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On the Water Waves Equations with Surface Tension
TL;DR: In this article, the smoothing effect for the surface tension water waves proved by H. Christianson, V. Hur, and G. Staffilani is shown to be a direct consequence of this reduction, which allows also to lower the regularity indexes of the initial data, and to obtain the natural weights in the estimates.
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Abstract: The purpose of this article is to clarify the Cauchy theory of the water waves equations as well in terms of regularity indexes for the initial conditions as for the smoothness of the bottom of the domain (namely no regularity assumption is assumed on the bottom). Our main result is that, following the approach developped by T. Alazard and G. M\'etivier in [1], after suitable paralinearizations, the system can be arranged into an explicit symmetric system of Schr\"odinger type. We then show that the smoothing effect for the (one dimensional) surface tension water waves proved by H. Christianson, V. M. Hur, and G. Staffilani in [9], is in fact a rather direct consequence of this reduction, which allows also to lower the regularity indexes of the initial data, and to obtain the natural weights in the estimates.
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A Long Wave Approximation for Capillary-Gravity Waves and an Effect of the Bottom
TL;DR: In this paper, the authors consider the case where the bottom is not flat and study an effect of the bottom to the long wave approximation, and derive a system of coupled KdV like equations and prove that the dynamics of the full problem can be described approximately by the solution of the coupled equations for a long time interval.