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# Structural stability for the splash singularities of the water waves problem

• In this paper we show a structural stability result for water waves. The main motivation for this result is that we aim to exhibit a water wavewhose interface starts as a graph and ends in a splash. Numerical simulations lead to an approximate solution with the desired behaviour. The stability result will conclude that near the approximate solution to water waves there is an exact solution.
Mathematics Subject Classification: Primary: 76B15, 35Q31; Secondary: 76E09.

 Citation:

•  [1] J. T. Beale, T. Y. Hou and J. Lowengrub, Convergence of a boundary integral method for water waves, SIAM J. Numer. Anal., 33 (1996), 1797-1843.doi: 10.1137/S0036142993245750. [2] A. Castro, D. Córdoba, C. Fefferman, F. Gancedo and J. Gómez-Serrano, Splash singularity for water waves, Proceedings of the National Academy of Sciences, 109 (2012), 733-738.doi: 10.1073/pnas.1115948108. [3] A. Castro, D. Córdoba, C. Fefferman, F. Gancedo and J. Gómez-Serrano, Finite time singularities for the free boundary incompressible Euler equations, Ann. of Math. (2), 178 (2013), 1061-1134.doi: 10.4007/annals.2013.178.3.6. [4] Á. Castro, D. Córdoba, C. Fefferman, F. Gancedo and M. López-Fernández, Rayleigh-Taylor breakdown for the Muskat problem with applications to water waves, Ann. of Math. (2), 175 (2012), 909-948.doi: 10.4007/annals.2012.175.2.9. [5] D. Coutand and S. Shkoller, On the finite-time splash and splat singularities for the 3-D free-surface Euler equations, Comm. Math. Phys., 325 (2014), 143-183.doi: 10.1007/s00220-013-1855-2. [6] C. Fefferman, A. D. Ionescu and V. Lie, On the absence of "splash'' singularities in the case of two-fluid interfaces, arXiv preprint arXiv:1312.2917, 2013. [7] G. B. Folland, Introduction to Partial Differential Equations, Princeton University Press, Princeton, NJ, second edition, 1995. [8] M. Joldes, Rigorous Polynomial Approximations and Applications, PhD thesis, École normale supérieure de Lyon, 2011. [9] D. Lannes, The Water Waves Problem: Mathematical Analysis and Asymptotics, Mathematical Surveys and Monographs. Amer Mathematical Society, 2013.
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