Low regularity Cauchy theory for the water-waves problem: canals and swimming pools

Alazard, T.; Burq, N.; Zuily, C.

Journées équations aux dérivées partielles (2011), article no. 3, 20 p.

The purpose of this talk is to present some recent results about the Cauchy theory of the gravity water waves equations (without surface tension). In particular, we clarify the theory as well in terms of regularity indexes for the initial conditions as fin terms of smoothness of the bottom of the domain (namely no regularity assumption is assumed on the bottom). Our main result is that, following the approach developed in [1, 2], after suitable para-linearizations, the system can be arranged into an explicit symmetric system of quasilinear waves equation type, and consequently can be solved at the usual levels of regularity (initial data in $H^s,s>1+d/2$). In particular, the system can be solved for initial surfaces having undounded curvature. As another illustration of this reduction, we show that in fact following the analysis by Bahouri-Chemin and Tataru for quasi-linear wave equations, using Strichartz estimates, the regularity threshold can be further lowered, which allows to obtain well posedness for non lipschitz initial velocity fields. We also take benefit from our low regularity result and an elementary (though seemingly yet unknown) observation to solve a question raised by Boussinesq on the water-wave system in a canal.

[1] Thomas Alazard and Guy Métivier. Paralinearization of the Dirichlet to Neumann operator, and regularity of three-dimensional water waves. Comm. Partial Differential Equations, 34(10-12):1632–1704, 2009. | MR | Zbl

[2] Thomas Alazard, Nicolas Burq, and Claude Zuily. On the water-wave equations with surface tension. Duke Math. J., 158(3):413–499, 2011. | MR

[3] Thomas Alazard, Nicolas Burq, and Claude Zuily. Strichartz estimates for water waves. Ann. Sci. Éc. Norm. Supér. (4), to appear. | MR

[4] Serge Alinhac. Paracomposition et opérateurs paradifférentiels. Comm. Partial Differential Equations, 11(1):87–121, 1986. | MR | Zbl

[5] Serge Alinhac. Interaction d’ondes simples pour des équations complètement non-linéaires. Ann. Sci. École Norm. Sup. (4), 21(1):91–132, 1988. | Numdam | MR | Zbl

[6] Serge Alinhac. Existence d’ondes de raréfaction pour des systèmes quasi-linéaires hyperboliques multidimensionnels. Comm. Partial Differential Equations, 14(2):173–230, 1989. | MR | Zbl

[7] Borys Alvarez-Samaniego and David Lannes. Large time existence for 3D water-waves and asymptotics. Invent. Math., 171(3):485–541, 2008. | MR | Zbl

[8] David M. Ambrose and Nader Masmoudi. The zero surface tension limit of two-dimensional water waves. Comm. Pure Appl. Math., 58(10):1287–1315, 2005. | MR | Zbl

[9] David M. Ambrose and Nader Masmoudi, The zero surface tension limit of three-dimensional water waves. Indiana Univ. Math. J., 58 no. 2: 479521, 2009, | MR | Zbl

[10] David M. Ambrose and Nader Masmoudi, Well-posedness of 3D vortex sheets with surface tension. Commun. Math. Sci. 5 no. 2: 391430, 2007 | MR | Zbl

[11] Hajer Bahouri and Jean-Yves Chemin. Équations d’ondes quasilinéaires et estimations de Strichartz. Amer. J. Math., 121(6):1337–1377, 1999. | MR | Zbl

[12] Hajer Bahouri, Jean-Yves Chemin, and Raphaël Danchin. Fourier analysis and nonlinear partial differential equations, volume 343 of Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences]. Springer, Heidelberg, 2011. | MR

[13] Claude Bardos and David Lannes. Mathematics for 2d interfaces. A paraître dans Panorama et Synthèses.

[14] J. Thomas Beale, Thomas Y. Hou, and John S. Lowengrub. Growth rates for the linearized motion of fluid interfaces away from equilibrium. Comm. Pure Appl. Math., 46(9):1269–1301, 1993. | MR | Zbl

[15] Klaus Beyer and Matthias Günther. On the Cauchy problem for a capillary drop. I. Irrotational motion. Math. Methods Appl. Sci., 21(12):1149–1183, 1998. | MR | Zbl

[16] Matthew Blair. Strichartz estimates for wave equations with coefficients of Sobolev regularity. Comm. Partial Differential Equations, 31(4-6):649–688, 2006. | MR | Zbl

[17] Jerry L. Bona, David Lannes, and Jean-Claude. Saut. Asymptotic models for internal waves. J. Math. Pures Appl. (9), 89(6):538–566, 2008. | MR | Zbl

[18] Jean-Michel Bony. Calcul symbolique et propagation des singularités pour les équations aux dérivées partielles non linéaires. Ann. Sci. École Norm. Sup. (4), 14(2):209–246, 1981. | Numdam | MR | Zbl

[19] Boussinesq, J. Sur une importante simplification de la théorie des ondes que produisent, à la surface d’un liquide, l’emersion d’un solide ou l’impulsion d’un coup de vent. Ann. Sci. École Norm. Sup. (3), 27, 942, 1910. | Numdam

