# American Institute of Mathematical Sciences

December  2014, 34(12): 4997-5043. doi: 10.3934/dcds.2014.34.4997

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

 1 Departamento de Matemáticas de la UAM, Instituto de Ciencias Matemáticas del CSIC, Campus de Cantoblanco, 28049 Madrid 2 Instituto de Ciencias Matemáticas, Consejo Superior de Investigaciones Científicas, C/ Nicolas Cabrera, 13-15, 28049 Madrid 3 Department of Mathematics, Princeton University, 1102 Fine Hall, Washington Road, Princeton, New Jersey 08544 4 Departamento de Análisis Matemático & IMUS, Universidad de Sevilla, Campus Reina Mercedes, 41012 Sevilla, Spain 5 Department of Mathematics, Princeton University, 1102 Fine Hall, Washington Rd, Princeton, NJ 08544

Received  January 2014 Revised  May 2014 Published  June 2014

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.
Citation: Angel Castro, Diego Córdoba, Charles Fefferman, Francisco Gancedo, Javier Gómez-Serrano. Structural stability for the splash singularities of the water waves problem. Discrete and Continuous Dynamical Systems, 2014, 34 (12) : 4997-5043. doi: 10.3934/dcds.2014.34.4997
##### References:
 [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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##### References:
 [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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