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Long Time Behavior of 2D Water Waves with Point Vortices

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Abstract

In this paper, we study the motion of the two dimensional inviscid incompressible, infinite depth water waves with point vortices in the fluid. We show that the Taylor sign condition \(-\frac{\partial P}{\partial \vec {n}}\geqslant 0\) can fail if the point vortices are sufficiently close to the free boundary, so the water waves could be subject to Taylor instability. Assuming the Taylor sign condition, we prove that the water wave system is locally wellposed in Sobolev spaces. Moreover, we show that if the water waves is symmetric with a symmetric vortex pair traveling downward initially, then the free interface remains smooth for a long time, and for initial data of size \(\epsilon \ll 1\), the lifespan is at least \(O(\epsilon ^{-2})\).

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Notes

  1. Indeed, \(a|z_{\alpha }|=-\frac{\partial P}{\partial \vec {n}}\Big |_{\Sigma (t)}\).

  2. The assumption that \((Z_{\alpha }-1, {\mathcal {D}}_tZ)\in H^3\times H^3\) is of course not optimal.

  3. This is a conditional result, we assume a priori the existence of the solution.

  4. \(f_1\) is not arbitrary, it must satisfy some compatible condition. See (5.23).

  5. Throughout this paper, by first order terms, we mean those quantities with size \(O(\epsilon )\); by second order terms, we mean quantities of size \(O(\epsilon ^2)\); and vice versa.

  6. The reason that we apply \(\frac{1}{2}(I+{\mathbb {H}})\) on both sides of (5.15) is that it is easier to prove the equivalence of (5.16) and (5.3). To show that (5.15) is equivalent to (5.3), we need to prove that a solution F to (5.15) satisfies \((I-{\mathbb {H}})F=0\), which means that \(F=(I+{\mathbb {H}})g\) for some function g.

  7. Note that \(\delta _0\geqslant 3\).

References

  1. Alazard, T., Delort, J.-M.: Global solutions and asymptotic behavior for two dimensional gravity water waves. Ann. Sci. Éc. Norm. Supér. (4) 48(5), 1149–1238 (2015)

    MathSciNet  MATH  Google Scholar 

  2. Alazard, T., Burq, N., Zuily, C.: On the Cauchy problem for gravity water waves. Invent. Math. 198(1), 71–163 (2014)

    ADS  MathSciNet  MATH  Google Scholar 

  3. Ambrose, D., Masmoudi, N.: The zero surface tension limit two-dimensional water waves. Commun. Pure Appl. Math. 58(10), 1287–1315 (2005)

    MathSciNet  MATH  Google Scholar 

  4. Beale, J.T., Hou, T.Y., Lowengrub, J.S.: Growth rates for the linearized motion of fluid interfaces away from equilibrium. Commun. Pure Appl. Math. 46(9), 1269–1301 (1993)

    MathSciNet  MATH  Google Scholar 

  5. Bieri, L., Miao, S., Shahshahani, S., Sijue, W.: On the motion of a self-gravitating incompressible fluid with free boundary. Commun. Math. Phys. 355(1), 161–243 (2017)

    ADS  MathSciNet  MATH  Google Scholar 

  6. Birkhoff, G.: Helmholtz and Taylor instability. Proc. Symp. Appl. Math 13, 55–76 (1962)

    MathSciNet  MATH  Google Scholar 

  7. Calderón, A.P.: Commutators of singular integral operators. Proc. Natl. Acad. Sci. U. S. A. 53(5), 1092 (1965)

    ADS  MathSciNet  MATH  Google Scholar 

  8. Castro, A., Córboda, D., Fefferman, C., Gancedo, F., Gómez-Serrano, J.: Finite time singularities for the free boundary incompressible euler equations. Ann. Math. 1061–1134 (2013)

  9. Castro, A., Córdoba, D., Fefferman, C., Gancedo, F., Gómez-Serrano, J.: Finite time singularities for water waves with surface tension. J. Math. Phys. 53(11), 115622 (2012)

    ADS  MathSciNet  MATH  Google Scholar 

  10. Chang, K.-A., Hsu, T.-J., Liu, P.L.-F.: Vortex generation and evolution in water waves propagating over a submerged rectangular obstacle: Part i. solitary waves. Coast. Eng. 44(1), 13–36 (2001)

    Google Scholar 

  11. Christodoulou, D., Lindblad, H.: On the motion of the free surface of a liquid. Commun. Pure Appl. Math. 53(12), 1536–1602 (2000)

    MathSciNet  MATH  Google Scholar 

  12. Coifman, R.R., David, G., Meyer, Y.: La solution des conjectures de calderón. Adv. Math. 48(2), 144–148 (1983)

    MathSciNet  MATH  Google Scholar 

  13. Coifman, R.R., McIntosh, A., Meyer, Y.: L’intégrale de cauchy définit un opérateur borné sur l2 pour les courbes lipschitziennes. Ann. Math. pp. 361–387 (1982)

