On the Cauchy problem for gravity water waves

We are interested in the system of gravity water waves equations without surface tension. Our purpose is to study the optimal regularity thresholds for the initial conditions. In terms of Sobolev embeddings, the initial surfaces we consider turn out to be only of $$C^{3/2+\epsilon }$$C3/2+ϵ-class for some $$\epsilon >0$$ϵ>0 and consequently have unbounded curvature, while the initial velocities are only Lipschitz. We reduce the system using a paradifferential approach.

[1]  S. Alinhac,et al.  Existence d'ondes de rarefaction pour des systems quasi‐lineaires hyperboliques multidimensionnels , 1989 .

[2]  Walter Craig,et al.  An existence theory for water waves and the boussinesq and korteweg-devries scaling limits , 1985 .

[3]  J. Chemin.  Calcul paradifférentiel précisé et applications à des équations aux dérivées partielles non semilinéaires , 1988 .

[4]  David Lannes,et al.  A Stability Criterion for Two-Fluid Interfaces and Applications , 2010, 1005.4565.

[5]  F. Rousset,et al.  Uniform Regularity and Vanishing Viscosity Limit for the Free Surface Navier–Stokes Equations , 2012, 1202.0657.

[6]  Sijue Wu.  On a class of self-similar 2D surface water waves , 2012, 1206.2208.

[7]  Daniel Tataru,et al.  Strichartz estimates for operators with nonsmooth coefficients and the nonlinear wave equation , 2000 .

[8]  Nader Masmoudi,et al.  Global Existence for Capillary Water Waves , 2012, 1210.1601.

[9]  S. Alinhac.  Paracomposition et operateurs paradifferentiels , 1986 .

[10]  Jean-Yves Chemin,et al.  Perfect Incompressible Fluids , 1998 .

[11]  Yan Guo,et al.  Local well-posedness of the viscous surface wave problem without surface tension , 2013 .

[12]  J. Boussinesq,et al.  Sur une importante simplification de la théorie des ondes que produisent, à la surface d'un liquide, l'émersion d'un solide ou l'impulsion d'un coup de vent , 1910 .

[13]  Klaus Beyer,et al.  On the Cauchy problem for a capillary drop. Part I: irrotational motion , 1998 .

[14]  Walter Craig,et al.  Numerical simulation of gravity waves , 1993 .

[15]  К Е Вейн,et al.  Математические аспекты поверхностных волн на воде@@@Mathematical aspects of surface water waves , 2007 .

[16]  G. Lebeau.  Régularité du problème de Kelvin–Helmholtz pour l’équation d’Euler 2D , 2001 .

[17]  C. Fefferman,et al.  Finite time singularities for the free boundary incompressible Euler equations , 2011, 1112.2170.

[18]  Sijue Wu,et al.  Global wellposedness of the 3-D full water wave problem , 2011 .

[19]  D. Christodoulou,et al.  S E M I N A I R E E quations aux , 2008 .

[20]  C. Bardos,et al.  Mathematics for 2d Interfaces , 2010, 1005.5329.

[21]  Walter Craig,et al.  Traveling Two and Three Dimensional Capillary Gravity Water Waves , 2000, SIAM J. Math. Anal..

[22]  Hideaki Yosihara,et al.  Gravity Waves on the Free Surface of an Incompressible Perfect Fluid of Finite Depth , 1982 .

[23]  R. Danchin,et al.  Fourier Analysis and Nonlinear Partial Differential Equations , 2011 .

[24]  Sijue Wu,et al.  Well-posedness in Sobolev spaces of the full water wave problem in 3-D , 1999 .

[25]  David Lannes,et al.  The Water Waves Problem: Mathematical Analysis and Asymptotics , 2013 .

[26]  Gerd Grubb,et al.  PROBLÉMES AUX LIMITES NON HOMOGÉNES ET APPLICATIONS , 1969 .

[27]  P. Secchi,et al.  A priori Estimates for 3D Incompressible Current-Vortex Sheets , 2011, 1102.2763.

