## A morawetz inequality for gravity-capillary water waves at low bond number.(English)Zbl 07460550

Summary: This paper is devoted to the 2D gravity-capillary water waves equations in their Hamiltonian formulation, addressing the general question of proving Morawetz inequalities. We continue the analysis initiated in our previous work, where we have established local energy decay estimates for gravity waves. Here we add surface tension and prove a stronger estimate with a local regularity gain, akin to the smoothing effect for dispersive equations. Our main result holds globally in time and holds for genuinely nonlinear waves, since we are only assuming some very mild uniform Sobolev bounds for the solutions. Furthermore, it is uniform both in the infinite depth limit and the zero surface tension limit.

### MSC:

 76B15 Water waves, gravity waves; dispersion and scattering, nonlinear interaction 35Q35 PDEs in connection with fluid mechanics 35R35 Free boundary problems for PDEs 76B45 Capillarity (surface tension) for incompressible inviscid fluids 35Q31 Euler equations

### Keywords:

water waves; gravity/capillary; local energy decay
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### References:

 [1] Agrawal, S.: Angled crested type water waves with surface tension: wellposedness of the problem (2019). arXiv:1909.09671 [2] Ai, A.: Low regularity solutions for gravity water waves II: the 2D case (2018). arXiv e-prints. arXiv:1811.10504. Annals of PDE, to appear [3] Ai, A., Low regularity solutions for gravity water waves, Water Waves, 1, 1, 145-215 (2019) · Zbl 1451.35125 [4] Alazard, T.; Burq, N.; Zuily, C., On the water-wave equations with surface tension, Duke Math. J., 158, 3, 413-499 (2011) · Zbl 1258.35043 [5] Alazard, T.; Delort, J-M, Global solutions and asymptotic behavior for two dimensional gravity water waves, Ann. Sci. Éc. Norm. Supér. (4), 48, 5, 1149-1238 (2015) · Zbl 1347.35198 [6] Alazard, T., Ifrim, M., Tataru, D.: A Morawetz inequality for water waves (2018). arXiv e-prints. arXiv:1806.08443 · Zbl 1458.35077 [7] Ambrose, DM; Masmoudi, N., The zero surface tension limit of two-dimensional water waves, Commun. Pure Appl. Math., 58, 10, 1287-1315 (2005) · Zbl 1086.76004 [8] Bahouri, H., Chemin, J.-Y., Danchin, R.: Fourier Analysis and Nonlinear Partial Differential Equations. Grundlehren der mathematischen Wissenschaften, vol. 343. Springer, Heidelberg (2011) · Zbl 1227.35004 [9] Benjamin, TB; Olver, PJ, Hamiltonian structure, symmetries and conservation laws for water waves, J. Fluid Mech., 125, 137-185 (1982) · Zbl 0511.76020 [10] Beyer, K.; Günther, M., On the Cauchy problem for a capillary drop. I. Irrotational motion, Math. Methods Appl. Sci., 21, 12, 1149-1183 (1998) · Zbl 0916.35141 [11] Castro, A.; Córdoba, D.; Fefferman, C.; Gancedo, F.; Gómez-Serrano, J., Finite time singularities for water waves with surface tension, J. Math. Phys., 53, 11, 115622 (2012) · Zbl 1328.76012 [12] Castro, A.; Córdoba, D.; Fefferman, C.; Gancedo, F.; Gómez-Serrano, J., Finite time singularities for the free boundary incompressible Euler equations, Ann. Math. (2), 178, 3, 1061-1134 (2013) · Zbl 1291.35199 [13] Chen, RM; Marzuola, J.; Spirn, D.; Wright, JD, On the regularity of the flow map for the gravity-capillary equations, J. Funct. Anal., 264, 3, 752-782 (2013) · Zbl 1270.35159 [14] Christianson, H.; Hur, VM; Staffilani, G., Strichartz estimates for the water-wave problem with surface tension, Commun. Partial Differ. Equ., 35, 12, 2195-2252 (2010) · Zbl 1280.35107 [15] Colliander, J.; Keel, M.; Staffilani, G.; Takaoka, H.; Tao, T., Global well-posedness and scattering for the energy-critical nonlinear Schrödinger equation in $${\mathbb{R}}^3$$, Ann. Math. (2), 167, 3, 767-865 (2008) · Zbl 1178.35345 [16] Constantin, P.; Saut, J-C, Local smoothing properties of dispersive equations, J. Am. Math. Soc., 1, 2, 413-439 (1988) · Zbl 0667.35061 [17] Coutand, D., Shkoller, S.: Well-posedness