#### Vol. 10, No. 4, 2017

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

### Xuecheng Wang

Vol. 10 (2017), No. 4, 893–928
##### 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