# 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.

show all references

##### 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.
 [1] Chiara Caracciolo, Ugo Locatelli. Computer-assisted estimates for Birkhoff normal forms. Journal of Computational Dynamics, 2020, 7 (2) : 425-460. doi: 10.3934/jcd.2020017 [2] Maxime Breden, Jean-Philippe Lessard. Polynomial interpolation and a priori bootstrap for computer-assisted proofs in nonlinear ODEs. Discrete and Continuous Dynamical Systems - B, 2018, 23 (7) : 2825-2858. doi: 10.3934/dcdsb.2018164 [3] Thomas Wanner. Computer-assisted equilibrium validation for the diblock copolymer model. Discrete and Continuous Dynamical Systems, 2017, 37 (2) : 1075-1107. doi: 10.3934/dcds.2017045 [4] István Balázs, Jan Bouwe van den Berg, Julien Courtois, János Dudás, Jean-Philippe Lessard, Anett Vörös-Kiss, JF Williams, Xi Yuan Yin. Computer-assisted proofs for radially symmetric solutions of PDEs. Journal of Computational Dynamics, 2018, 5 (1&2) : 61-80. doi: 10.3934/jcd.2018003 [5] A. Aschwanden, A. Schulze-Halberg, D. Stoffer. Stable periodic solutions for delay equations with positive feedback - a computer-assisted proof. Discrete and Continuous Dynamical Systems, 2006, 14 (4) : 721-736. doi: 10.3934/dcds.2006.14.721 [6] Piotr Zgliczyński. Steady state bifurcations for the Kuramoto-Sivashinsky equation: A computer assisted proof. Journal of Computational Dynamics, 2015, 2 (1) : 95-142. doi: 10.3934/jcd.2015.2.95 [7] Anwar Ja'afar Mohamad Jawad, Mohammad Mirzazadeh, Anjan Biswas. Dynamics of shallow water waves with Gardner-Kadomtsev-Petviashvili equation. Discrete and Continuous Dynamical Systems - S, 2015, 8 (6) : 1155-1164. doi: 10.3934/dcdss.2015.8.1155 [8] Maciej J. Capiński, Emmanuel Fleurantin, J. D. Mireles James. Computer assisted proofs of two-dimensional attracting invariant tori for ODEs. Discrete and Continuous Dynamical Systems, 2020, 40 (12) : 6681-6707. doi: 10.3934/dcds.2020162 [9] Lorenzo Valvo, Ugo Locatelli. Hamiltonian control of magnetic field lines: Computer assisted results proving the existence of KAM barriers. Journal of Computational Dynamics, 2022  doi: 10.3934/jcd.2022002 [10] Elena Kartashova. Nonlinear resonances of water waves. Discrete and Continuous Dynamical Systems - B, 2009, 12 (3) : 607-621. doi: 10.3934/dcdsb.2009.12.607 [11] Robert McOwen, Peter Topalov. Asymptotics in shallow water waves. Discrete and Continuous Dynamical Systems, 2015, 35 (7) : 3103-3131. doi: 10.3934/dcds.2015.35.3103 [12] Mouhamadou Aliou M. T. Baldé, Diaraf Seck. Coupling the shallow water equation with a long term dynamics of sand dunes. Discrete and Continuous Dynamical Systems - S, 2016, 9 (5) : 1521-1551. doi: 10.3934/dcdss.2016061 [13] Guy V. Norton, Robert D. Purrington. The Westervelt equation with a causal propagation operator coupled to the bioheat equation.. Evolution Equations and Control Theory, 2016, 5 (3) : 449-461. doi: 10.3934/eect.2016013 [14] Yongcai Geng. Singularity formation for relativistic Euler and Euler-Poisson equations with repulsive force. Communications on Pure and Applied Analysis, 2015, 14 (2) : 549-564. doi: 10.3934/cpaa.2015.14.549 [15] Walter A. Strauss. Vorticity jumps in steady water waves. Discrete and Continuous Dynamical Systems - B, 2012, 17 (4) : 1101-1112. doi: 10.3934/dcdsb.2012.17.1101 [16] Jerry L. Bona, Henrik Kalisch. Models for internal waves in deep water. Discrete and Continuous Dynamical Systems, 2000, 6 (1) : 1-20. doi: 10.3934/dcds.2000.6.1 [17] Vera Mikyoung Hur. On the formation of singularities for surface water waves. Communications on Pure and Applied Analysis, 2012, 11 (4) : 1465-1474. doi: 10.3934/cpaa.2012.11.1465 [18] Martina Chirilus-Bruckner, Guido Schneider. Interaction of oscillatory packets of water waves. Conference Publications, 2015, 2015 (special) : 267-275. doi: 10.3934/proc.2015.0267 [19] Ying Sui, Huimin Yu. Singularity formation for compressible Euler equations with time-dependent damping. Discrete and Continuous Dynamical Systems, 2021, 41 (10) : 4921-4941. doi: 10.3934/dcds.2021062 [20] Emile Franc Doungmo Goufo, Abdon Atangana. Dynamics of traveling waves of variable order hyperbolic Liouville equation: Regulation and control. Discrete and Continuous Dynamical Systems - S, 2020, 13 (3) : 645-662. doi: 10.3934/dcdss.2020035

2020 Impact Factor: 1.392