On the Finite-Time Splash and Splat Singularities for the 3-D Free-Surface Euler Equations

Communications in Mathematical Physics

Volumn 325, Issue 1, 2014, Pages 143-183

  Coutand, Daniel ^a   Shkoller, Steve ^b  

^a HERIOT WATT UNIVERSITY   (United Kingdom)

^b UNIVERSITY OF CALIFORNIA   (United States)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 84891662791     PISSN: 00103616     EISSN: 14320916     Source Type: Journal    
DOI: 10.1007/s00220-013-1855-2     Document Type: Article

References (34)

1. Adams R. A.: Sobolev Spaces. Academic Press, London-New York (1978).
2. Alazard, T., Burq, N., Zuily, C.: On the Cauchy problem for gravity water waves. http://arxiv. org/abs/1212. 0626v1 [math. AP], 2012.
3. Alvarez-Samaniego B., Lannes D.: Large time existence for 3D water-waves and asymptotics. Invent. Math. 171, 485-541 (2008).
4. Ambrose D. M., Masmoudi N.: The zero surface tension limit of two-dimensional water waves. Commun. Pure Appl. Math. 58, 1287-1315 (2005).
5. Ambrose D., Masmoudi N.: The zero surface tension limit of three-dimensional water waves. Indiana Univ. Math. J. 58, 479-521 (2009).
6. Beale J. T., Hou T., Lowengrub J.: Growth rates for the linearized motion of fluid interfaces away from equilibrium. Commun. Pure Appl. Math. 46, 1269-1301 (1993).
7. Castro A., Córdoba D., Fefferman C., Gancedo F., López-Fernández M.: Rayleigh-Taylor breakdown for the Muskat problem with applications to water waves. Ann. Math. 175, 909-948 (2012).
8. Castro A., Córdoba D., Fefferman C., Gancedo F., López-Fernández M.: Turning waves and breakdown for incompressible flows. Proc. Natl. Acad. Sci. 108, 4754-4759 (2011).
9. Castro, A., Córdoba, D., Fefferman, C., Gancedo, F., Gómez-Serrano, M.: Splash singularity for water waves. http://arxiv. org/abs/1106. 2120v2 [math. AP], 2011.
10. Castro, A., Córdoba, D., Fefferman, C., Gancedo, F., Gómez-Serrano, M.: Finite time singularities for the free boundary incompressible Euler equations. Ann. Math. (2013) (to appear). http://arxiv. org/abs/1112. 2170v3 [math. AP].
11. Craig W.: An existence theory for water waves and the Boussinesq and Korteweg-de Vries scaling limits. Commun. Part. Differ. Eqs. 10(8), 787-1003 (1985).
12. Christodoulou D., Lindblad H.: On the motion of the free surface of a liquid. Commun. Pure Appl. Math. 53, 1536-1602 (2000).
13. Cheng A., Coutand D., Shkoller S.: On the motion of vortex sheets with surface tension in the 3D euler equations with vorticity. Commun. Pure Appl. Math. 61, 1715-1752 (2008).
14. Coutand D., Shkoller S.: Well-posedness of the free-surface incompressible Euler equations with or without surface tension. J. Am. Math. Soc. 20, 829-930 (2007).
15. Coutand D., Shkoller S.: A simple proof of well-posedness for the free-surface incompressible Euler equations. Disc. Cont. Dyn. Syst. Ser. S 3, 429-449 (2010).
16. Coutand D., Shkoller S.: Well-posedness in smooth function spaces for the moving-boundary 3-D compressible Euler equations in physical vacuum. Arch. Rational Mech. Anal. 206, 515-616 (2012).
17. Ebin D.: The equations of motion of a perfect fluid with free boundary are not well posed. Comm. Partial Differ. Equ. 10, 1175-1201 (1987).
18. Germain P., Masmoudi N., Shatah J.: Global solutions for the gravity water waves equation in dimension 3. C. R. Math. Acad. Sci. Paris 347, 897-902 (2009).
19. Howison S. D., Ockendon J. R., Oliver J. M., Purvis R., Smith F. T.: Droplet impact on a thin fluid layer. J. Fluid Mech. 542, 1-23 (2005).
20. Lannes D.: Well-posedness of the water-waves equations. J. Am. Math. Soc. 18, 605-654 (2005).
21. Lindblad H.: Well-posedness for the motion of an incompressible liquid with free surface boundary. Ann. Math. 162, 109-194 (2005).
22. Nalimov V. I.: The Cauchy-Poisson problem (in Russian), Dynamika Splosh. Sredy 18, 104-210 (1974).
23. Og̃uz N. H., Prosperetti A.: Bubble entrainment by the impact of drops on liquid surfaces. J. Fluid Mech. 219, 143-179 (1990).
24. Rayleigh L.: On the instability of jets. Proc. Lond. Math. Soc. s1-10(1), 4-13 (1878).
25. Temam, R.: Navier-Stokes equations. Theory and numerical analysis. 3rd edn. In: Studies in Mathematics and its Applications, Vol. 2. Amsterdam: North-Holland Publishing Co., 1984.
26. Shatah J., Zeng C.: Geometry and a priori estimates for free boundary problems of the Euler equation. Commun. Pure Appl. Math. 61, 698-744 (2008).
27. Taylor G.: The instability of liquid surfaces when accelerated in a direction perpendicular to their planes I. Proc. R. Soc. Lond. Ser. A. 201, 192-196 (1950).
28. Worthington, M.: The splash of a drop and allied phenomena. Washington, DC: US Govt. Printing Office, Smithsonian Report, 1894. (Reprinted with additions: A Study of Splashes. London: Macmillan, 1963).
29. Wu S.: Well-posedness in Sobolev spaces of the full water wave problem in 2-D. Invent. Math. 130, 39-72 (1997).
30. Wu S.: Well-posedness in Sobolev spaces of the full water wave problem in 3-D. J. Am. Math. Soc. 12, 445-495 (1999).
31. Wu S.: Almost global wellposedness of the 2-D full water wave problem. Invent. Math. 177, 45-135 (2009).
32. Wu S.: Global wellposedness of the 3-D full water wave problem. Invent. Math. 184, 125-220 (2011).
33. Yosihara H.: Gravity waves on the free surface of an incompressible perfect fluid. Publ. RIMS Kyoto Univ. 18, 49-96 (1982).
34. Zhang P., Zhang Z.: On the free boundary problem of three-dimensional incompressible Euler equations. Commun. Pure Appl. Math. 61, 877-940 (2008).