[20] Nicolas Burq, Patrick Gérard, and Nikolay Tzvetkov. Strichartz inequalities and the nonlinear Schrödinger equation on compact manifolds. Amer. J. Math., 126(3):569–605, 2004. | MR | Zbl

[21] Jean-Yves Chemin. Calcul paradifférentiel précisé et applications à des équations aux dérivées partielles non semilinéaires. Duke Math. J., 56(3):431–469, 1988. | Zbl

[22] Jean-Yves Chemin. Perfect incompressible fluids, volume 14 of Oxford Lecture Series in Mathematics and its Applications. The Clarendon Press Oxford University Press, New York, 1998. Translated from the 1995 French original by Isabelle Gallagher and Dragos Iftimie. | Zbl

[23] Hans Christianson, Vera Hur and Gigliola Staffilani, Strichartz estimates for the water-wave problem with surface tension. Comm. Partial Differential Equations 35, no. 12: 21952252, 2010. | MR

[24] Demetrios Christodoulou and Hans Lindblad. On the motion of the free surface of a liquid. Comm. Pure Appl. Math., 53(12):1536–1602, 2000. | MR | Zbl

[25] Angel Castro, Diego Córdoba, Charles, L. Fefferman, Francisco Gancedo and Maria López-Fernández Turning waves and breakdown for incompressible flows. To appear, Annals of Math., 2011. | MR

[26] Angel Castro, Diego Córdoba, Charles, L. Fefferman, Francisco Gancedo and Javier Gómez-Serrano Splash singularity for water waves. preprint, 2011

[27] Daniel Coutand and Steve Shkoller. Well-posedness of the free-surface incompressible Euler equations with or without surface tension. J. Amer. Math. Soc., 20(3):829–930 (electronic), 2007. | MR | Zbl

[28] W. Craig, C. Sulem, and P.-L. Sulem. Nonlinear modulation of gravity waves: a rigorous approach. Nonlinearity, 5(2):497–522, 1992. | MR | Zbl

[29] Walter Craig. An existence theory for water waves and the Boussinesq and Korteweg-deVries scaling limits. Communications in Partial Differential Equations, 10(8):787–1003, 1985. | MR | Zbl

[30] Walter Craig, Ulrich Schanz, and Catherine Sulem. The modulational regime of three-dimensional water waves and the Davey-Stewartson system. Ann. Inst. H. Poincaré Anal. Non Linéaire, 14(5):615–667, 1997. | Numdam | MR | Zbl

[31] Walter Craig and C.Eugene WayneKre?g. Mathematical aspects of surface waves on water. Uspekhi Mat. Nauk 62:3 (375), 95–116, 2007 translation in Russian Math. Surveys 62 (3), 453473, 2007 | MR | Zbl

[32] Pierre Germain, Nader Masmoudi, and Jalal Shatah. Global solutions for the gravity water waves equation in dimension 3. Prépulication 2009. | MR | Zbl

[33] Matthias Günther and Georg Prokert. On a Hele-Shaw type domain evolution with convected surface energy density: the third-order problem. SIAM J. Math. Anal., 38(4):1154–1185 (electronic), 2006. | MR | Zbl

[34] Lars Hörmander. The boundary problems of physical geodesy. Arch. Rational Mech. Anal., 62(1):1–52, 1976. | MR | Zbl

[35] Lars Hörmander. Lectures on nonlinear hyperbolic differential equations, volume 26 of Mathématiques & Applications (Berlin) [Mathematics & Applications]. Springer-Verlag, Berlin, 1997. | MR | Zbl

[36] Thomas Y. Hou, Zhen-huan Teng, and Pingwen Zhang. Well-posedness of linearized motion for $3$-D water waves far from equilibrium. Comm. Partial Differential Equations, 21(9-10):1551–1585, 1996. | MR | Zbl

[37] Tatsuo Iguchi, A long wave approximation for capillary-gravity waves and an effect of the bottom. (English summary) Comm. Partial Differential Equations 32, no. 1-3 : 3785, 2007. | MR | Zbl

[38] Tatsuo Iguchi, Well-posedness of the initial value problem for capillary-gravity waves. Funkcial. Ekvac. 44,no. 2: 219241, 2001. | MR | Zbl

[39] Gérard Iooss and Pavel Plotnikov, Asymmetrical three-dimensional travelling gravity waves. Arch. Ration. Mech. Anal. 200, no. 3: 789880, 2011, | MR

[40] David Lannes. A stability criterion for two-fluid interfaces and applications. Prépublication 2010.

[41] David Lannes. Water waves: mathematical analysis and asymptotics. to appear.

