Constructs small-amplitude traveling quasi-periodic electron-layers in 1D space-periodic Vlasov-Poisson equations near symmetric flat velocity strips for most strip areas via Nash-Moser and reducibility methods.
Time Quasi-Periodic Three-dimensional Traveling Gravity Water Waves
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abstract
Starting with the pioneering computations of Stokes in 1847, the search of traveling waves in fluid mechanics has always been a fundamental topic, since they can be seen as building blocks to determine the long time dynamics (which is a widely open problem). In this paper we prove the existence of time quasi-periodic traveling wave solutions for three-dimensional pure gravity water waves in finite depth, on flat tori, with an arbitrary number of speeds of propagation. These solutions are global in time, they do not reduce to stationary solutions in any moving reference frame and they are approximately given by finite sums of Stokes waves traveling with rationally independent speeds of propagation. This is a very hard small divisors problem for Partial Differential Equations due to the fact that one deals with a dispersive quasi-linear PDE in higher dimension with a very complicated geometry of the resonances. Our result is the first KAM (Kolmogorov-Arnold-Moser) result for an autonomous, dispersive, quasi-linear PDE in dimension greater than one and it is the first example of global solutions, which do not reduce to steady ones in any moving reference frame, for 3D water waves equations on compact domains.
fields
math.AP 5years
2026 5representative citing papers
Gravity water waves with constant nonzero vorticity γ and rational γ²/g admit smooth small-amplitude solutions whose high Sobolev norms grow arbitrarily large while low norms stay small.
Multiple distinct 3D Stokes waves bifurcate from non-resonant 2D ones via equivariant critical point theory on a symmetry-invariant space.
Non-rigid time-periodic vortex patch solutions bifurcating from translating symmetric dipoles are constructed for the 2D Euler equations using Lyapunov-Schmidt reduction and Nash-Moser methods.
Initial data close to large quasi-periodic traveling waves in the β-plane equation remain close for arbitrary long times independent of wave size, yielding almost global existence for open sets of large initial data.
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Long time dynamics close to large amplitude quasi-periodic traveling waves in two dimensional forced rotating fluids
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