On the 3-dimensional water waves system above a flat bottom

Wang, Xuecheng

doi:10.2140/apde.2017.10.893

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On the 3-dimensional water waves system above a flat bottom

Xuecheng Wang

Vol. 10 (2017), No. 4, 893–928

DOI: 10.2140/apde.2017.10.893

Abstract

As a starting point for studying the long-time behavior of the 3-dimensional water waves system in the flat bottom setting, we try to improve the understanding of the Dirichlet–Neumann operator in this set-up. As an application, we study the 3-dimensional gravity waves system and derive a new energy estimate of $L^{2} - L^{\infty}$ type, which has good structure in the $L^{\infty}$ -type space. This has been used in our Ph.D. thesis (2016) to prove the global regularity of the 3-dimensional gravity waves system for suitably small initial data.

Keywords

3-dimensional water waves, finite depth, flat bottom, new energy estimate

Mathematical Subject Classification 2010

Primary: 35Q35, 76B15

Milestones

Received: 18 February 2016

Revised: 16 January 2017

Accepted: 25 February 2017

Published: 9 May 2017

Authors

Xuecheng Wang
Mathematics Department Princeton University Princeton, NJ 08544 United States

Vol. 10, No. 4, 2017