The Cauchy problem on large time for the Water Waves equations with large topography variations

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• 作者：Benoî ; t Mé ; sognon-Gireau ^{benoit.mesognon-gireau@ens.fr}
• 关键词：Water Waves ; Cauchy problem ; Large time ; Bathymetric
• 刊名：Annales de l'Institut Henri Poincare (C) Non Linear Analysis
• 出版年：2017
• 出版时间：January-February 2017
• 年：2017
• 卷：34
• 期：1
• 页码：89-118
• 全文大小：497 K
• 卷排序：34

文摘

This paper shows that the long time existence of solutions to the Water Waves equations remains true with a large topography in presence of surface tension. More precisely, the dimensionless equations depend strongly on three parameters ε,μ,βε,μ,β measuring the amplitude of the waves, the shallowness and the amplitude of the bathymetric variations respectively. In [2], the local existence of solutions to this problem is proved on a time interval of size 1max⁡(β,ε) and uniformly with respect to μ  . In presence of large bathymetric variations (typically β≫εβ≫ε), the existence time is therefore considerably reduced. We remove here this restriction and prove the local existence on a time interval of size 1ε under the constraint that the surface tension parameter must be at the same order as the shallowness parameter μ. We also show that the result of [5] dealing with large bathymetric variations for the Shallow Water equations can be viewed as a particular endpoint case of our result.