Cauchy theory for the gravity water waves system with non localized initial data

2013/05/02 by Alazard, Thomas, Burq, Nicolas, Zuily, Claude · 1 citation
#Analysis of PDEs (math.AP) #FOS: Mathematics

paper · doi:10.48550/arxiv.1305.0457

Abstract

In this article, we develop the local Cauchy theory for the gravity water waves system, for rough initial data which do not decay at infinity. We work in the context of L²-based uniformly local Sobolev spaces introduced by Kato. We prove a classical well-posedness result (without loss of derivatives). Our result implies also a local well-posedness result in Hölder spaces (with loss of d/2 derivatives). As an illustration, we solve a question raised by Boussinesq on the water waves problem in a canal. We take benefit of an elementary observation to show that the strategy suggested by Boussinesq does indeed apply to this setting.

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• Two-dimensional water waves with constant vorticity and general bottom topography

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