On the local well-posedness for a full-dispersion Boussinesq system with surface tension

Kalisch, Henrik; Pilod, Didier

doi:10.1090/proc/14397

On the local well-posedness for a full-dispersion Boussinesq system with surface tension
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by Henrik Kalisch and Didier Pilod PDF

Proc. Amer. Math. Soc. 147 (2019), 2545-2559 Request permission

Abstract:

In this note, we prove local-in-time well-posedness for a fully dispersive Boussinesq system arising in the context of free surface water waves in two and three spatial dimensions. Those systems can be seen as a weak nonlocal dispersive perturbation of the shallow-water system. Our method of proof relies on energy estimates and a compactness argument. However, due to the lack of symmetry of the nonlinear part, those traditional methods have to be supplemented with the use of a modified energy in order to close the a priori estimates.

References

P. Aceves-Sánchez, A. A. Minzoni, and P. Panayotaros, Numerical study of a nonlocal model for water-waves with variable depth, Wave Motion 50 (2013), no. 1, 80–93. MR 2991247, DOI 10.1016/j.wavemoti.2012.07.002
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), no. 1287, 555–601. MR 385355, DOI 10.1098/rsta.1975.0035
Cosmin Burtea, New long time existence results for a class of Boussinesq-type systems, J. Math. Pures Appl. (9) 106 (2016), no. 2, 203–236 (English, with English and French summaries). MR 3515301, DOI 10.1016/j.matpur.2016.02.008
John D. Carter, Bidirectional Whitham equations as models of waves on shallow water, Wave Motion 82 (2018), 51–61. MR 3844340, DOI 10.1016/j.wavemoti.2018.07.004
L. Dawson, H. McGahagan, and G. Ponce, On the decay properties of solutions to a class of Schrödinger equations, Proc. Amer. Math. Soc. 136 (2008), no. 6, 2081–2090. MR 2383514, DOI 10.1090/S0002-9939-08-09355-6
Evgueni Dinvay, On well-posedness of a dispersive system of the Whitham-Boussinesq type, Appl. Math. Lett. 88 (2019), 13–20. MR 3862707, DOI 10.1016/j.aml.2018.08.005
Evgueni Dinvay, Daulet Moldabayev, Denys Dutykh, and Henrik Kalisch, The Whitham equation with surface tension, Nonlinear Dynam. 88 (2017), no. 2, 1125–1138. MR 3628376, DOI 10.1007/s11071-016-3299-7
V. Duchêne, S. Israwi, and R. Talhouk, A new class of two-layer Green-Naghdi systems with improved frequency dispersion, Stud. Appl. Math. 137 (2016), no. 3, 356–415. MR 3564304, DOI 10.1111/sapm.12125
Mats Ehrnström, Mark D. Groves, and Erik Wahlén, On the existence and stability of solitary-wave solutions to a class of evolution equations of Whitham type, Nonlinearity 25 (2012), no. 10, 2903–2936. MR 2979975, DOI 10.1088/0951-7715/25/10/2903
M. Ehrnström, L. Pei, and Y. Wang, A conditional well-posedness result for the bidirectional Whitham equation, preprint, arXiv:1708.0455 (2017).
M. Ehrnström and E. Wahlén, On Whitham’s conjecture of a highest cusped wave for a nonlocal shallow water wave equation, to appear in Ann. Inst. H. Poincaré, Anal. Nonlin., preprint, arXiv:1602.05384 (2016).
John K. Hunter, Mihaela Ifrim, Daniel Tataru, and Tak Kwong Wong, Long time solutions for a Burgers-Hilbert equation via a modified energy method, Proc. Amer. Math. Soc. 143 (2015), no. 8, 3407–3412. MR 3348783, DOI 10.1090/proc/12215
Vera Mikyoung Hur, Wave breaking in the Whitham equation, Adv. Math. 317 (2017), 410–437. MR 3682673, DOI 10.1016/j.aim.2017.07.006
