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Propagation of singularities for gravity-capillary water waves

Hui Zhu

Vol. 17 (2024), No. 1, 281–344
Abstract

We obtain two results of propagation for the gravity-capillary water wave system. The first result shows the propagation of oscillations and the spatial decay at infinity; the second result shows a microlocal smoothing effect under the nontrapping condition of the initial free surface. These results extend the works of Craig, Kappeler and Strauss (1995), Wunsch (1999) and Nakamura (2005) to quasilinear dispersive equations. These propagation results are stated for water waves with asymptotically flat free surfaces, of which we also obtain the existence. To prove these results, we generalize the paradifferential calculus of Bony (1979) to weighted Sobolev spaces and develop a semiclassical paradifferential calculus. We also introduce the quasihomogeneous wavefront sets which characterize, in a general manner, the oscillations and the spatial growth/decay of distributions.

Keywords
water wave, smoothing effect, propagation of singularity, wavefront set
Mathematical Subject Classification
Primary: 35A01, 35A18, 35A21, 35S50, 76B15
Milestones
Received: 13 December 2021
Accepted: 16 June 2022
Published: 5 February 2024
Authors
Hui Zhu
Department of Mathematics
University of Michigan
Ann Arbor, MI
United States
Laboratoire de Mathématiques d’Orsay
Université Paris-Sud, CNRS
Université Paris-Saclay
Orsay
France

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