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Water Waves With or Without Surface Tension

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Handbook of Mathematical Analysis in Mechanics of Viscous Fluids

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Abstract

In this survey article, we present results from the last several decades on several systems of PDE that model water waves. Some of those results provide short-time existence of smooth solutions; other results establish existence of smooth solutions for all time; still other results assert that singularities form in finite time.

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References

  1. H. Abels, The initial-value problem for the Navier-Stokes equations with a free surface in L q-Sobolev spaces. Adv. Differ. Equ. 10(1), 4564 (2005)

    Google Scholar 

  2. T. Alazard, N. Burq, C. Zuily, On the water-wave equations with surface tension. Duke Math. J. 158(3), 413499 (2011)

    Google Scholar 

  3. T. Alazard, N. Burq, C. Zuily, Strichartz estimates for water waves. Ann. Sci. Ec. Norm. Super. (4) 44(5), 855–903 (2011)

    Google Scholar 

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

    Google Scholar 

  5. T. Alazard, J.M. Delort, Sobolev estimates for two dimensional gravity water waves. Asterisque N. 374(viii), p. 241 (2015)

    Google Scholar 

  6. D.M. Ambrose, N. Masmoudi, The zero surface tension limit of two-dimensional water waves. Commun. Pure Appl. Math. 58(10), 12871315 (2005)

    Google Scholar 

  7. C.J. Amick, L.E. Fraenkel, J.F. Toland, On the Stokes conjecture for the wave of extreme form. Acta Math. 148, 193–214 (1982)

    Article  MathSciNet  MATH  Google Scholar 

  8. J. Beale, The initial value problem for the Navier-Stokes equations with a free surface. Commun. Pure Appl. Math. 34(3), 359–392 (1981)

    Article  MathSciNet  MATH  Google Scholar 

  9. J. Beale, Large-time regularity of viscous surface waves. Arch. Ration. Mech. Anal. 84(4), 307–352 (1984)

    Article  MathSciNet  MATH  Google Scholar 

  10. J. Beale, T. Nishida, Large-time behavior of viscous surface waves, in Recent Topics in Nonlinear PDE, II, Sendai, 1984. North-Holland Mathematics Studies, vol. 128 (North-Holland, Amsterdam, 1985), pp. 1–14

    Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  12. J.T. Beale, T.Y. Hou, J. Lowengrub, Convergence of a boundary integral method for water waves. SIAM J. Numer. Anal. 33(5), 1797–1843 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  13. A. Castro, D. Córdoba, C. Fefferman, F. Gancedo, J. Gómez-Serrano, Splash singularity for water waves. Proc. Natl. Acad. Sci. 109(3), 733–738 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  14. A. Castro, D. Córdoba, C.L. Fefferman, F. Gancedo, J. Gómez-Serrano, Finite time singularities for water waves with surface tension. J. Math. Phys. 53(11), 115622,26 (2012)

    Google Scholar 

  15. A. Castro, D. Córdoba, C. Fefferman, F. Gancedo, M. López-Fernández, Rayleigh-Taylor breakdown for the Muskat problem with applications to water waves. Ann. Math. 175(2), 909–948 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  16. A. Castro, D. Córdoba, C. Fefferman, F. Gancedo, J. Gómez-Serrano, Finite time singularities for the free boundary incompressible Euler equations. Ann. Math. 178(3), 1061–1134 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  17. A. Castro, D. Córdoba, C. Fefferman, F. Gancedo, J. Gómez-Serrano. Splash singularities for the free boundary Navier-Stokes equations. Preprint arXiv:1504.02775

    Google Scholar 

  18. A. Castro, D. Córdoba, C.L. Fefferman, F. Gancedo, J. Gómez-Serrano, Structural stability for the splash singularities of the water wave problem. Discrete Contin. Dyn. Syst. 34(12), 4997–5043 (2014)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  20. P. Constantin, On the Euler equations of incompressible fluids. Bull. Am. Math. Soc. (N.S.) 44(4), 603–621 (2007)

    Google Scholar 

  21. D. Córdoba, T. Pernas-Casta\(\tilde{n}\) o, Non-splat singularity for the one-phase Muskat problem. Trans. Am. Math. Soc. 369(1), 711–754 (2017, to appear)

