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The Cauchy problem on large time for the water waves equations with large topography variations. (English) Zbl 1359.35154

Summary: This paper shows that the long time existence of solutions to the Water Waves equations remains true with a large topography in presence of surface tension. More precisely, the dimensionless equations depend strongly on three parameters \(\varepsilon\), \(\mu\), \(\beta\) measuring the amplitude of the waves, the shallowness and the amplitude of the bathymetric variations respectively. In [B. Alvarez-Samaniego and D. Lannes, Invent. Math. 171, No. 3, 485–541 (2008; Zbl 1131.76012)], the local existence of solutions to this problem is proved on a time interval of size \(\frac{1}{\max(\beta, \varepsilon)}\) and uniformly with respect to \(\mu\). In presence of large bathymetric variations (typically \(\beta \gg \varepsilon\)), the existence time is therefore considerably reduced. We remove here this restriction and prove the local existence on a time interval of size \(\frac{1}{\varepsilon}\) under the constraint that the surface tension parameter must be at the same order as the shallowness parameter \(\mu\). We also show that the result of D. Bresch and G. Métivier [AMRX, Appl. Math. Res. Express 2010, 119–141 (2010; Zbl 1204.35117)] dealing with large bathymetric variations for the Shallow Water equations can be viewed as a particular endpoint case of our result.

MSC:

35Q35 PDEs in connection with fluid mechanics
76B15 Water waves, gravity waves; dispersion and scattering, nonlinear interaction
76M45 Asymptotic methods, singular perturbations applied to problems in fluid mechanics
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[1] Alazard, Thomas; Burq, Nicolas; Zuily, Claude, The Water-Wave Equations: From Zakharov to Euler, Progr. Nonlinear Differential Equations Appl., vol. 84 (2013), Birkhäuser/Springer: Birkhäuser/Springer New York · Zbl 1273.35220
[2] Borys, Alvarez-Samaniego; Lannes, David, Large time existence for 3D water-waves and asymptotics, Invent. Math., 171, 3, 485-541 (2008) · Zbl 1131.76012
[3] Boussinesq, Joseph, Théorie générale des mouvements qui sont propagés dans un canal rectangulaire horizontal, C. R. Séances Acad. Sci., 256-260 (1871) · JFM 03.0486.02
[4] Boussinesq, Joseph, Théorie des ondes et des remous qui se propagent le long d’un canal rectangulaire horizontal, en communiquant au liquide contenu dans ce canal des vitesses sensiblement pareilles de la surface au fond, J. Math. Pures Appl., 55-108 (1872) · JFM 04.0493.04
[5] Bresch, D.; Métivier, G., Anelastic limits for Euler-type systems, Appl. Math. Res. Express, 2010, 2, 119-141 (2010) · Zbl 1204.35117
[6] Craig, W.; Sulem, C., Numerical simulation of gravity waves, J. Comput. Phys., 108, 1, 73-83 (1993) · Zbl 0778.76072
[7] Craig, W.; Sulem, C.; Sulem, P.-L., Nonlinear modulation of gravity waves: a rigorous approach, Nonlinearity, 5, 2, 497-522 (1992) · Zbl 0742.76012
[8] Deny, J.; Lions, J. L., Les espaces du type de Beppo Levi, Ann. Inst. Fourier, 5, 305-370, 1953-1954 (1955) · Zbl 0065.09903
[9] Ebin, D. G., The equations of motion of a perfect fluid with free boundary are not well posed, Commun. Partial Differ. Equ., 1987, 12, 1175-1201 (1987) · Zbl 0631.76018
[10] Iguchi, Tatsuo, A shallow water approximation for water waves, J. Math. Kyoto Univ., 49, 1, 13-55 (2009) · Zbl 1421.76020
[11] Lannes, David, A stability criterion for two-fluid interfaces and applications, Arch. Ration. Mech. Anal., 208, 2, 481-567 (2013) · Zbl 1278.35194
[12] Lannes, David, The Water Waves Problem (2013), American Mathematical Society · Zbl 1410.35003
[14] Lannes, David; Saut, Jean-Claude, Weakly transverse Boussinesq systems and the KP approximation, Nonlinearity, 2853-2875 (2006) · Zbl 1122.35114
[16] Métivier, G.; Schochet, S., The incompressible limit of the non-isentropic euler equations, Arch. Ration. Mech. Anal., 158, 1, 61-90 (2001) · Zbl 0974.76072
[17] Peregrine, D. Howell, Long waves on a beach, J. Fluid Mech., 815-827 (1967) · Zbl 0163.21105
[18] Rousset, Frederic; Tzvetkov, Nikolay, Transverse instability of the line solitary water-waves, Invent. Math., 184, 2, 257-388 (2011) · Zbl 1225.35024
[19] Taylor, G., The instability of liquid surfaces when accelerated in a direction perpendicular to their planes, Proc. R. Soc. Lond. Ser. A, 1950, 201, 192-196 (1950) · Zbl 0038.12201
[20] Taylor, M. E., Partial Differential Equations III, vol. 117 (1997), Springer
[22] Zakharov, V. E., Stability of periodic waves of finite amplitude on the surface of a deep fluid, J. Appl. Mech. Tech. Phys., 1968, 9, 190-194 (1968)
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