Book contents
- Frontmatter
- Contents
- Contributors
- Preface
- 1 High-Order Perturbation of Surfaces Short Course: Boundary Value Problems
- 2 High-Order Perturbation of Surfaces Short Course: Traveling Water Waves
- 3 High-Order Perturbation of Surfaces Short Course: Analyticity Theory
- 4 High-Order Perturbation of Surfaces Short Course: Stability of Traveling Water Waves
- 5 A Novel Non-Local Formulation of Water Waves
- 6 The Dimension-Breaking Route to Three-Dimensional Solitary Gravity-Capillary Water Waves
- 7 Validity and Non-Validity of the Nonlinear Schrödinger Equation as a Model for Water Waves
- 8 Vortex Sheet Formulations and Initial Value Problems: Analysis and Computing
- 9 Wellposedness and Singularities of the WaterWave Equations
- 10 Conformal Mapping and Complex Topographies
- 11 Variational Water Wave Modelling: from Continuum to Experiment
- 12 Symmetry, Modulation, and Nonlinear Waves
- References
9 - Wellposedness and Singularities of the WaterWave Equations
Published online by Cambridge University Press: 05 February 2016
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Book contents
- Frontmatter
- Contents
- Contributors
- Preface
- 1 High-Order Perturbation of Surfaces Short Course: Boundary Value Problems
- 2 High-Order Perturbation of Surfaces Short Course: Traveling Water Waves
- 3 High-Order Perturbation of Surfaces Short Course: Analyticity Theory
- 4 High-Order Perturbation of Surfaces Short Course: Stability of Traveling Water Waves
- 5 A Novel Non-Local Formulation of Water Waves
- 6 The Dimension-Breaking Route to Three-Dimensional Solitary Gravity-Capillary Water Waves
- 7 Validity and Non-Validity of the Nonlinear Schrödinger Equation as a Model for Water Waves
- 8 Vortex Sheet Formulations and Initial Value Problems: Analysis and Computing
- 9 Wellposedness and Singularities of the WaterWave Equations
- 10 Conformal Mapping and Complex Topographies
- 11 Variational Water Wave Modelling: from Continuum to Experiment
- 12 Symmetry, Modulation, and Nonlinear Waves
- References
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- Lectures on the Theory of Water Waves , pp. 171 - 202Publisher: Cambridge University PressPrint publication year: 2016
References
[1] The instability of liquid surfaces when accelerated in a direction perpendicular to their planes I.Proc. Roy. Soc. London A 201 (1950), pp.192-196Google Scholar
[2] Helmholtz and Taylor instabilityProc. Symp. in Appl. Math. XIII, pp. 55-76.
[3] , & Growth rates for the linearized motion of fluid interfaces away from equilibriumComm. Pure Appl. Math. 46 (1993), no. 9, pp. 1269-1301.Google Scholar
[4] On the theory of oscillatory waves.Trans. Cambridge Philos. Soc., 8 (1847), pp. 441-455.Google Scholar
[5] . Determination rigoureuse des ondes permanentes d'ampleurfinie.Math. Ann., 93(1), 1925. pp. 264-314Google Scholar
[6] The Cauchy-Poisson problem (in Russian), Dynamika Splosh. Sredy 18, 1974, pp. 104-210.Google Scholar
[7] Gravity waves on the free surface of an incompressible perfect fluid of finite depth, RIMS Kyoto 18 (1982), pp. 49-96Google Scholar
[8] An existence theory for water waves and the Boussinesq and Korteweg-devries scaling limitsComm. in P. D. E. 10(8) (1985), pp. 787-1003Google Scholar
[9] Well-posedness in Sobolev spaces of the full water wave problem in 2-DInvent. Math. 130 (1997), pp. 39-72Google Scholar
[10] Well-posedness in Sobolev spaces of the full water wave problem in 3-DJ. Amer. Math. Soc. 12 no. 2 (1999), pp. 445-495.Google Scholar
[11] , The zero surface tension limit of two-dimensional water wavesComm. Pure Appl. Math. 58 (2005), no. 10, pp. 1287-1315Google Scholar
[12] , On the motion of the free surface of a liquidComm. Pure Appl. Math. 53 (2000), no. 12, pp. 1536-1602Google Scholar
[13] , Wellposedness of the free-surface incompressible Euler equations with or without surface tensionJ. AMS. 20 (2007), no. 3, pp. 829-930.Google Scholar
[14] Well-posedness of the initial value problem for capillary-gravity wavesFunkcial. Ekvac. 44 (2001) no. 2, pp. 219-241.Google Scholar
