Global wellposedness of the 3-D full water wave problem

We consider the problem of global in time existence and uniqueness of solutions of the 3-D infinite depth full water wave problem, in the setting that the interface tends to the horizontal plane, the velocity and acceleration on the interface tend to zero at spatial infinity. We show that the nature of the nonlinearity of the water wave equation is essentially of cubic and higher orders. For any initial interface that is sufficiently small in its steepness and velocity, we show that there exists a unique smooth solution of the full water wave problem for all time, and the solution decays at the rate 1/t.

[1]  S. Klainerman.  Weighted L∞ and L1 estimates for solutions to the classical wave equation in three space dimensions , 1984 .

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

[3]  Nader Masmoudi,et al.  Global Solutions for 3D Quadratic Schrödinger Equations , 2008, 1001.5158.

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

[5]  Sergiu Klainerman,et al.  Uniform decay estimates and the lorentz invariance of the classical wave equation , 1985 .

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

[7]  Jalal Shatah,et al.  Normal forms and quadratic nonlinear Klein‐Gordon equations , 1985 .

[8]  E. Stein.  Singular Integrals and Di?erentiability Properties of Functions , 1971 .

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

[10]  Catherine Sulem,et al.  The nonlinear Schrödinger equation , 2012 .

[11]  Y. Meyer,et al.  La solution des conjectures de Calderón , 1983 .

[12]  Tatsuo Iguchi,et al.  A shallow water approximation for water waves , 2009 .

[13]  C. Sulem,et al.  The nonlinear Schrödinger equation : self-focusing and wave collapse , 2004 .

[14]  David Lannes,et al.  Large time existence for 3D water-waves and asymptotics , 2007, math/0702015.

[15]  Walter A. Strauss,et al.  Nonlinear Wave Equations , 1990 .

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

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

[18]  Sergiu Klainerman,et al.  Global existence of small amplitude solutions to nonlinear klein-gordon equations in four space-time dimensions , 1985 .

[19]  J. Craggs.  Applied Mathematical Sciences , 1973 .

[20]  Walter Craig,et al.  Non–existence of solitary water waves in three dimensions , 2002, Philosophical Transactions of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences.

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

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

[23]  Carlos E. Kenig,et al.  ELLIPTIC BOUNDARY VALUE PROBLEMS ON LIPSCHITZ DOMAINS , 1987 .

[24]  Guido Schneider,et al.  The long‐wave limit for the water wave problem I. The case of zero surface tension , 2000 .

[25]  Christopher D. Sogge,et al.  Lectures on Nonlinear Wave Equations , 2005 .

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

[27]  Thomas Y. Hou,et al.  Growth rates for the linearized motion of fluid interfaces away from equilibrium , 1993 .

[28]  Y. Meyer,et al.  L'integrale de Cauchy Definit un Operateur Borne sur L 2 Pour Les Courbes Lipschitziennes , 1982 .

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

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

[31]  G. Taylor.  The instability of liquid surfaces when accelerated in a direction perpendicular to their planes. I , 1950, Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences.

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

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

[34]  Yi Zhou,et al.  Decay rate of solutions to hyperbolic system of first order , 1999 .

[35]  S. Sun.  Some analytical properties of capillary-gravity waves in two-fluid flows of infinite depth , 1997, Proceedings of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences.

[36]  J. Gilbert,et al.  Clifford Algebras and Dirac Operators in Harmonic Analysis , 1991 .

[37]  Elias M. Stein.  Beijing lectures in harmonic analysis , 1986 .

[38]  Masao Ogawa,et al.  FREE BOUNDARY PROBLEM FOR AN INCOMPRESSIBLE IDEAL FLUID WITH SURFACE TENSION , 2002 .