## On the water-wave equations with surface tension.(English)Zbl 1258.35043

Summary: The purpose of this article is to clarify the Cauchy theory of the water-wave equations in terms of regularity indexes for the initial conditions, as well as for the smoothness of the bottom of the domain. (Namely, no regularity assumption is assumed on the bottom.) Our main result is that, after suitable paralinearization, the system can be arranged into an explicit symmetric system of Schrödinger type. We then show that the smoothing effect for the (one-dimensional) surface-tension water waves is in fact a rather direct consequence of this reduction, and following this approach, we are able to obtain a sharp result in terms of regularity of the indexes of the initial data and weights in the estimates.

### MSC:

 35B65 Smoothness and regularity of solutions to PDEs 35Q35 PDEs in connection with fluid mechanics
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### References:

 [1] T. Alazard, N. Burq, and C. Zuily, Strichartz estimates for water waves , to appear in Ann. Sci. Éc. Norm. Supér. (4), preprint,\arxiv1002.0323[math.AP] [2] -, Local well posedness for the gravity water waves system , in preparation. [3] T. Alazard and G. Métivier, Paralinearization of the Dirichlet to Neumann operator, and regularity of three-dimensional water waves , Comm. Partial Differential Equations 34 (2009), 1632-1704. · Zbl 1207.35082 [4] S. Alinhac, Existence d’ondes de raréfaction pour des systèmes quasi-linéaires hyperboliques multidimensionnels , Comm. Partial Differential Equations 14 (1989), 173-230. · Zbl 0692.35063 [5] B. Alvarez-Samaniego and D. Lannes, Large time existence for 3D water-waves and asymptotics , Invent. Math. 171 (2008), 485-541. · Zbl 1131.76012 [6] D. M. Ambrose and N. Masmoudi, The zero surface tension limit of two-dimensional water waves , Comm. Pure Appl. Math. 58 (2005), 1287-1315. · Zbl 1086.76004 [7] K. Beyer and M. Günther, On the Cauchy problem for a capillary drop, I: Irrotational motion , Math. Methods Appl. Sci. 21 (1998), 1149-1183. · Zbl 0916.35141 [8] J. L. Bona, D. Lannes, and J.-C. Saut, Asymptotic models for internal waves , J. Math. Pures Appl. (9) 89 (2008), 538-566. · Zbl 1138.76028 [9] J. L. Bona and R. Smith, The initial-value problem for the Korteweg-de Vries equation , Philos. Trans. Roy. Soc. London Ser. A 278 (1975), 555-601. JSTOR: · Zbl 0306.35027 [10] J.-M. Bony, Calcul symbolique et propagation des singularités pour les équations aux dérivées partielles non linéaires , Ann. Sci. Éc. Norm. Supér. (4) 14 (1981), 209-246. · Zbl 0495.35024 [11] J.-Y. Chemin, Calcul paradifférentiel précisé et applications à des équations aux dérivées partielles non semilinéaires , Duke Math. J. 56 (1988), 431-469. · Zbl 0676.35009 [12] H. Christianson, V. M. Hur, and G. Staffilani, Strichartz estimates for the water-wave problem with surface tension , Comm. Partial Differential Equations 35 (2010), 2195-2252. · Zbl 1280.35107 [13] D. Coutand and S. Shkoller, Well-posedness of the free-surface incompressible Euler equations with or without surface tension , J. Amer. Math. Soc. 20 (2007), 829-930. · Zbl 1123.35038 [14] S.-I. Doi, On the Cauchy problem for Schrödinger type equations and the regularity of solutions , J. Math. Kyoto Univ. 34 (1994), 319-328. · Zbl 0807.35026 [15] -, Remarks on the Cauchy problem for Schrödinger-type equations , Comm. Partial Differential Equations 21 (1996), 163-178. · Zbl 0853.35025 [16] L. Hörmander, Lectures on Nonlinear Hyperbolic Differential Equations , Math. Appl. (Berlin) 26 , Springer, Berlin, 1997. [17] T. Iguchi, A long wave approximation for capillary-gravity waves and an effect of the bottom , Comm. Partial Differential Equations 