2022/09/20 by Alberto Enciso, Daniel Peralta‐Salas, Enciso, Alberto +3 · 2 citations
Mathematics · #Analysis of PDEs (math.AP) #FOS: Mathematics #Navier-Stokes equation solutions
paper · pdf · doi:10.48550/arxiv.2209.09812
openalex publication_date 2022/09/20 · openalex created_date 2022/09/23 · openalex updated_date 2026/07/28
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.