The NLS approximation for two dimensional deep gravity waves Dedicated to Professor Jean-Yves Chemin on the Occasion of His 60th Birthday
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摘要
This article is concerned with in?nite depth gravity water waves in two space dimensions. We consider this system expressed in position-velocity potential holomorphic coordinates. Our goal is to study this problem with small wave packet data, and to show that this is well approximated by the cubic nonlinear Schr¨odinger equation(NLS) on the natural cubic time scale.
This article is concerned with in?nite depth gravity water waves in two space dimensions. We consider this system expressed in position-velocity potential holomorphic coordinates. Our goal is to study this problem with small wave packet data, and to show that this is well approximated by the cubic nonlinear Schr¨odinger equation(NLS) on the natural cubic time scale.
This article is concerned with in?nite depth gravity water waves in two space dimensions. We consider this system expressed in position-velocity potential holomorphic coordinates. Our goal is to study this problem with small wave packet data, and to show that this is well approximated by the cubic nonlinear Schr¨odinger equation(NLS) on the natural cubic time scale.
引文
1 Alvarez-Samaniego B,Lannes D.Large time existence for 3D water-waves and asymptotics.Invent Math,2008,171:485-541
2 Chirilus-Bruckner M,D¨ull W-P,Schneider G.NLS approximation of time oscillatory long waves for equations with quasilinear quadratic terms.Math Nachr,2015,288:158-166
3 D¨ull W-P,He?M.Existence of long time solutions and validity of the nonlinear Schr¨odinger approximation for a quasilinear dispersive equation.J Differential Equations,2018,264:2598-2632
4 D¨ull W-P,Schneider G,Wayne C E.Justification of the nonlinear Schr¨odinger equation for the evolution of gravity driven 2D surface water waves in a canal of finite depth.Arch Ration Mech Anal,2016,220:543-602
5 Dyachenko A I,Kuznetsov E A,Spector M D,et al.Analytical description of the free surface dynamics of an ideal fluid(canonical formalism and conformal mapping).Phys Lett A,1996,221:73-79
6 Hasimoto H,Ono H.Nonlinear modulation of gravity waves.J Phys Soc Japan,1972,33:805-811
7 Hunter J K,Ifrim M,Tataru D.Two dimensional water waves in holomorphic coordinates.Comm Math Phys,2016,346:483-552
8 Ifrim M,Tataru D.Two dimensional water waves in holomorphic coordinates II:Global solutions.Bull Soc Math France,2016,144:369-394
9 Ifrim M,Tataru D.The lifespan of small data solutions in two dimensional capillary water waves.Arch Ration Mech Anal,2017,225:1279-1346
10 Ifrim M,Tataru D.Two-dimensional gravity water waves with constant vorticity I:Cubic lifespan.Anal PDE,2019,12:903-967
11 Koch H,Tataru D.Conserved energies for the cubic nonlinear Schr¨odinger equation in one dimension.Duke Math J,2018,167:3207-3313
12 Ovsjannikov L V.To the shallow water theory foundation.Arch Mech(Arch Mech Stos),1974,26:407-422
13 Schneider G.Validity and non-validity of the nonlinear Schr¨odinger equation as a model for water waves.In:Lectures on the Theory of Water Waves.London Mathematical Society Lecture Note Series,vol.426.Cambridge:Cambridge University Press,2016,121-139
14 Schneider G,Wayne C E.Justification of the NLS approximation for a quasilinear water wave model.J Differential Equations,2011,251:238-269
15 Schneider G,Wayne C E.Corrigendum:The long-wave limit for the water wave problem I.The case of zero surface tension.Comm Pure Appl Math,2012,65:587-591
16 Totz N,Wu S J.A rigorous justification of the modulation approximation to the 2D full water wave problem.Comm Math Phys,2012,310:817-883
17 Wu S J.Almost global wellposedness of the 2-D full water wave problem.Invent Math,2009,177:45-135
18 Zakharov V E.Stability of periodic waves of finite amplitude on the surface of a deep fluid.J Appl Mech Tech Phys,1968,9:190-194
2 Chirilus-Bruckner M,D¨ull W-P,Schneider G.NLS approximation of time oscillatory long waves for equations with quasilinear quadratic terms.Math Nachr,2015,288:158-166
3 D¨ull W-P,He?M.Existence of long time solutions and validity of the nonlinear Schr¨odinger approximation for a quasilinear dispersive equation.J Differential Equations,2018,264:2598-2632
4 D¨ull W-P,Schneider G,Wayne C E.Justification of the nonlinear Schr¨odinger equation for the evolution of gravity driven 2D surface water waves in a canal of finite depth.Arch Ration Mech Anal,2016,220:543-602
5 Dyachenko A I,Kuznetsov E A,Spector M D,et al.Analytical description of the free surface dynamics of an ideal fluid(canonical formalism and conformal mapping).Phys Lett A,1996,221:73-79
6 Hasimoto H,Ono H.Nonlinear modulation of gravity waves.J Phys Soc Japan,1972,33:805-811
7 Hunter J K,Ifrim M,Tataru D.Two dimensional water waves in holomorphic coordinates.Comm Math Phys,2016,346:483-552
8 Ifrim M,Tataru D.Two dimensional water waves in holomorphic coordinates II:Global solutions.Bull Soc Math France,2016,144:369-394
9 Ifrim M,Tataru D.The lifespan of small data solutions in two dimensional capillary water waves.Arch Ration Mech Anal,2017,225:1279-1346
10 Ifrim M,Tataru D.Two-dimensional gravity water waves with constant vorticity I:Cubic lifespan.Anal PDE,2019,12:903-967
11 Koch H,Tataru D.Conserved energies for the cubic nonlinear Schr¨odinger equation in one dimension.Duke Math J,2018,167:3207-3313
12 Ovsjannikov L V.To the shallow water theory foundation.Arch Mech(Arch Mech Stos),1974,26:407-422
13 Schneider G.Validity and non-validity of the nonlinear Schr¨odinger equation as a model for water waves.In:Lectures on the Theory of Water Waves.London Mathematical Society Lecture Note Series,vol.426.Cambridge:Cambridge University Press,2016,121-139
14 Schneider G,Wayne C E.Justification of the NLS approximation for a quasilinear water wave model.J Differential Equations,2011,251:238-269
15 Schneider G,Wayne C E.Corrigendum:The long-wave limit for the water wave problem I.The case of zero surface tension.Comm Pure Appl Math,2012,65:587-591
16 Totz N,Wu S J.A rigorous justification of the modulation approximation to the 2D full water wave problem.Comm Math Phys,2012,310:817-883
17 Wu S J.Almost global wellposedness of the 2-D full water wave problem.Invent Math,2009,177:45-135
18 Zakharov V E.Stability of periodic waves of finite amplitude on the surface of a deep fluid.J Appl Mech Tech Phys,1968,9:190-194