Well-posedness of two-dimensional hydroelastic waves with mass

Journal of Differential Equations

Volumn 262, Issue 9, 2017, Pages 4656-4699

  Liu, Shunlian ^a   Ambrose, David M ^a  

^a DREXEL UNIVERSITY   (United States)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 85008506415     PISSN: 00220396     EISSN: 10902732     Source Type: Journal    
DOI: 10.1016/j.jde.2016.12.016     Document Type: Article

References (32)

1. [1] Alben, S., Shelley, M.J., Flapping states of a flag in an inviscid fluid: bistability and the transition to chaos. Phys. Rev. Lett., 100(7), 2008, 074301.
2. [2] Ambrose, D.M., Well-posedness of Vortex Sheets with Surface Tension. 2002, ProQuest LLC, Ann Arbor, MI Ph.D. thesis, Duke University.
3. [3] Ambrose, D.M., Well-posedness of vortex sheets with surface tension. SIAM J. Math. Anal. 35:1 (2003), 211–244 (electronic).
4. [4] Ambrose, D.M., Well-posedness of two-phase Hele–Shaw flow without surface tension. European J. Appl. Math. 15:5 (2004), 597–607.
5. [5] Ambrose, D.M., The zero surface tension limit of two-dimensional interfacial Darcy flow. J. Math. Fluid Mech. 16:1 (2014), 105–143.
6. [6] Ambrose, D.M., Masmoudi, N., The zero surface tension limit of two-dimensional water waves. Comm. Pure Appl. Math. 58:10 (2005), 1287–1315.
7. [7] Ambrose, D.M., Siegel, M., Well-posedness of two-dimensional hydroelastic waves. Proc. Roy. Soc. Edinburgh Sect. A, 2018 in press.
8. [8] Baker, G.R., Meiron, D.I., Orszag, S.A., Generalized vortex methods for free-surface flow problems. J. Fluid Mech. 123 (1982), 477–501.
9. [9] Baldi, P., Toland, J.F., Bifurcation and secondary bifurcation of heavy periodic hydroelastic travelling waves. Interfaces Free Bound. 12:1 (2010), 1–22.
10. [10] Beale, J.T., Hou, T.Y., Lowengrub, J.S., Growth rates for the linearized motion of fluid interfaces away from equilibrium. Comm. Pure Appl. Math. 46:9 (1993), 1269–1301.
11. [11] Christianson, H., Hur, V.M., Staffilani, G., Strichartz estimates for the water-wave problem with surface tension. Comm. Partial Differential Equations 35:12 (2010), 2195–2252.
12. [12] Córdoba, A., Córdoba, D., Gancedo, F., Interface evolution: water waves in 2-D. Adv. Math. 223:1 (2010), 120–173.
13. [13] Córdoba, A., Córdoba, D., Gancedo, F., Interface evolution: the Hele–Shaw and Muskat problems. Ann. of Math. (2) 173:1 (2011), 477–542.
14. [14] Düll, W.-P., Validity of the Korteweg–de Vries approximation for the two-dimensional water wave problem in the arc length formulation. Comm. Pure Appl. Math. 65:3 (2012), 381–429.
15. [15] Guo, Y., Hallstrom, C., Spirn, D., Dynamics near unstable, interfacial fluids. Comm. Math. Phys. 270:3 (2007), 635–689.
16. [16] Guyenne, P., Părău, E.I., Computations of fully nonlinear hydroelastic solitary waves on deep water. J. Fluid Mech. 713 (2012), 307–329.
17. [17] Helson, H., Harmonic Analysis. 1983, Addison–Wesley Publishing Company Advanced Book Program, Reading, MA.
18. [18] Hou, T.Y., Lowengrub, J.S., Shelley, M.J., Removing the stiffness from interfacial flows with surface tension. J. Comput. Phys. 114:2 (1994), 312–338.
19. [19] Hou, T.Y., Lowengrub, J.S., Shelley, M.J., The long-time motion of vortex sheets with surface tension. Phys. Fluids 9:7 (1997), 1933–1954.
20. [20] Hunter, J.K., Nachtergaele, B., Applied Analysis. 2001, World Scientific Publishing Co., Inc., River Edge, NJ.
21. [21] Majda, A.J., Bertozzi, A.L., Vorticity and Incompressible Flow. Cambridge Texts Appl. Math., vol. 27, 2002, Cambridge University Press, Cambridge.
22. [22] Milewski, P.A., Wang, Z., Three dimensional flexural-gravity waves. Stud. Appl. Math. 131:2 (2013), 135–148.
23. [23] Plotnikov, P.I., Toland, J.F., Modelling nonlinear hydroelastic waves. Philos. Trans. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci. 369:1947 (2011), 2942–2956.
24. [24] Squire, V.A., Dugan, J.P., Wadhams, P., Rottier, P.J., Liu, A.K., Of ocean waves and sea ice. Annu. Rev. Fluid Mech. 27:1 (1995), 115–168.
25. [25] Taylor, M.E., Partial Differential Equations III. Nonlinear Equations. second ed. Appl. Math. Sci., vol. 117, 2011, Springer, New York.
26. [26] Toland, J.F., Heavy hydroelastic travelling waves. Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci. 463:2085 (2007), 2371–2397.
27. [27] Toland, J.F., Steady periodic hydroelastic waves. Arch. Ration. Mech. Anal. 189:2 (2008), 325–362.
28. [28] Wang, Z., Părău, E.I., Milewski, P.A., Vanden-Broeck, J.-M., Numerical study of interfacial solitary waves propagating under an elastic sheet. Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci., 470(2168), 2014, 20140111.
29. [29] Wang, Z., Vanden-Broeck, J.-M., Milewski, P.A., Two-dimensional flexural-gravity waves of finite amplitude in deep water. IMA J. Appl. Math. 78:4 (2013), 750–761.
30. [30] Wu, S., Well-posedness in Sobolev spaces of the full water wave problem in 2-D. Invent. Math. 130:1 (1997), 39–72.
31. [31] Ye, J., Tanveer, S., Global existence for a translating near-circular Hele–Shaw bubble with surface tension. SIAM J. Math. Anal. 43:1 (2011), 457–506.
32. [32] Ye, J., Tanveer, S., Global solutions for a two-phase Hele–Shaw bubble for a near-circular initial shape. Complex Var. Elliptic Equ. 57:1 (2012), 23–61.