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Water Waves

, Volume 1, Issue 1, pp 145–215 | Cite as

Low Regularity Solutions for Gravity Water Waves

  • Albert AiEmail author
Original Article
  • 46 Downloads

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.

Keywords

Gravity water waves Strichartz estimates Local smoothing estimates Low regularity solutions Wave packet parametrix 

Notes

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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Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.University of California, BerkeleyBerkeleyUSA

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