Strichartz estimates for water waves
[Estimées de Strichartz pour les ondes de surface]
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 (η=0,ψ=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 (η=0,ψ=0)).

DOI : 10.24033/asens.2156
Classification : 35Bxx, 35Lxx, 35Sxx, 35Jxx
Keywords: Euler equation, free boundary problems, water-waves, Cauchy theory, dispersive estimates
Mot clés : Équation d'Euler, problèmes à frontière libre, ondes de surfaces, théorie de Cauchy, estimées dispersives
@article{ASENS_2011_4_44_5_855_0,
     author = {Alazard, Thomas and Burq, Nicolas and Zuily, Claude},
     title = {Strichartz estimates for water waves},
     journal = {Annales scientifiques de l'\'Ecole Normale Sup\'erieure},
     pages = {855--903},
     publisher = {Soci\'et\'e math\'ematique de France},
     volume = {Ser. 4, 44},
     number = {5},
     year = {2011},
     doi = {10.24033/asens.2156},
     mrnumber = {2931520},
     zbl = {1260.35140},
     language = {en},
     url = {http://archive.numdam.org/articles/10.24033/asens.2156/}
}
TY  - JOUR
AU  - Alazard, Thomas
AU  - Burq, Nicolas
AU  - Zuily, Claude
TI  - Strichartz estimates for water waves
JO  - Annales scientifiques de l'École Normale Supérieure
PY  - 2011
SP  - 855
EP  - 903
VL  - 44
IS  - 5
PB  - Société mathématique de France
UR  - http://archive.numdam.org/articles/10.24033/asens.2156/
DO  - 10.24033/asens.2156
LA  - en
ID  - ASENS_2011_4_44_5_855_0
ER  - 
%0 Journal Article
%A Alazard, Thomas
%A Burq, Nicolas
%A Zuily, Claude
%T Strichartz estimates for water waves
%J Annales scientifiques de l'École Normale Supérieure
%D 2011
%P 855-903
%V 44
%N 5
%I Société mathématique de France
%U http://archive.numdam.org/articles/10.24033/asens.2156/
%R 10.24033/asens.2156
%G en
%F ASENS_2011_4_44_5_855_0
Alazard, Thomas; Burq, Nicolas; Zuily, Claude. Strichartz estimates for water waves. Annales scientifiques de l'École Normale Supérieure, Série 4, Tome 44 (2011) no. 5, pp. 855-903. doi : 10.24033/asens.2156. http://archive.numdam.org/articles/10.24033/asens.2156/

[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

Cité par Sources :