V. M. Hur and A. K. Pandey, Modulational instability in a full-dispersion shallow water model, preprint, arXiv:1608.04685 (2016).
Vera Mikyoung Hur and Mathew A. Johnson, Modulational instability in the Whitham equation with surface tension and vorticity, Nonlinear Anal. 129 (2015), 104–118. MR 3414922, DOI 10.1016/j.na.2015.08.019
Tosio Kato and Gustavo Ponce, Commutator estimates and the Euler and Navier-Stokes equations, Comm. Pure Appl. Math. 41 (1988), no. 7, 891–907. MR 951744, DOI 10.1002/cpa.3160410704
Carlos E. Kenig and Didier Pilod, Local well-posedness for the KdV hierarchy at high regularity, Adv. Differential Equations 21 (2016), no. 9-10, 801–836. MR 3513119
Carlos E. Kenig, Gustavo Ponce, and Luis Vega, Well-posedness and scattering results for the generalized Korteweg-de Vries equation via the contraction principle, Comm. Pure Appl. Math. 46 (1993), no. 4, 527–620. MR 1211741, DOI 10.1002/cpa.3160460405
Christian Klein, Felipe Linares, Didier Pilod, and Jean-Claude Saut, On Whitham and related equations, Stud. Appl. Math. 140 (2018), no. 2, 133–177. MR 3763731, DOI 10.1111/sapm.12194
Soonsik Kwon, On the fifth-order KdV equation: local well-posedness and lack of uniform continuity of the solution map, J. Differential Equations 245 (2008), no. 9, 2627–2659. MR 2455780, DOI 10.1016/j.jde.2008.03.020
David Lannes, The water waves problem, Mathematical Surveys and Monographs, vol. 188, American Mathematical Society, Providence, RI, 2013. Mathematical analysis and asymptotics. MR 3060183, DOI 10.1090/surv/188
David Lannes, A stability criterion for two-fluid interfaces and applications, Arch. Ration. Mech. Anal. 208 (2013), no. 2, 481–567. MR 3035985, DOI 10.1007/s00205-012-0604-6
Felipe Linares, Didier Pilod, and Jean-Claude Saut, Well-posedness of strongly dispersive two-dimensional surface wave Boussinesq systems, SIAM J. Math. Anal. 44 (2012), no. 6, 4195–4221. MR 3023445, DOI 10.1137/110828277
Fabrice Planchon, Nikolay Tzvetkov, and Nicola Visciglia, On the growth of Sobolev norms for NLS on 2- and 3-dimensional manifolds, Anal. PDE 10 (2017), no. 5, 1123–1147. MR 3668586, DOI 10.2140/apde.2017.10.1123
Filippo Remonato and Henrik Kalisch, Numerical bifurcation for the capillary Whitham equation, Phys. D 343 (2017), 51–62. MR 3606200, DOI 10.1016/j.physd.2016.11.003
Nathan Sanford, Keri Kodama, John D. Carter, and Henrik Kalisch, Stability of traveling wave solutions to the Whitham equation, Phys. Lett. A 378 (2014), no. 30-31, 2100–2107. MR 3226084, DOI 10.1016/j.physleta.2014.04.067
Jean-Claude Saut and Li Xu, The Cauchy problem on large time for surface waves Boussinesq systems, J. Math. Pures Appl. (9) 97 (2012), no. 6, 635–662. MR 2921604, DOI 10.1016/j.matpur.2011.09.012
Jean-Claude Saut, Chao Wang, and Li Xu, The Cauchy problem on large time for surface-waves-type Boussinesq systems II, SIAM J. Math. Anal. 49 (2017), no. 4, 2321–2386. MR 3668593, DOI 10.1137/15M1050203
S. Trillo, M. Klein, G. F. Clauss, and M. Onorato, Observation of dispersive shock waves developing from initial depressions in shallow water, Phys. D 333 (2016), 276–284. MR 3523508, DOI 10.1016/j.physd.2016.01.007
Whitham, G. B. Variational Methods and Applications to Water Waves, Proceedings of the Royal Society of London. Series A, Mathematical and Physical Sciences 299, no. 1456 (1967): 6-25. http://www.jstor.org/stable/2415780.
Li Xu, Intermediate long wave systems for internal waves, Nonlinearity 25 (2012), no. 3, 597–640. MR 2887985, DOI 10.1088/0951-7715/25/3/597