Almost global wellposedness of the 2-D full water wave problem

By Sijue Wu. 2009

249 citations recorded by the source corpus.

We consider the problem of global in time existence and uniqueness of solutions of the 2-D infinite depth full water wave equation. It is known that this equation has a solution for a time period [0,T/ε] for initial data of the form εΨ, where T depends only on Ψ. In this paper, we show that for such data there exists a unique solution for a time period [0,eT/ε]. This is achieved by better understandings of the nature of the nonlinearity of the full water wave equation.

Subfields: Applied Mathematics, Mathematical Physics, Oceanography

View Paper: https://www.semanticscholar.org/paper/611796e064a88093f0933de8b9cf74b0ca5b2ae2

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In Applied Mathematics

In Mathematical Physics

In Oceanography

Citations in the corpus

Nonlinear fractional Schrödinger equations in one dimension (2012, methodology): “This is consistent with the almost global existence result of Wu [28], in the case of the irrotational t wo-dimensional water wave problem.”
Global solutions for the gravity water waves system in 2d (2013, methodology): “The following local existence result holds: Proposition 2.1(Local existence in Lagrangian coordinates [Wu09, Theorem5.”
Two dimensional water waves in holomorphic coordinates II: global solutions (2014, unknown): “(iii) almost global well-posedness for small localized data, refining and simplifying Wu’s approach in [24].”
Local well-posedness of the viscous surface wave problem without surface tension (2013, methodology): “Local well-posedness in this framework was established by Wu [31, 32] and Lannes [22], an almost-global well-posedness result was then proved by Wu [33] for the 2D problem, and global well-posedness was proved by Wu [34] and Germain-Masmoudi-Shatah [17] in 3D.”
A geometric approach to generalized Stokes conjectures (2009, background): “While recent years have seen great progress in the study of the initial-value problem (see [40] for large-time well-posedness for small data, and the references therein for short-time well-posedness for arbitrary data), in the present paper we confine ourselves to traveling-wave solutions of the abo”
Global wellposedness of the 3-D full water wave problem (2011, methodology): “6Formulas for b and A similar to those in 2D [32] are also available and can be obtained in a similar way: (I − H)b = (I − H)([∂t + b ·∇⊥, K]ze3 − [∂t + b ·∇⊥, H]ze3 + Ku3e3), (I − H)(Ae3) = e3 + [∂t + b ·∇⊥, H]u + [AN ×∇, H]λ∗ + (I − H)(−Aζβ × (∂αKze3) + Aζα × (∂β Kze3) + A∂αλ× ∂βλ).”
Finite time singularities for the free boundary incompressible Euler equations (2011, background): “By taking advantage of the dispersive properties of the water-wave system, Wu [32] proved exponentially large time of existence for small initial data.”
Global solutions and asymptotic behavior for two dimensional gravity water waves (2013, background): “In her breakthrough result [65], Wu proved that the maximal time of existence Tε is larger or equal to ec/ε for d = 1.”