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On the wave turbulence theory of 2D gravity waves, II: propagation of randomness
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This is the second part of our work initiating the rigorous study of wave turbulence for water waves equations. We combine energy estimates, normal forms, and probabilistic and combinatorial arguments to complete the construction of long-time solutions with random initial data for the 2d (1d interface) gravity water waves system on large tori. This is the first long-time regularity result for solutions of water waves systems with large energy (but small local energy), which is the correct setup for applications to wave turbulence. Such a result is only possible in the presence of randomness.
Forward citations
Cited by 3 Pith papers
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Transfer of energy for pure-gravity water waves with constant vorticity
Smooth small solutions to gravity water waves with constant vorticity show arbitrary growth in high Sobolev norms, proving energy transfer to high frequencies and weak turbulence while the flow remains smooth.
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Rogue waves and large deviations for 2D pure gravity deep water waves
For the 2D pure-gravity deep-water water-wave equation, a crest of size λ0 ε^{1-δ} appears with probability exp(-λ0^2 ε^{-2δ}/(2σ^2)) up to the optimal nonlinear time scale.
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Rigorous Derivation of the Wave Kinetic Equation for $\beta$-FPUT System
For a reduced beta-FPUT model, the wave kinetic equation is rigorously derived under the scaling beta=N^{-gamma} up to times of order N^{-epsilon} min(N, N^{5gamma/4}).
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