On the absence of splash singularities in the case of two-fluid interfaces

Duke Mathematical Journal

Volumn 165, Issue 3, 2016, Pages 417-462

  Fefferman, Charles ^a   Ionescu, Alexandru D ^a   Lie, Victor ^b  

^a PRINCETON UNIVERSITY   (United States)

^b PURDUE UNIVERSITY   (United States)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 84958781337     PISSN: 00127094     EISSN: None     Source Type: Journal    
DOI: 10.1215/00127094-3166629     Document Type: Article

References (35)

1. T. ALAZARD, N. BURQ, and C. ZUILY, On the water-wave equations with surface tension, Duke Math. J. 158 (2011), 413-499. MR 2805065. DOI 10.1215/00127094-1345653.
2. T. ALAZARD and J.-M. DELORT, Global solutions and asymptotic behavior for two dimensional gravity water waves, preprint, arXiv:1305.4090v2 [math.AP].
3. T. ALAZARD, N. BURQ, and C. ZUILY Sobolev estimates for two dimensional gravity water waves, preprint, arXiv:1307.3836v1 [math.AP].
4. D. M. AMBROSE, Well-posedness of vortex sheets with surface tension, SIAM J. Math. Anal. 35 (2003), 211-244. MR 2001473. DOI 10.1137/S0036141002403869.
5. D. AMBROSE and N. MASMOUDI, Well-posedness of 3D vortex sheets with surface tension, Commun. Math. Sci. 5 (2007), 391-430. MR 2334849.
6. A. CASTRO, D. CÓRDOBA, C. FEFFERMAN, F. GANCEDO, and J. GÓMEZ-SERRANO, Finite time singularities for water waves with surface tension, J. Math. Phys. 53 (2012), art. 115622. MR 3026567. DOI 10.1063/1.4765339.
7. 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. MR 3092476. DOI 10.4007/annals.2013.178.3.6.
8. A. CHENG, D. COUTAND, and S. SHKOLLER, On the motion of vortex sheets with surface tension in three-dimensional Euler equations with vorticity, Comm. Pure Appl. Math. 61 (2008), 1715-1752. MR 2456184. DOI 10.1002/cpa.20240.
9. H. CHRISTIANSON, V. HUR, and G. STAFFILANI, Strichartz estimates for the water-wave problem with surface tension, Comm. Partial Differential Equations 35 (2010), 2195-2252. MR 2763354. DOI 10.1080/03605301003758351.
10. D. CHRISTODOULOU and H. LINDBLAD, On the motion of the free surface of a liquid, Comm. Pure Appl. Math. 53 (2000), 1536-1602. MR 1780703. DOI 10.1002/1097-0312(200012)53:12<1536::AID-CPA2>3.3.CO;2-H.
11. D. COUTAND and S. SHKOLLER, Well-posedness of the free-surface incompressible Euler equations with or without surface tension, J. Amer.Math. Soc. 20 (2007), 829-930. MR 2291920. DOI 10.1090/S0894-0347-07-00556-5.
12. 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. MR 3147437. DOI 10.1007/s00220-013-1855-2.
13. W. CRAIG, An existence theory for water waves and the Boussinesq and Korteweg-de Vries scaling limits, Comm. Partial Differential Equations 10 (1985), 787-1003. MR 0795808. DOI 10.1080/03605308508820396.
14. F. GANCEDO and R. STRAIN, Absence of splash singularities for surface quasi-geostrophic sharp fronts and the Muskat problem, Proc. Natl. Acad. Sci. USA 111 (2014), 635-639. MR 3181769.
15. P. GERMAIN, N. MASMOUDI, and J. SHATAH, Global solutions for the gravity water waves equation in dimension 3, Ann. of Math. (2) 175 (2012), 691-754. MR 2993751. DOI 10.4007/annals.2012.175.2.6.
16. P. GERMAIN, N. MASMOUDI, and J. SHATAH Global existence for capillary water waves, Comm. Pure Appl. Math. 68 (2015), 625-687. MR 3318019. DOI 10.1002/cpa.21535.
