Strichartz estimates for water waves

Alazard, Thomas; Burq, Nicolas; Zuily, Claude

Annales scientifiques de l'École Normale Supérieure, Série 4, Tome 44 (2011) no. 5, pp. 855-903.

Nous nous intéressons dans cet article aux propriétés dispersives du système des ondes de surface en dimension $2$, avec tension de surface. Nous démontrons tout d’abord des estimées de Strichartz, avec pertes de dérivées, au niveau de régularité où nous avons construit des solutions dans [3]. Ensuite, pour des données initiales plus régulières, nous démontrons les estimées de Strichartz optimales (i.e. sans perte de régularité par rapport à celles du système linéarisé en ($\eta =0,\psi =0$)).

In this paper we investigate the dispersive properties of the solutions of the two dimensional water-waves system with surface tension. First we prove Strichartz type estimates with loss of derivatives at the same low level of regularity we were able to construct the solutions in [3]. On the other hand, for smoother initial data, we prove that the solutions enjoy the optimal Strichartz estimates (i.e, without loss of regularity compared to the system linearized at ($\eta =0,\psi =0$)).

[1] T. Alazard, N. Burq & C. Zuily, On the Cauchy problem for water gravity waves, preprint, 2011. | Zbl

[2] T. Alazard, N. Burq & C. Zuily, On the water-wave equations with surface tension, Duke Math. J. 158 (2011), 413-499. | MR | Zbl

[3] T. Alazard & G. Métivier, Paralinearization of the Dirichlet to Neumann operator, and regularity of three-dimensional water waves, Comm. Partial Differential Equations 34 (2009), 1632-1704. | MR | Zbl

[4] S. Alinhac, Paracomposition et opérateurs paradifférentiels, Comm. Partial Differential Equations 11 (1986), 87-121. | MR | Zbl

[5] S. Alinhac, Existence d'ondes de raréfaction pour des systèmes quasi-linéaires hyperboliques multidimensionnels, Comm. Partial Differential Equations 14 (1989), 173-230. | MR | Zbl

[6] D. M. Ambrose & N. Masmoudi, The zero surface tension limit of two-dimensional water waves, Comm. Pure Appl. Math. 58 (2005), 1287-1315. | MR | Zbl

[7] H. Bahouri & J.-Y. Chemin, Équations d'ondes quasilinéaires et estimations de Strichartz, Amer. J. Math. 121 (1999), 1337-1377. | MR | Zbl

[8] M. S. Berger, Nonlinearity and functional analysis, Academic Press, 1977. | MR | Zbl

[9] J. Bergh & J. Löfström, Interpolation spaces. An introduction, Grundl. Math. Wiss. 223, Springer, 1976. | MR | Zbl

[10] K. Beyer & M. Günther, On the Cauchy problem for a capillary drop. I. Irrotational motion, Math. Methods Appl. Sci. 21 (1998), 1149-1183. | MR | Zbl

[11] M. Blair, Strichartz estimates for wave equations with coefficients of Sobolev regularity, Comm. Partial Differential Equations 31 (2006), 649-688. | MR | Zbl

[12] J.-M. Bony, Calcul symbolique et propagation des singularités pour les équations aux dérivées partielles non linéaires, Ann. Sci. École Norm. Sup. 14 (1981), 209-246. | EuDML | Numdam | MR | Zbl

[13] N. Burq, P. Gérard & N. Tzvetkov, Strichartz inequalities and the nonlinear Schrödinger equation on compact manifolds, Amer. J. Math. 126 (2004), 569-605. | MR | Zbl

[14] N. Burq & F. Planchon, On well-posedness for the Benjamin-Ono equation, Math. Ann. 340 (2008), 497-542. | MR | Zbl

[15] J.-Y. Chemin, Fluides parfaits incompressibles, Astérisque 230 (1995). | Numdam | MR | Zbl

[16] H. Christianson, V. M. Hur & G. Staffilani, Strichartz estimates for the water-wave problem with surface tension, Comm. Partial Differential Equations 35 (2010), 2195-2252. | MR | Zbl

[17] D. Coutand & S. Shkoller, Well-posedness of the free-surface incompressible Euler equations with or without surface tension, J. Amer. Math. Soc. 20 (2007), 829-930. | MR | Zbl

[18] P. Germain, N. Masmoudi & J. Shatah, Global solutions for the gravity water waves equation in dimension 3, C. R. Math. Acad. Sci. Paris 347 (2009), 897-902. | MR | Zbl

[19] T. Iguchi, A long wave approximation for capillary-gravity waves and an effect of the bottom, Comm. Partial Differential Equations 32 (2007), 37-85. | MR | Zbl

[20] H. Koch & D. Tataru, Dispersive estimates for principally normal pseudodifferential operators, Comm. Pure Appl. Math. 58 (2005), 217-284. | MR | Zbl

[21] G. Métivier, Para-differential calculus and applications to the Cauchy problem for nonlinear systems, Centro di Ricerca Matematica Ennio De Giorgi (CRM) Series 5, Edizioni della Normale, Pisa, 2008. | MR | Zbl

[22] M. Ming & Z. Zhang, Well-posedness of the water-wave problem with surface tension, J. Math. Pures Appl. 92 (2009), 429-455. | MR | Zbl

[23] F. Rousset & N. Tzvetkov, On the transverse instability of one dimensional capillary-gravity waves, Discrete Contin. Dyn. Syst. Ser. B 13 (2010), 859-872. | MR | Zbl

[24] B. Schweizer, On the three-dimensional Euler equations with a free boundary subject to surface tension, Ann. Inst. H. Poincaré Anal. Non Linéaire 22 (2005), 753-781. | EuDML | Numdam | MR | Zbl

[25] J. Shatah & C. Zeng, Geometry and a priori estimates for free boundary problems of the Euler equation, Comm. Pure Appl. Math. 61 (2008), 698-744. | MR | Zbl

[26] H. F. Smith, A parametrix construction for wave equations with $C^{1,1}$ coefficients, Ann. Inst. Fourier (Grenoble) 48 (1998), 797-835. | EuDML | Numdam | MR | Zbl

[27] G. Staffilani & D. Tataru, Strichartz estimates for a Schrödinger operator with nonsmooth coefficients, Comm. Partial Differential Equations 27 (2002), 1337-1372. | MR | Zbl

[28] D. Tataru, Strichartz estimates for operators with nonsmooth coefficients and the nonlinear wave equation, Amer. J. Math. 122 (2000), 349-376. | MR | Zbl

[29] D. Tataru, Strichartz estimates for second order hyperbolic operators with nonsmooth coefficients. II, Amer. J. Math. 123 (2001), 385-423. | MR | Zbl

[30] S. Wu, Almost global wellposedness of the 2-D full water wave problem, Invent. Math. 177 (2009), 45-135. | MR | Zbl

[31] S. Wu, Global well-posedness of the 3D full water wave problem, preprint arXiv:0910.2473. | Zbl