  14. Coutand, D., Shkoller, S.: Well-posedness of the free-surface incompressible Euler equations with or without surface tension. J. Am. Math. Soc. 20(3), 829–930 (2007)

    MathSciNet  MATH  Google Scholar 

  15. Coutand, D., Shkoller, S.: On the finite-time splash and splat singularities for the 3-d free-surface Euler equations. Commun. Math. Phys. 325(1), 143–183 (2014)

    ADS  MathSciNet  MATH  Google Scholar 

  16. Coutand, D., Shkoller, S.: On the impossibility of finite-time splash singularities for vortex sheets. Arch. Ration. Mech. Anal. 221(2), 987–1033 (2016)

    MathSciNet  MATH  Google Scholar 

  17. Craig, W.: An existence theory for water waves and the Boussinesq and Korteweg-Devries scaling limits. Commun. P.D.E 10(8), 787–1003 (1985)

    MATH  Google Scholar 

  18. Curtis, C.W., Kalisch, H.: Vortex dynamics in nonlinear free surface flows. Phys. Fluids 29(3), 032101 (2017)

    ADS  Google Scholar 

  19. Dalrymple, R.A., Rogers, B.D.: Numerical modeling of water waves with the sph method. Coast. Eng. 53(2–3), 141–147 (2006)

    Google Scholar 

  20. David, G.: Opérateurs intégraux singuliers sur certaines courbes du plan complexe. Annales scientifiques de l’École Normale Supérieure 17, 157–189 (1984)

    MathSciNet  MATH  Google Scholar 

  21. Ebin, D.G.: The equations of motion of a perfect fluid with free boundary are not well posed. Commun. Part. Diff. Equ. 12(10), 1175–1201 (1987)

    MathSciNet  MATH  Google Scholar 

  22. Fish, S.: Vortex dynamics in the presence of free surface waves. Phys. Fluids A Fluid Dyn. 3(4), 504–506 (1991)

    ADS  Google Scholar 

  23. Germain, P., Masmoudi, N., Shatah, J.: Global solutions for the gravity water waves equation in dimension 3. Ann. Math. pp. 691–754 (2012)

  24. Ginsberg, D.: On the lifespan of three-dimensional gravity water waves with vorticity. arXiv preprint arXiv:1812.01583, (2018)

  25. Hill, F.M.: A numerical study of the descent of a vortex pair in a stably stratified atmosphere. J. Fluid Mech. 71(1), 1–13 (1975)

    ADS  MATH  Google Scholar 

  26. Hunter, J.K., Ifrim, M., Tataru, D.: Two dimensional water waves in holomorphic coordinates. Commun. Math. Phys. 346(2), 483–552 (2016)

    ADS  MathSciNet  MATH  Google Scholar 

  27. Ifrim, M., Tataru, D.: Two dimensional water waves in holomorphic coordinates II: global solutions. arXiv preprint arXiv:1404.7583 (2014)

  28. Ifrim, M. Tataru, D.: Two dimensional gravity water waves with constant vorticity: I. cubic lifespan. arXiv preprint arXiv:1510.07732 (2015)

  29. Iguchi, T., Tanaka, N., Tani, A.: On a free boundary problem for an incompressible ideal fluid in two space dimensions. Adv. Math. Sci. Appl. 9, 415–472 (1999)

    MathSciNet  MATH  Google Scholar 

  30. Iguchi, T.: Well-posedness of the initial value problem for capillary-gravity waves. Funkcialaj Ekvacioj Serio Internacia 44(2), 219–242 (2001)

    MathSciNet  MATH  Google Scholar 

  31. Ionescu, A.D., Pusateri, F.: Global solutions for the gravity water waves system in 2d. Invent. Math. 199(3), 653–804 (2015)

    ADS  MathSciNet  MATH  Google Scholar 

  32. Lannes, D.: Well-posedness of the water-waves equations. J. Am. Math. Soc. 18(3), 605–654 (2005)

    MathSciNet  MATH  Google Scholar 

  33. Lindblad, H.: Well-posedness for the motion of an incompressible liquid with free surface boundary. Ann. Math. pp. 109–194 (2005)

  34. Marchioro, C., Pulvirenti, M.: Mathematical Theory of Incompressible Nonviscous Fluids, vol. 96. Springer, Berlin (2012)

    MATH  Google Scholar 

  35. Marcus, D.L., Berger, S.A.: The interaction between a counter-rotating vortex pair in vertical ascent and a free surface. Phys. Fluids A Fluid Dyn. 1(12), 1988–2000 (1989)

    ADS  MATH  Google Scholar 

  36. Nalimov, V.I.: The Cauchy–Poisson problem (in russian). Dynamika Splosh Sredy 18, 104–210 (1974)

    Google Scholar 

  37. Ogawa, M., Tani, A.: Free boundary problem for an incompressible ideal fluid with surface tension. Math. Models Methods Appl. Sci. 12(12), 1725–1740 (2002)

    MathSciNet  MATH  Google Scholar 

  38. Ogawa, M., Tani, A.: Incompressible perfect fluid motion with free boundary of finite depth. Adv. Math. Sci. Appl. 13(1), 201–223 (2003)

