Spatially quasi-periodic solutions of the Euler equation

TL;DR

This paper establishes a framework for quasi-periodic solutions of the Euler equation, proving local well-posedness and preservation of quasi-periodicity in the initial data within a new functional setting.

Contribution

It introduces a novel approach to analyze quasi-periodic solutions of the Euler equation and proves local well-posedness in this context.

Findings

Euler equation preserves quasi-periodicity of initial data

Solutions depend analytically on time and initial conditions

Framework applicable to quasi-periodic maps and diffeomorphisms

Abstract

We develop a framework for studying quasi-periodic maps and diffeomorphisms on $\mathbb{R}^n$. As an application, we prove that the Euler equation is locally well posed in a space of quasi-periodic vector fields on $\mathbb{R}^n$. In particular, the equation preserves the spatial quasi-periodicity of the initial data. Several results on the analytic dependence of solutions on the time and the initial data are proved.