TY - JOUR

T1 - Global solutions of the gravity-capillary water-wave system in three dimensions

AU - Deng, Yu

AU - Ionescu, Alexandru D.

AU - Pausader, Benoît

AU - Pusateri, Fabio

N1 - Funding Information:
Y. Deng was supported in part by a Jacobus Fellowship from Princeton University. A. D. Ionescu was supported in part by NSF grant DMS-1265818 and NSF-FRG grant DMS-1463753. B. Pausader was supported in part by NSF grant DMS-1362940, and a Sloan fellowship. F. Pusateri was supported in part by NSF grant DMS-1265875.

PY - 2017/12

Y1 - 2017/12

N2 - In this paper we prove global regularity for the full water-wave system in three dimensions for small data, under the influence of both gravity and surface tension. This problem presents essential difficulties which were absent in all of the earlier global regularity results for other water-wave models. To construct global solutions, we use a combination of energy estimates and matching dispersive estimates. There is a significant new difficulty in proving energy estimates in our problem, namely the combination of slow pointwise decay of solutions (no better than (Formula Presented)) and the presence of a large, codimension-1, set of quadratic time-resonances. To deal with such a situation, we propose here a new mechanism, which exploits a non-degeneracy property of the time-resonant hypersurfaces and some special structure of the quadratic part of the non-linearity, connected to the conserved energy of the system. The dispersive estimates rely on analysis of the Duhamel formula in the Fourier space. The main contributions come from the set of space-time resonances, which is a large set of dimension 1. To control the corresponding bilinear interactions, we use harmonic analysis techniques, such as orthogonality arguments in the Fourier space and atomic decompositions of functions. Most importantly, we construct and use a refined norm which is well adapted to the geometry of the problem.

AB - In this paper we prove global regularity for the full water-wave system in three dimensions for small data, under the influence of both gravity and surface tension. This problem presents essential difficulties which were absent in all of the earlier global regularity results for other water-wave models. To construct global solutions, we use a combination of energy estimates and matching dispersive estimates. There is a significant new difficulty in proving energy estimates in our problem, namely the combination of slow pointwise decay of solutions (no better than (Formula Presented)) and the presence of a large, codimension-1, set of quadratic time-resonances. To deal with such a situation, we propose here a new mechanism, which exploits a non-degeneracy property of the time-resonant hypersurfaces and some special structure of the quadratic part of the non-linearity, connected to the conserved energy of the system. The dispersive estimates rely on analysis of the Duhamel formula in the Fourier space. The main contributions come from the set of space-time resonances, which is a large set of dimension 1. To control the corresponding bilinear interactions, we use harmonic analysis techniques, such as orthogonality arguments in the Fourier space and atomic decompositions of functions. Most importantly, we construct and use a refined norm which is well adapted to the geometry of the problem.

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U2 - 10.4310/ACTA.2017.v219.n2.a1

DO - 10.4310/ACTA.2017.v219.n2.a1

M3 - Article

AN - SCOPUS:85046663702

VL - 219

SP - 213

EP - 402

JO - Acta Mathematica

JF - Acta Mathematica

SN - 0001-5962

IS - 2

ER -