[28]  Hans Lindblad.  Well-posedness for the motion of an incompressible liquid with free surface boundary , 2005 .

[29]  Jalal Shatah,et al.  Geometry and a priori estimates for free boundary problems of the Euler's equation , 2006 .

[30]  Daniel Coutand,et al.  Well-posedness of the free-surface incompressible Euler equations with or without surface tension , 2005 .

[31]  M. Safonov.  Boundary estimates for positive solutions to second order elliptic equations , 2008, 0810.0522.

[32]  J. Reeder,et al.  Three-dimensional, nonlinear wave interaction in water of constant depth☆ , 1981 .

[33]  M. Sablé-Tougeron,et al.  Régularité microlocale pour des problèmes aux limites non linéaires , 1984 .

[34]  J. Shatah,et al.  Local Well-Posedness for Fluid Interface Problems , 2011 .

[35]  J. Chemin,et al.  Équations d'ondes quasilinéaires et estimations de Strichartz , 1999 .

[36]  Guy Metivier,et al.  Para-Differential Calculus and Applications to the Cauchy Problem for Nonlinear Systems , 2008 .

[37]  Sijue Wu,et al.  Almost global wellposedness of the 2-D full water wave problem , 2009, 0910.2473.

[38]  Vladimir E. Zakharov,et al.  Stability of periodic waves of finite amplitude on the surface of a deep fluid , 1968 .

[39]  T. Alazard,et al.  The Water-Wave Equations: From Zakharov to Euler , 2012, 1212.0632.

[40]  J. Bony,et al.  Calcul symbolique et propagation des singularités pour les équations aux dérivées partielles non linéaires , 1980 .

[41]  W. Craig,et al.  Mathematical aspects of surface water waves , 2007 .

[42]  Nader Masmoudi,et al.  Global solutions for the gravity water waves equation in dimension 3 , 2009 .

[43]  S. Shkoller,et al.  On the Finite-Time Splash and Splat Singularities for the 3-D Free-Surface Euler Equations , 2012, 1201.4919.

[44]  J. Marzuola,et al.  On the regularity of the flow map for the gravity-capillary equations , 2011, 1111.5361.

[45]  S. Alinhac,et al.  Interaction d'ondes simples pour des équations complètement non linéaires , 1988 .

[46]  Thomas Alazard,et al.  On the water-wave equations with surface tension , 2009, 0906.4406.

[47]  T. Alazard,et al.  STRICHARTZ ESTIMATES FOR WATER WAVES by T. Alazard, N. Burq & C. Zuily , 2010 .

[48]  Sijue Wu,et al.  Well-posedness in Sobolev spaces of the full water wave problem in 2-D , 1997 .

[49]  T. Alazard,et al.  Paralinearization of the Dirichlet to Neumann Operator, and Regularity of Three-Dimensional Water Waves , 2009, 0901.2888.

[50]  Nader Masmoudi,et al.  The zero surface tension limit two‐dimensional water waves , 2005 .

[51]  C. Fefferman,et al.  Turning waves and breakdown for incompressible flows , 2010, Proceedings of the National Academy of Sciences.

[52]  Walter Craig,et al.  The modulational regime of three-dimensional water waves and the Davey-Stewartson system , 1997 .

[53]  L. Hörmander,et al.  Lectures on Nonlinear Hyperbolic Differential Equations , 1997 .

[54]  Ping Zhang,et al.  On the free boundary problem of three‐dimensional incompressible Euler equations , 2008 .

[55]  A. Córdoba,et al.  The Rayleigh-Taylor condition for the evolution of irrotational fluid interfaces , 2009, Proceedings of the National Academy of Sciences.

[56]  L. N. Sretenskii.  A Cauchy-Poisson problem , 1971 .

[57]  A priori estimates for fluid interface problems , 2006, math/0609542.

[58]  David Lannes,et al.  Well-posedness of the water-waves equations , 2005 .