Quasi-periodic solutions to the incompressible Euler equations in dimensions two and higher

TL;DR

This paper constructs smooth, high-dimensional quasi-periodic solutions to the incompressible Euler equations in multiple dimensions, demonstrating their density in phase space and extending previous 2D results to higher dimensions.

Contribution

It introduces the first known high-dimensional quasi-periodic solutions to the Euler equations, extending 2D results to 3D and higher, and shows their density in the phase space.

Findings

Existence of smooth, high-dimensional quasi-periodic solutions in 3D and higher.

Density of these solutions in the phase space of the Euler equations.

Approximation of initial data by solutions dense on tori of arbitrary dimension.

Abstract

Building on the work of Crouseilles and Faou on the 2D case, we construct $C^\infty$ quasi-periodic solutions to the incompressible Euler equations with periodic boundary conditions in dimension 3 and in any even dimension. These solutions are genuinely high-dimensional, which is particularly interesting because there are extremely few examples of high-dimensional initial data for which global solutions are known to exist. These quasi-periodic solutions can be engineered so that they are dense on tori of arbitrary dimension embedded in the space of solenoidal vector fields. Furthermore, in the two-dimensional case we show that quasi-periodic solutions are dense in the phase space of the Euler equations. More precisely, for any integer $N\geq 1$ we prove that any $L^q$ initial stream function can be approximated in $L^q$ (strongly when $1\leq q< \infty$ and weak-* when $q=\infty$) by smooth initial data whose solutions are dense on $N$-dimensional tori.