[42] David Lannes. Well-posedness of the water-waves equations. J. Amer. Math. Soc., 18(3):605–654 (electronic), 2005. | MR | Zbl

[43] Gilles Lebeau Singularités des solutions d’équations d’ondes semi-linéaires. Ann. Sci. École Norm. Sup. (4) 25- 2: 201231, 1992. | Numdam | MR | Zbl

[44] Gilles Lebeau. Régularité du problème de Kelvin-Helmholtz pour l’équation d’Euler 2d. ESAIM Control Optim. Calc. Var., 8:801–825 (electronic), 2002. A tribute to J. L. Lions. | Numdam | MR | Zbl

[45] Hans Lindblad. Well posedness for the motion of a compressible liquid with free surface boundary. Comm. Math. Phys., 260(2):319–392, 2005. | MR | Zbl

[46] Hans Lindblad. Well-posedness for the motion of an incompressible liquid with free surface boundary. Ann. of Math. (2), 162(1):109–194, 2005. | MR | Zbl

[47] Vladimir G. Maz’ya and Tatyana O. Shaposhnikova. Theory of Sobolev multipliers, volume 337 of Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences]. Springer-Verlag, Berlin, 2009. With applications to differential and integral operators. | MR | Zbl

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

[49] Guy Métivier and Kevin Zumbrun. Large viscous boundary layers for noncharacteristic nonlinear hyperbolic problems. Mem. Amer. Math. Soc., 175(826):vi+107, 2005. | MR | Zbl

[50] Yves Meyer. Remarques sur un théorème de J.-M. Bony. In Proceedings of the Seminar on Harmonic Analysis (Pisa, 1980), number suppl. 1, pages 1–20, 1981. | MR | Zbl

[51] V. I. Nalimov. The Cauchy-Poisson problem. Dinamika Splošn. Sredy, (Vyp. 18 Dinamika Zidkost. so Svobod. Granicami):104–210, 254, 1974. | MR

[52] Frédéric Rousset and Nikolay Tzvetkov. Transverse instability of the line solitary water-waves. Prépublication 2009.

[53] John, Reeder and Marvin Shinbrot. Three-dimensional, nonlinear wave interaction in water of constant depth. Nonlinear Anal. 5 (3): 303323, 1981. | MR | Zbl

[54] Guido Schneider and C. Eugene Wayne. The long-wave limit for the water wave problem. I. The case of zero surface tension. Comm. Pure Appl. Math. 53 (12): 14751535, 2000. | MR | Zbl

[55] Ben Schweizer. On the three-dimensional Euler equations with a free boundary subject to surface tension. Ann. Inst. H. Poincaré Anal. Non Linéaire, 22 (6): 753781, 2005. | Numdam | MR | Zbl

[56] Monique Sablé-Tougeron. Régularité microlocale pour des problèmes aux limites non linéaires. Ann. Inst. Fourier (Grenoble), 36(1):39–82, 1986. | Numdam | MR | Zbl

[57] Jalal Shatah and Chongchun Zeng. Geometry and a priori estimates for free boundary problems of the Euler equation. Comm. Pure Appl. Math., 61(5):698–744, 2008. | MR | Zbl

[58] Jalal Shatah and Chongchun Zeng. A priori estimates for fluid interface problems. Comm. Pure Appl. Math., 61(6):848–876, 2008. | MR | Zbl

[59] Hart F. Smith. A parametrix construction for wave equations with $C^{1,1}$ coefficients. Ann. Inst. Fourier (Grenoble), 48(3):797–835, 1998. | Numdam | MR | Zbl

[60] Gigliola Staffilani and Daniel Tataru, Strichartz estimates for a Schrödinger operator with nonsmooth coefficients. Comm. Partial Differential Equations 27, no. 7-8: 13371372, 2002. | MR | Zbl

[61] Daniel Tataru. Strichartz estimates for operators with nonsmooth coefficients and the nonlinear wave equation. Amer. J. Math., 122(2):349–376, 2000. | MR | Zbl

[62] Daniel Tataru. Strichartz estimates for second order hyperbolic operators with nonsmooth coefficients. II. Amer. J. Math., 123(3):385–423, 2001. | MR | Zbl

[63] Daniel Tataru. Strichartz estimates for second order hyperbolic operators with nonsmooth coefficients. III. J. Amer. Math. Soc., 15(2):419–442 (electronic), 2002. | MR | Zbl

[64] Michael E. Taylor. Pseudodifferential operators and nonlinear PDE, volume 100 of Progress in Mathematics. Birkhäuser Boston Inc., Boston, MA, 1991. | MR | Zbl

[65] Yuri Trakhinin. Local existence for the free boundary problem for nonrelativistic and relativistic compressible Euler equations with a vacuum boundary condition. Comm. Pure Appl. Math., 62(11):1551–1594, 2009. | MR | Zbl

[66] Sijue Wu. Global well-posedness of the 3-d full water wave problem. Prépulication 2010.

[67] Sijue Wu. Well-posedness in Sobolev spaces of the full water wave problem in 2-D. Invent. Math., 130(1):39–72, 1997. | MR | Zbl

[68] Sijue Wu. Well-posedness in Sobolev spaces of the full water wave problem in 3-D. J. Amer. Math. Soc., 12(2):445–495, 1999. | MR | Zbl

[69] Sijue Wu. Almost global wellposedness of the 2-D full water wave problem. Invent. Math., 177(1):45–135, 2009. | MR | Zbl

[70] Hideaki Yosihara. Gravity waves on the free surface of an incompressible perfect fluid of finite depth. Publ. Res. Inst. Math. Sci., 18(1):49–96, 1982. | MR | Zbl