    MathSciNet  MATH  Google Scholar 

  39. Wu, S., Kinsey, R.: A priori estimates for two-dimensional water waves with angled crests. preprint 2014, arXiv:1406.7573

  40. Shatah, J., Zeng, C.: Geometry and a priori estimates for free boundary problems of the Euler’s equation. arXiv preprint arXiv:math/0608428 (2006)

  41. Taylor, G.I.: The instability of liquid surfaces when accelerated in a direction perpendicular to their planes. I. Proc. R. Soc. Lond. A 201(1065), 192–196 (1950)

    ADS  MathSciNet  MATH  Google Scholar 

  42. Taylor, M.E.: Tools for PDE: Pseudodifferential Operators, Paradifferential Operators, and Layer Potentials, vol. 81. American Mathematical Society, Providence (2007)

    MATH  Google Scholar 

  43. Telste, J.G.: Potential flow about two counter-rotating vortices approaching a free surface. J. Fluid Mech. 201, 259–278 (1989)

    ADS  MathSciNet  Google Scholar 

  44. Totz, N., Sijue, W.: A rigorous justification of the modulation approximation to the 2d full water wave problem. Commun. Math. Phys. 310(3), 817–883 (2012)

    ADS  MathSciNet  MATH  Google Scholar 

  45. Verchota, G.: Layer potentials and regularity for the Dirichlet problem for Laplace’s equation in lipschitz domains. J. Funct. Anal. 59(3), 572–611 (1984)

    MathSciNet  MATH  Google Scholar 

  46. Wang, X.: Global infinite energy solutions for the 2d gravity water waves system. Commun. Pure Appl. Math. 71(1), 90–162 (2018)

    MathSciNet  MATH  Google Scholar 

  47. Willmarth, W.W., Tryggvason, G., Hirsa, A., Yu, D.: Vortex pair generation and interaction with a free surface. Phys. Fluids A Fluid Dyn. 1(2), 170–172 (1989)

    ADS  Google Scholar 

  48. Wu, S.: A blow-up criteria and the existence of 2d gravity water waves with angled crests. preprint 2015, arXiv:1502.05342

  49. Wu, S.: On a class of self-similar 2d surface water waves. preprint 2012, arXiv:1206.2208

  50. Wu, S.: Wellposedness of the 2d full water wave equation in a regime that allows for non-\(c^1\) interfaces. arXiv:1803.08560

  51. Sijue, W.: Well-posedness in sobolev spaces of the full water wave problem in 2-d. Invent. Math. 130(1), 39–72 (1997)

    MathSciNet  MATH  Google Scholar 

  52. Sijue, W.: Well-posedness in sobolev spaces of the full water wave problem in 3-d. J. Am. Math. Soc. 12, 445–495 (1999)

    MathSciNet  MATH  Google Scholar 

  53. Sijue, W.: Almost global wellposedness of the 2-d full water wave problem. Invent. Math. 177(1), 45 (2009)

    MathSciNet  MATH  Google Scholar 

  54. Sijue, W.: Global wellposedness of the 3-d full water wave problem. Invent. Math. 184(1), 125–220 (2011)

    MathSciNet  MATH  Google Scholar 

  55. Sijue, W.: Well-posedness and singularities of the water wave equations. Lect. Theory Water Waves 426, 171 (2016)

    MATH  Google Scholar 

  56. Sijue, W.: Wellposedness and singularities of the water wave equations. Lect. Theory Water Waves 426, 171–202 (2016)

    MathSciNet  MATH  Google Scholar 

  57. Yosihara, H.: Gravity waves on the free surface of an incompressible perfect fluid of finite depth. RIMS Kyoto 18, 49–96 (1982)

    MathSciNet  MATH  Google Scholar 

  58. Zhang, P., Zhang, Z.: On the free boundary problem of three-dimensional incompressible Euler equations. Commun. Pure Appl. Math. 61(7), 877–940 (2008)

    MathSciNet  MATH  Google Scholar 

  59. Zheng, F.: Long-term regularity of 3d gravity water waves. arXiv preprint arXiv:1910.01912 (2019)

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Acknowledgements

The author would like to thank his Ph.D advisor, Prof. Sijue Wu, for introducing him this topic, for many helpful discussions and invaluable comments on this problem. The author would like to thank the referee for many helpful suggestions. The author also would like to thank Prof. Robert Krasny, Prof. Charles Doering, Prof. Stefan Llewellyn Smith, and Prof. Christopher Curtis for providing references on water waves with point vortices. The author would like to Thank Prof. Susan Friedlander for improving the presentation of this manuscript. This work is partially supported by NSF Grant DMS-1361791.

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  1. Department of Mathematics, University of Southern California, Los Angeles, CA, 90089, USA

    Qingtang Su

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Su, Q. Long Time Behavior of 2D Water Waves with Point Vortices. Commun. Math. Phys. 380, 1173–1266 (2020). https://doi.org/10.1007/s00220-020-03885-z

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