We show existence of global solutions for the gravity water waves equation in dimension 3, in the case of small data. The proof combines energy estimates, which yield control of L 2 related norms, with dispersive estimates, which give decay in L ∞ . To obtain these dispersive estimates, we use an analysis in Fourier space; the study of space and time resonances is then the crucial point.
We study Sobolev regularity disturbances to the periodic, plane Couette flow in the 3D incompressible Navier-Stokes equations at high Reynolds number Re. Our goal is to estimate how the stability threshold scales in Re: the largest the initial perturbation can be while still resulting in a solution that does not transition away from Couette flow. In this work we prove that initial data which satisfies u in H σ ≤ δRe −3/2 for any σ > 9/2 and some δ = δ(σ) > 0 depending only on σ, is global in time, remains within O(Re −1/2 ) of the Couette flow in L 2 for all time, and converges to the class of "2.5 dimensional" streamwise-independent solutions referred to as streaks for times t Re 1/3 . Numerical experiments performed by Reddy et. al. with "rough" initial data estimated a threshold of ∼ Re −31/20 , which shows very close agreement with our estimate.
Abstract. We show existence of global solutions for the gravity water waves equation in dimension 3, in the case of small data. The proof combines energy estimates, which yield control of L 2 related norms, with dispersive estimates, which give decay in L ∞ . To obtain these dispersive estimates, we use an analysis in Fourier space; the study of space and time resonances is then the crucial point.
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