Quasi-periodic solutions to the incompressible Euler equations in dimensions two and higher
Alberto Enciso, Daniel Peralta‐Salas, Francisco Torres de Lizaur
Abstract
Building on the work of Crouseilles and Faou on the 2D case, we construct C∞ 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⩾1 we prove that any Lq initial stream function can be approximated in Lq (strongly when 1⩽q<∞ and weak-⁎ when q=∞) by smooth initial data whose solutions are dense on N-dimensional tori.