial growth/decay at infinity as $|x|^\beta$ with $\beta<1/2$. In particular, we show that the solution of the Euler equation generically develops an asymptotic expansion at infinity with non-vanishing asymptotic terms that depend analytically on time and the initial data. We identify the evolution space for initial data in the Schwartz class with a certain space of symbols.