Two Dimensional Water Waves in Holomorphic Coordinates
By J. K. Hunter, Mihaela Ifrim, D. Tataru. Communications in Mathematical Physics, 2014
180 citations recorded by the source corpus.
This article is concerned with the infinite depth water wave equation in two space dimensions. We consider this problem expressed in position-velocity potential holomorphic coordinates. Viewing this problem as a quasilinear dispersive equation, we establish two results: (i) local well-posedness in Sobolev spaces, and (ii) almost global solutions for small localized data. Neither of these results are new; they have been recently obtained by Alazard–Burq–Zuily (Invent Math 198:71–163, 2014), respectively by Wu (Invent Math 177:45–135, 2009) using different coordinates and methods. Instead our goal is to improve the understanding of this problem by providing a single setting for both problems, by proving sharper versions of the above results, as well as presenting new, simpler proofs. This article is self contained.
Subfields: Mathematical Physics, Oceanography, Statistical and Nonlinear Physics
View Paper: https://www.semanticscholar.org/paper/28b8058e7d2589cc5eee1ab4fbcab8548871001e
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Citations in the corpus
- A Paradifferential Approach for Well-Posedness of the Muskat Problem (2019, background): “This seems a rather formidable problem for water-waves since the mechanism of dispersion is harder to properly pin down in the quasilinear case (see [1, 2, 3, 4, 32, 46, 54, 61, 62]), but becomes much more tractable in the case of the Muskat problem due to its parabolicity.”