[15] Well-posedness of the water-wave equationsJ. Amer. Math. Soc. 18 (2005), pp. 605-654Google Scholar
[16] Well-posedness for the motion of an incompressible liquid with free surface boundaryAnn. of Math. 162 (2005), no. 1, pp. 109-194.Google Scholar
[17] , Free boundary problem for an incompressible ideal fluid with surface tensionMath. Models Methods Appl. Sci. 12, (2002), no. 12, pp. 1725-1740.Google Scholar
[18] , Geometry and a priori estimates for free boundary problems of the Euler's equationComm. Pure Appl. Math. V. 61. no. 5 (2008), pp. 698-744.Google Scholar
[19] , On the free boundary problem of 3-D incompressible Euler equations.Comm. Pure. Appl. Math. V. 61. no. 7 (2008), pp. 877-940Google Scholar
[20] , & On the Cauchy problem for gravity water wavesInvent. Math. 198 (2014), pp.71-163Google Scholar
[21] , & Strichartz estimates and the Cauchy problem for the gravity water waves equations Preprint 2014, arXiv: 1404.4276
[22] Almost global wellposedness of the 2-D full water wave problemInvent. Math, 177 (2009), no. 1, pp. 45-135.Google Scholar
[23] Global wellposedness of the 3-D full water wave problemInvent. Math. 184 (2011), no. 1, pp.125-220.Google Scholar
[24] , , & Global solutions of the gravity water wave equation in dimension 3Ann. of Math (2). 175 (2012), no. 2, pp. 691-754.Google Scholar
[25] & . Global solutions for the gravity water waves system in 2dInvent. Math. 199 (2015), no. 3, p. 653804Google Scholar
[26] & Global solutions and asymptotic behavior for two dimensional gravity water waves Ann. Sci. Ec. Norm. Super., to appear
[27] , & Two dimensional water waves in holomorphic coordinates Preprint 2014, arXiv:1401.1252
[28] & Two dimensional water waves in holomorphic coordinates II: global solutions Preprint 2014, arXiv: 1404.7583
[29] On a class of self-similar 2d surface water waves Preprint 2012 arXiv1 206:2208
[30] & A priori estimates for two-dimensional water waves with angled crests Preprint 2014, arXiv1406:7573
[31] A blow-up criteria and the existence of 2d gravity water waves with angled crests Preprint 2015 arXiv1 502:05342
[32] Calderon-Zygmund operators, pseudo-differential operators and the Cauchy integral of calderon, vol. 994, Lecture Notes in Math.Springer, 1983.Google Scholar
[33] , , , & Finite time singularities for the free boundary incompressible Euler equationsAnn. of Math. (2) 178 (2013), no. 3, pp. 1061-1134Google Scholar
[35] Elliptic boundary value problems on Lipschitz domainsBeijing Lectures in Harmonic Analysis, ed. by , Princeton Univ. Press, 1986, p. 131-183.Google Scholar
[36] & Clifford algebras and Dirac operators in harmonic analysisCambridge University Press, 1991Google Scholar
[37] A wave operator for a nonlinear Klein-Gordon equation, Letters. Math. Phys., 7 (1983), no. 5, pp. 387-398.Google Scholar
[38] Normal forms and quadratic nonlinear Klein-Gordon equationsComm. Pure Appl. Math. 38 (1985), pp. 685-696.Google Scholar
[39] & A Rigorous justification of the modulation approximation to the 2D full water wave problemComm. Math. Phys. 310 (2012), pp. 817-883Google Scholar
[40] & Enhanced lifespan of smooth solutions of a Burgers-Hilbert equationSIAM J. Math. Anal, 44 (3) 2012, pp. 1279-2235.Google Scholar
[41] The null condition and global existence to nonlinear wave equationsLectures in Appl. Math. 23 (1986), pp. 293-325.Google Scholar
[42] A Rigorous justification of the modulation approximation to the 3D full water wave problem accepted by Comm. Math. Phys.
[43] Nonstandard estimates for a class of 1D dispersive equations and applications to linearized water waves Preprint (2014) arXiv: 1409.8088
[44] , and Lintegrale de Cauchy definit un operateur borne sur L2 pour les courbes lipschitziennesAnnals of Math. 116 (1982), pp. 361-387.Google Scholar
[45] & Decay rate of solutions to hyperbolic system of first orderActa. Math. Sinica, English series. 15 (1999), no. 4, pp. 471-484Google Scholar
[46] , and La solution des conjectures de CalderónAdv. in Math. 48 (1983), pp.144-148.Google Scholar
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