32 (2007), 37-85. · Zbl 1136.35081 [18] G. Iooss and P. I. Plotnikov, Small Divisor Problem in the Theory of Three-Dimensional Water Gravity Waves , Mem. Amer. Math. Soc. 200 , Amer. Math. Soc., Providence, 2009. · Zbl 1172.76005 [19] T. Kano and T. Nishida, Sur les ondes de surface de l’eau avec une justification math$$\acute{e}$$matique des $$\acute{e}$$quations des ondes en eau peu profonde , J. Math. Kyoto Univ. 19 (1979), 335-370. · Zbl 0419.76013 [20] T. Kato, The Cauchy problem for quasi-linear symmetric hyperbolic systems , Arch. Ration. Mech. Anal. 58 (1975), 181-205. · Zbl 0343.35056 [21] D. Lannes, Well-posedness of the water-waves equations , J. Amer. Math. Soc. 18 (2005), 605-654. · Zbl 1123.35038 [22] G. Métivier, Para-Differential Calculus and Applications to the Cauchy Problem for Nonlinear Systems , CRM Series 5 , Ed. Norm., Pisa, 2008. [23] G. Métivier and J. Rauch, Dispersive Stabilization , Bull. Lond. Math. Soc. 42 (2010), 250-262. · Zbl 1192.35165 [24] G. Métivier and K. Zumbrun, Large Viscous Boundary Layers for Noncharacteristic Nonlinear Hyperbolic Problems , Mem. Amer. Math. Soc. 175 , Amer. Math. Soc., Providence, 2005. · Zbl 1074.35066 [25] Y. Meyer, “Remarques sur un théorème de J.-M. Bony” in Proceedings of the Seminar on Harmonic Analysis (Pisa, 1980) , Rend. Circ. Mat. Palermo (2) 1981 , Springer Italia, Milan, 1981, 1-20. · Zbl 0473.35021 [26] M. Ming and Z. Zhang, Well-posedness of the water-wave problem with surface tension , J. Math. Pures Appl. (9) 92 (2009), 429-455. · Zbl 1190.35186 [27] F. Rousset and N. Tzvetkov, Transverse instability of the line solitary water-waves , Invent. Math. 184 (2011), 257-388. · Zbl 1225.35024 [28] G. Schneider and C. E. Wayne, The rigorous approximation of long-wavelength capillary-gravity waves , Arch. Ration. Mech. Anal. 162 (2002), 247-285. · Zbl 1055.76006 [29] B. Schweizer, On the three-dimensional Euler equations with a free boundary subject to surface tension , Ann. Inst. H. Poincar$$\acute{\mathrm e}$$ Anal. Non Lin$$\acute{\mathrm e}$$aire 22 (2005), 753-781. · Zbl 1069.35056 [30] J. Shatah and C. Zeng, Geometry and a priori estimates for free boundary problems of the Euler equation , Comm. Pure Appl. Math. 61 (2008), 698-744. · Zbl 1174.76001 [31] D. Tataru, On the Fefferman-Phong inequality and related problems , Comm. Partial Differential Equations 27 (2002), 2101-2138. · Zbl 1045.35115 [32] M. E. Taylor, Pseudodifferential Operators and Nonlinear PDE , Progr. Math. 100 , Birkhäuser Boston, Boston, 1991. · Zbl 0746.35062 [33] Y. 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 (2009), 1551-1594. · Zbl 1185.35342 [34] N. Tzvetkov, “Ill-posedness issues for nonlinear dispersive equations” in Lectures on Nonlinear Dispersive Equations , GAKUTO Internat. Ser. Math. Sci. Appl. 27 , Gakk$$\grave{\mathrm o}$$tosho, Tokyo, 2006, 63-103. · Zbl 1158.35081 [35] S. Wu, Well-posedness in Sobolev spaces of the full water wave problem in $$2$$-D , Invent. Math. 130 (1997), 39-72. · Zbl 0892.76009 [36] -, Well-posedness in Sobolev spaces of the full water wave problem in 3-D , J. Amer. Math. Soc. 12 (1999), 445-495. JSTOR: · Zbl 0921.76017 [37] H. Yoshihara, Gravity waves on the free surface of an incompressible perfect fluid of finite depth , Publ. Res. Inst. Math. Sci. 18 (1982), 49-96. · Zbl 0493.76018 [38] -, Capillary-gravity waves for an incompressible ideal fluid , J. Math. Kyoto Univ. 23 (1983), 649-694. · Zbl 0548.76018 [39] V. E. Zakharov, “Weakly nonlinear waves on the surface of an ideal finite depth fluid” in Nonlinear Waves and Weak Turbulence , Amer. Math. Soc. Transl. Ser. 2 182 , Amer. Math. Soc., Providence, 1998, 167-197. · Zbl 0914.76015
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