of the free-surface incompressible Euler equations with or without surface tension. J. Am. Math. Soc. 20(3), 829-930 (2007) (electronic) · Zbl 1123.35038 [18] Coutand, D.; Shkoller, S., On the finite-time splash and splat singularities for the 3-D free-surface Euler equations, Commun. Math. Phys., 325, 1, 143-183 (2014) · Zbl 1285.35071 [19] Craig, W.; Nicholls, D., Travelling two and three dimensional capillary gravity water waves, SIAM J. Math. Anal., 32, 2, 323-359 (2000) · Zbl 0976.35049 [20] Craig, W.; Sulem, C.; Sulem, P-L, Nonlinear modulation of gravity waves: a rigorous approach, Nonlinearity, 5, 2, 497-522 (1992) · Zbl 0742.76012 [21] de Poyferré, T., A priori estimates for water waves with emerging bottom, Arch. Ration. Mech. Anal., 232, 2, 763-812 (2019) · Zbl 07041263 [22] de Poyferré, T.; Nguyen, Q-H, Strichartz estimates and local existence for the gravity-capillary waves with non-Lipschitz initial velocity, J. Differ. Equ., 261, 1, 396-438 (2016) · Zbl 1344.35113 [23] de Poyferré, T.; Nguyen, Q-H, A paradifferential reduction for the gravity-capillary waves system at low regularity and applications, Bull. Soc. Math. Fr., 145, 4, 643-710 (2017) · Zbl 1397.35237 [24] Deng, Y.; Ionescu, A.; Pausader, B.; Pusateri, F., Global solutions of the gravity-capillary water-wave system in three dimensions, Acta Math., 219, 2, 213-402 (2017) · Zbl 1397.35190 [25] Dyachenko, AI; Kuznetsov, EA; Spector, MD; Zakharov, VE, Analytical description of the free surface dynamics of an ideal fluid (canonical formalism and conformal mapping), Phys. Lett. A, 221, 1-2, 73-79 (1996) [26] Fefferman, C.; Ionescu, AD; Lie, V., On the absence of splash singularities in the case of two-fluid interfaces, Duke Math. J., 165, 3, 417-462 (2016) · Zbl 1346.35152 [27] Germain, P.; Masmoudi, N.; Shatah, J., Global solutions for the gravity water waves equation in dimension 3, Ann. Math. (2), 175, 2, 691-754 (2012) · Zbl 1241.35003 [28] Germain, P.; Masmoudi, N.; Shatah, J., Global existence for capillary water waves, Commun. Pure Appl. Math., 68, 4, 625-687 (2015) · Zbl 1314.35100 [29] Guo, Y.; Tice, I., Decay of viscous surface waves without surface tension, Anal. PDE, 6, 6, 1429-1533 (2013) · Zbl 1292.35206 [30] Harrop-Griffiths, B.; Ifrim, M.; Tataru, D., The lifespan of small data solutions to the KP-I, Int. Math. Res. Not., 1, 1-28 (2017) · Zbl 1405.35185 [31] Hunter, J.; Ifrim, M.; Tataru, D., Two dimensional water waves in holomorphic coordinates, Commun. Math. Phys., 346, 2, 483-552 (2016) · Zbl 1358.35121 [32] Ifrim, M.; Tataru, D., Two dimensional water waves in holomorphic coordinates II: global solutions, Bull. Soc. Math. Fr., 144, 2, 369-394 (2016) · Zbl 1360.35179 [33] Ifrim, M.; Tataru, D., The lifespan of small data solutions in two dimensional capillary water waves, Arch. Ration. Mech. Anal., 225, 3, 1279-1346 (2017) · Zbl 1375.35347 [34] Ifrim, M., Tataru, D.: No solitary waves in 2-d gravity and capillary waves in deep water. Nonlinearity 33(10), 5457-5477 (2020) · Zbl 1459.76029 [35] Iguchi, T., A long wave approximation for capillary-gravity waves and an effect of the bottom, Commun. Part. Differ. Equ., 32, 1-3, 37-85 (2007) · Zbl 1136.35081 [36] Ionescu, AD; Pusateri, F., Global solutions for the gravity water waves system in 2d, Invent. Math., 199, 3, 653-804 (2015) · Zbl 1325.35151 [37] Ionescu, A.D., Pusateri, F.: Global regularity for 2D water waves with surface tension. Mem. Am. Math. Soc. 256(1227), v+124 (2018) · Zbl 1435.76002 [38] Kenig, CE; Ponce, G.; Vega, L., The Cauchy problem for quasi-linear Schrödinger equations, Invent. Math., 158, 2, 343-388 (2004) · Zbl 1177.35221 [39] Lannes, D., A stability criterion for two-fluid interfaces and applications, Arch. Ration. Mech. Anal., 208, 2, 481-567 (2013) · Zbl 1278.35194 [40] Lannes, D.: Water Waves: Mathematical Analysis and Asymptotics. Mathematical Surveys and Monographs, vol. 188. American Mathematical Society, Providence (2013) · Zbl 