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A Morawetz Inequality for Gravity-Capillary Water Waves at Low Bond Number

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Abstract

This paper is devoted to the 2D gravity-capillary water waves equations in their Hamiltonian formulation, addressing the general question of proving Morawetz inequalities. We continue the analysis initiated in our previous work, where we have established local energy decay estimates for gravity waves. Here we add surface tension and prove a stronger estimate with a local regularity gain, akin to the smoothing effect for dispersive equations. Our main result holds globally in time and holds for genuinely nonlinear waves, since we are only assuming some very mild uniform Sobolev bounds for the solutions. Furthermore, it is uniform both in the infinite depth limit and the zero surface tension limit.

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Notes

  1. See, e.g., [8, 41].

  2. This should be interpreted as \(\kappa \le c g\) for a small universal constant c,  which in particular does not depend on T and h.

  3. These two operators coincide in the infinite depth setting.

  4. Here for the last term in \({\mathrm{LE}}_\eta \) we also use \(\int \theta _y \theta \mathrm{d}y = \frac{1}{2} \eta ^2\).

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Acknowledgements

T. Alazard was partially supported by the SingFlows project, Grant ANR-18-CE40-0027 of the French National Research Agency (ANR). M. Ifrim was partially was supported by a Luce Assistant Professorship, by the Sloan Foundation, and by an NSF CAREER Grant DMS-1845037. D. Tataru was partially supported by the NSF Grant DMS-1800294 as well as by a Simons Investigator grant from the Simons Foundation. The authors are also very grateful to the anonymous referee for carefully reading the manuscript and for many very helpful questions and comments.

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  1. CNRS and CMLA, École Normale Supérieure de Paris-Saclay, Cachan, France

    Thomas Alazard

  2. Department of Mathematics, University of Wisconsin, Madison, USA

    Mihaela Ifrim

  3. Department of Mathematics, University of California, Berkeley, USA

    Daniel Tataru

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Alazard, T., Ifrim, M. & Tataru, D. A Morawetz Inequality for Gravity-Capillary Water Waves at Low Bond Number. Water Waves 3, 429–472 (2021). https://doi.org/10.1007/s42286-020-00044-8

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