    Google Scholar 

  22. A. Córdoba, D. Córdoba, F. Gancedo, Interface evolution: water waves in 2-D. Adv. Math. 223(1), 120–173 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  23. D. Córdoba, A. Enciso, N. Grubic, On the existence of stationary splash singularities for the Euler equations. Adv. Math. 288, 922–941 (2016)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  26. D. Coutand, S. Shkoller, On the splash singularity for the free-surface of a Navier-Stokes fluid. Arxiv preprint arXiv:1505.01929

    Google Scholar 

  27. D. Coutand, S. Shkoller, On the impossibility of finite-time splash singularities for vortex sheets. Arxiv preprint arXiv:1407.1479 (2014)

    Google Scholar 

  28. W. Craig, An existence theory for water waves and the Boussinesq and Korteweg-de Vries scaling limits. Commun. Partial Differ. Equ. 10(8), 787–1003 (1985)

    Article  MathSciNet  MATH  Google Scholar 

  29. G.D. Crapper, An exact solution for progressive capillary waves of arbitrary amplitude. J. Fluid Mech. 2, 532–540 (1957)

    Article  MathSciNet  MATH  Google Scholar 

  30. Y. Deng, A.D. Ionescu, B. Pausader, F. Pusateri, Global solutions of the gravity-capillary water waves system in 3D. Preprint arXiv:1601.05685

    Google Scholar 

  31. C. Fefferman, A. Ionescu, V. Lie, On the absence of splash singularities in the case of two-fluid interfaces. Duke Math. 163(3), 417–462 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  32. P. Germain, N. Masmoudi, J. Shatah, Global solutions for the gravity water waves equation in dimension 3. C. R. Math. Acad. Sci. Paris 347(15–16), 897–902 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  33. P. Germain, N. Masmoudi, J. Shatah, Global solutions for capillary waves equation in dimension 3. Commun. Pure Appl. Math. 68(4), 625–687 (2015)

    Article  MATH  Google Scholar 

  34. Y. Guo, I. Tice, Local well-posedness of the viscous surface wave problem without surface tension. Anal. PDE 6(2), 287–369 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  35. Y. Guo, I. Tice, Decay of viscous surface waves without surface tension in horizontally infinite domains. Anal. PDE 6(6), 1429–1533 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  36. Y. Guo, I. Tice, Almost exponential decay of periodic viscous surface waves without surface tension. ARch. Ration. Mech. Anal. 207(2), 459–531 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  37. Y. Hataya, Globlal solution of two-layer Navier-Stokes flow. Nonlinear Anal. 63, 1409–1420 (2005)

    Article  Google Scholar 

  38. Y. Hataya, Decaying solution of a Navier-Stokes flow without surface tension. J. Math. Kyoto Univ. 49(4), 691–717 (2009)

    MathSciNet  MATH  Google Scholar 

  39. M. Ifrim, D. Tataru, Two dimensional water waves in holomorphic coordinates II: global solutions. Bull. Soc. Math. Fr. 144(2), 369–394 (2016)

    MathSciNet  MATH  Google Scholar 

  40. M. Ifrim, D. Tataru, The lifespan of small data solutions in two dimensional capillary water waves. Preprint arXiv:1406.5471

    Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  43. D. Lannes, The Water Waves Problem: Mathematical Analysis and Asymptotics. Mathematical Surveys and Monographs (American Mathematical Society, Providence, 2013)

    Book  MATH  Google Scholar 

  44. J. Leray, Sur le mouvement d’un liquide visqueux emplissant l’espace. Acta Math. 63(1), 193–248 (1934)

    Article  MathSciNet  Google Scholar 

  45. T. Levi-Civita, Determinationrigoureuse des ondes permanentes d’ampleur finie. Math. Ann. 93, 264–314 (1925)

    Article  MathSciNet  MATH  Google Scholar 

  46. H. Lindblad, Well-posedness for the motion of an incompressible liquid with free surface boundary. Ann. Math. (2) 162(1), 109–194 (2005)

    Google Scholar 

  47. R. Moore, F. Bierbaum, Methods and Applications of Interval Analysis, vol. 2 (Society for Industrial & Applied Mathematics, Philadelphia, 1979)