1410.35003 [41] Littlewood, JE; Paley, REAC, Theorems on Fourier series and power series, J. Lond. Math. Soc., 6, 3, 230-233 (1931) · JFM 57.0318.01 [42] Marzuola, J.; Metcalfe, J.; Tataru, D., Strichartz estimates and local smoothing estimates for asymptotically flat Schrödinger equations, J. Funct. Anal., 255, 6, 1497-1553 (2008) · Zbl 1180.35187 [43] Metcalfe, J.; Tataru, D., Global parametrices and dispersive estimates for variable coefficient wave equations, Math. Ann., 353, 4, 1183-1237 (2012) · Zbl 1259.35006 [44] Métivier, G.: Para-Differential Calculus and Applications to the Cauchy Problem for Nonlinear Systems. Centro di Ricerca Matematica Ennio De Giorgi (CRM) Series, vol. 5. Edizioni della Normale, Pisa (2008) · Zbl 1156.35002 [45] Ming, M., Wang, C.: Water waves problem with surface tension in a corner domain II: the local well-posedness (2018). arXiv:1812.09911 · Zbl 1471.35239 [46] Morawetz, CS, Time decay for the nonlinear Klein-Gordon equations, Proc. R. Soc. Ser. A, 306, 291-296 (1968) · Zbl 0157.41502 [47] Nalimov, V.I.: The Cauchy-Poisson problem. Dinamika Splošn. Sredy (Vyp. 18 Dinamika Zidkost. so Svobod. Granicami) 254, 104-210 (1974) [48] Nguyen, H.Q.: A sharp Cauchy theory for the 2D gravity-capillary waves. Ann. Inst. H. Poincaré Anal. Non Linéaire 34(7), 1793-1836 (2017) · Zbl 1451.76028 [49] Ozawa, T.; Rogers, KM, A sharp bilinear estimate for the Klein-Gordon equation in $${\mathbb{R}}^{1+1}$$, Int. Math. Res. Not., 5, 1367-1378 (2014) · Zbl 1296.42014 [50] Planchon, F., Vega, L.: Bilinear virial identities and applications. Ann. Sci. Éc. Norm. Supér. (4) 42(2):261-290 (2009) · Zbl 1192.35166 [51] Rousset, F., Tzvetkov, N.: Transverse instability of the line solitary water-waves. Invent. Math. 184(2):257-388 (2011) [Prépublication (2009)] · Zbl 1225.35024 [52] Schneider, G.; Wayne, CE, The rigorous approximation of long-wavelength capillary-gravity waves, Arch. Ration. Mech. Anal., 162, 3, 247-285 (2002) · Zbl 1055.76006 [53] Schweizer, B.: On the three-dimensional Euler equations with a free boundary subject to surface tension. Ann. Inst. H. Poincaré Anal. Non Linéaire 22(6):753-781 (2005) · Zbl 1148.35071 [54] Shatah, J.; Zeng, C., A priori estimates for fluid interface problems, Commun. Pure Appl. Math., 61, 6, 848-876 (2008) · Zbl 1143.35347 [55] Vega, L., Schrödinger equations: pointwise convergence to the initial data, Proc. Am. Math. Soc., 102, 4, 874-878 (1988) · Zbl 0654.42014 [56] Wang, X.: Global regularity for the 3d finite depth capillary water waves (2016). arXiv:1611.05472 [57] Wu, S., Well-posedness in Sobolev spaces of the full water wave problem in 2-D, Invent. Math., 130, 1, 39-72 (1997) · Zbl 0892.76009 [58] Wu, S., Well-posedness in Sobolev spaces of the full water wave problem in 3-D, J. Am. Math. Soc., 12, 2, 445-495 (1999) · Zbl 0921.76017 [59] Wu, S., Almost global wellposedness of the 2-D full water wave problem, Invent. Math., 177, 1, 45-135 (2009) · Zbl 1181.35205 [60] Wu, S., Global wellposedness of the 3-D full water wave problem, Invent. Math., 184, 1, 125-220 (2011) · Zbl 1221.35304 [61] Yosihara, H., Gravity waves on the free surface of an incompressible perfect fluid of finite depth, Publ. Res. Inst. Math. Sci., 18, 1, 49-96 (1982) · Zbl 0493.76018 [62] Zakharov, VE, Stability of periodic waves of finite amplitude on the surface of a deep fluid, J. Appl. Mech. Tech. Phys., 9, 2, 190-194 (1968) [63] Zakharov, V.E.: Weakly nonlinear waves on the surface of an ideal finite depth fluid. In: Nonlinear Waves and Weak Turbulence. American Mathematical Society Translations: Series 2, vol. 182, pp. 167-197. American Mathematical Society, Providence (1998) · Zbl 0914.76015 [64] Zhu, H., Control of three dimensional water waves, Arch. Ration. Mech. Anal., 236, 2, 893-966 (2020) · Zbl 1431.76034 [65] Zhu, H.: Propagation of singularities for gravity-capillary water waves (2018). arXiv:1810.09339
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