Quasi-periodic solutions of the 2D Euler equation

TL;DR

This paper constructs quasi-periodic solutions to the 2D Euler equation featuring localized traveling profiles over stationary states, revealing new dynamic behaviors in fluid flow models.

Contribution

It introduces a method to explicitly construct quasi-periodic solutions with localized traveling profiles in the 2D Euler equation, expanding understanding of fluid dynamics solutions.

Findings

Existence of quasi-periodic solutions with localized traveling profiles

Solutions propagate orthogonally to stationary state variable

Frequencies determined by locally constant velocities

Abstract

We consider the two-dimensional Euler equation with periodic boundary conditions. We construct time quasi-periodic solutions of this equation made of localized travelling profiles with compact support propagating over a stationary state depending on only one variable. The direction of propagation is orthogonal to this variable, and the support is concentrated on flat strips of the stationary state. The frequencies of the solution are given by the locally constant velocities associated with the stationary state.