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Low Regularity Solutions for Gravity Water Waves

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Abstract

We prove local well-posedness for the gravity water waves equations without surface tension, with initial velocity field in \(H^s\), \(s > \frac{d}{2} + 1 - \mu \), where \(\mu = \frac{1}{10}\) in the case \(d = 1\) and \(\mu = \frac{1}{5}\) in the case \(d \ge 2\), extending previous results of Alazard–Burq–Zuily. The improvement primarily arises in two areas. First, we perform an improved analysis of the regularity of the change of variables from Eulerian to Lagrangian coordinates. Second, we perform a time-interval length optimization of the localized Strichartz estimates.

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Acknowledgments

The author would like to thank his advisor, Daniel Tataru, for introducing him to this research area and for many helpful discussions.

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Authors and Affiliations

  1. University of California, Berkeley, Berkeley, USA

    Albert Ai

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  1. Albert Ai
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Correspondence to Albert Ai.

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The author was supported by the Department of Defense (DoD) through the National Defense Science and Engineering Graduate Fellowship (NDSEG) Program, as well as by the Simons Foundation.

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Ai, A. Low Regularity Solutions for Gravity Water Waves. Water Waves 1, 145–215 (2019). https://doi.org/10.1007/s42286-019-00002-z

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