17. T. IGUCHI, N. TANAKA, and A. TANI, On the two-phase free boundary problem for two-dimensional water waves, Math. Ann. 309 (1997), 199-223. MR 1474190. DOI 10.1007/s002080050110.
18. A. IONESCU and F. PUSATERI, Nonlinear fractional Schrödinger equations in one dimension, J. Funct. Anal. 266 (2014), 139-176. MR 3121725. DOI 10.1016/j.jfa.2013.08.027.
19. A. IONESCU and F. PUSATERI Global solutions for the gravity water waves system in 2d, Invent. Math. 199 (2015), 653-804. MR 3314514. DOI 10.1007/s00222-014-0521-4.
20. D. LANNES, Well-posedness of the water-waves equations, J. Amer.Math. Soc. 18 (2005), 605-654. MR 2138139. DOI 10.1090/S0894-0347-05-00484-4.
21. D. LANNES, A stability criterion for two-fluid interfaces and applications, Arch. Ration. Mech. Anal. 208 (2013), 481-567. MR 3035985. DOI 10.1007/s00205-012-0604-6.
22. D. LANNES, The Water Waves Problem: Mathematical Analysis and Asymptotics. Math. Surveys Monogr. 188, Amer. Math. Soc., Providence, 2013. MR 3060183. DOI 10.1090/surv/188.
23. H. LINDBLAD, Well-posedness for the motion of an incompressible liquid with free surface boundary, Ann. of Math. (2) 162 (2005), 109-194. MR 2178961. DOI 10.4007/annals.2005.162.109.
24. V. I. NALIMOV, The Cauchy-Poisson Problem, Dynamika Splosh. Sredy 18 (1974), 104-210. MR 0609882.
25. F. PUSATERI, On the limit as the surface tension and density ratio tend to zero for the two-phase Euler equations, J. Hyperbolic Differ. Equ. 8 (2011), 347-373. MR 2812146. DOI 10.1142/S021989161100241X.
26. J. SHATAH and C. ZENG, Geometry and a priori estimates for free boundary problems of the Euler equation, Comm. Pure Appl. Math. 61 (2008), 698-744. MR 2388661. DOI 10.1002/cpa.20213.
27. J. SHATAH and C. ZENG, A priori estimates for fluid interface problems, Comm. Pure Appl. Math. 61 (2008), 848-876. MR 2400608. DOI 10.1002/cpa.20241.
28. J. SHATAH and C. ZENG, Local well-posedness for fluid interface problems, Arch. Ration. Mech. Anal. 199 (2011), 653-705. MR 2763036. DOI 10.1007/s00205-010-0335-5.
29. C. SULEM, P.-L. SULEM, C. BARDOS, and U. FRISCH, Finite time analyticity for the twoand three-dimensional Kelvin-Helmholtz instability, Comm. Math. Phys. 80 (1981), 485-516. MR 0628507.
30. C. SULEM and P.-L. SULEM, Finite time analyticity for the two- and three-dimensional Rayleigh-Taylor instabilit, Trans. Amer. Math. Soc. 287, no. 1 (1981), 127-160. MR 0766210. DOI 10.2307/2000401.
31. S. WU, Well-posedness in Sobolev spaces of the full water wave problem in 2-D, Invent. Math. 130 (1997), 39-72. MR 1471885. DOI 10.1007/s002220050177.
32. S. WU, Well-posedness in Sobolev spaces of the full water wave problem in 3-D, J. Amer. Math. Soc. 12 (1999), 445-495. MR 1641609. DOI 10.1090/S0894-0347-99-00290-8.
33. S. WU, Almost global wellposedness of the 2-D full water wave problem, Invent. Math. 177 (2009), 45-135. MR 2507638. DOI 10.1007/s00222-009-0176-8.
34. S. WU, Global wellposedness of the 3-D full water wave problem, Invent. Math. 184 (2011), 125-220. MR 2782254. DOI 10.1007/s00222-010-0288-1.
35. H. YOSIHARA, Gravity waves on the free surface of an incompressible perfect fluid of finite depth, Publ. Res. Inst.Math. Sci. 18 (1982), 49-96. MR 0660822. DOI 10.2977/prims/1195184016. (419)