Long-term regularity of 3D gravity water waves

Abstract

We study a fundamental model in fluid mechanics--the 3D gravity water wave equation, in which an incompressible fluid occupying half the 3D space flows under its own gravity. In this paper we show long-term regularity of solutions whose initial data is small but not localized. Our results include: almost global wellposedness for unweighted Sobolev initial data and global wellposedness for weighted Sobolev initial data with weight $|x{|}^{\alpha }$, for any $\alpha >0$. In the periodic case, if the initial data lives on an $R$ by $R$ torus, and $\epsilon$ close to the constant solution, then the life span of the solution is at least $R/({\epsilon }^2(\log R{)}^2)$.

arXiv comment88 pages

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Cited by in corpus (4)

• The quartic integrability and long time existence of steep water waves in 2D
• On the transition of the Rayleigh-Taylor instability in 2d water waves
• Nonlinear modulational instabililty of the Stokes waves in 2d full water waves
• Cancellations of Resonances and Long Time Dynamics of Cubic Schrödinger Equation on $\mathbb{T}$