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

Structural stability for the splash singularities of the water waves problem

Angel Castro 1, , Diego Córdoba 2, , Charles Fefferman 3, , Francisco Gancedo 4, and  Javier Gómez-Serrano 5,

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.
Keywords: splash singularity, contour dynamics, Water waves, Euler equation., computer-assisted.
Mathematics Subject Classification: Primary: 76B15, 35Q31; Secondary: 76E0.
Citation: Angel Castro, Diego Córdoba, Charles Fefferman, Francisco Gancedo, Javier Gómez-Serrano. Structural stability for the splash singularities of the water waves problem. Discrete & 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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. Google Scholar

[7]

G. B. Folland, Introduction to Partial Differential Equations, Princeton University Press, Princeton, NJ, second edition, 1995.  Google Scholar

[8]

M. Joldes, Rigorous Polynomial Approximations and Applications, PhD thesis, École normale supérieure de Lyon, 2011. Google Scholar

[9]

D. Lannes, The Water Waves Problem: Mathematical Analysis and Asymptotics, Mathematical Surveys and Monographs. Amer Mathematical Society, 2013.  Google Scholar

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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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. Google Scholar

[7]

G. B. Folland, Introduction to Partial Differential Equations, Princeton University Press, Princeton, NJ, second edition, 1995.  Google Scholar

[8]

M. Joldes, Rigorous Polynomial Approximations and Applications, PhD thesis, École normale supérieure de Lyon, 2011. Google Scholar

[9]

D. Lannes, The Water Waves Problem: Mathematical Analysis and Asymptotics, Mathematical Surveys and Monographs. Amer Mathematical Society, 2013.  Google Scholar

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