    Book  Google Scholar 

  48. M. Muskat, Two fluid systems in porous media. The encroachment of water into an oil sand. Physics 5, 250–264 (1934)

    MATH  Google Scholar 

  49. V.I. Nalimov, The Cauchy-Poisson problem. Dinamika Splo \(\breve{s}\) n. Sredy, (Vyp. 18 Dinamika Zidkost. so Svobod. Granicami), 104–210 (1974)

    Google Scholar 

  50. A.I. Nekrasov, Tocnaya teoriya voln ustanovivsegocya vida na poverhnosti tyazeloy zidkosti. Izdat. Akad. Nauk SSSR, Moscow (1951)

    Google Scholar 

  51. P.I. Plotnikov, Justification of the Stokes conjecture in the theory of surface waves. Dinamika Sploshn. Sredy 57, 41–76 (1982)

    MathSciNet  MATH  Google Scholar 

  52. P.G. Saffman, G. Taylor, The penetration of a fluid into a porous medium or Hele-Shaw cell containing a more viscous liquid. Proc. R. Soc. Lond. Ser. A 245, 312–329 (1958)

    Article  MathSciNet  MATH  Google Scholar 

  53. V.A. Solonnikov, Solvability of a problem on the motion of a viscous incompressible fluid that is bounded by a free surface. Maht. USSR-Izu 6, 1323–1358 (1978)

    MATH  Google Scholar 

  54. V.A. Solonnikov, On an initial boundary value problem for the Stokes systems arising in the study of a problem with a free boundary. Proc. Steklov Inst. Maht. 3, 191–239 (1991)

    MATH  Google Scholar 

  55. V.A. Solonnikov, Solvability of the problem of evolution of a viscous incompressible fluid bounded by a free surface on a finite time interval. St. Petersburg Math. J. 3(1), 189–220 (1992)

    MathSciNet  MATH  Google Scholar 

  56. G.G. Stokes, Considerations relative to the greatest height of oscillatory irrota- tional waves which can be propagated without change of form, in Mathematical and Physical Papers, ed. by G.G. Stokes, J. Larmor, J.W. Strutt Rayleigh (Cambridge University Press, Cambridge, 1880)

    Google Scholar 

  57. D.L.G. Sylvester, Large time existence of small viscous surface waves without surface tension. Commun. Partial Differ. Equ. 15(6), 823–903 (1990)

    Article  MathSciNet  MATH  Google Scholar 

  58. E. Varvaruca, G.S. Weiss, A geometric approach to generalized Stokes conjectures. Acta Math. 206(2), 363–403 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  59. S. Wu, Well-posedness in Sobolev spaces of the full water wave problem in 2-D. Invent. Math. 130(1), 39–72 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  60. S. Wu, Well-posedness in Sobolev spaces of the full water wave problem in 3-D. J. Am. Math. Soc. 12(2), 445–495 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  61. S. Wu, Almost global wellposedness of the 2-D full water wave problem. Invent. Math. 177(1), 45–135 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  62. S. Wu, Global wellposedness of the 3-D full water wave problem. Invent. Math. 184(1), 125–220 (2011)

    Article  MathSciNet  MATH  Google Scholar 

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

    Google Scholar 

  64. H. Yosihara, Gravity waves on the free surface of an incompressible perfect fluid of finite depth. Publ. Res. Inst. Math. Sci. 18(1), 49–96 (1982)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

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Acknowledgements

We thank Tania Pernas for providing the figures of the paper. DC was partially supported by the grants MTM2014-59488-P (Spain) and SEV-2015-556. CF was partially supported by NSF grant DMS-1265524, AFOSR grant FA9550-12-1-0425, and Grant No 2014055 from the United States-Israel Binational Science Foundation (BSF).

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Authors and Affiliations

  1. Instituto de Ciencias Matemáticas, Consejo Superior de Investigaciones Científicas, C/ Nicolas Cabrera, 13-15, 28049, Madrid, Spain

    Diego Córdoba

  2. Department of Mathematics, Princeton University, 1102 Fine Hall, Washington Road, 08544-1000, Princeton, NJ, USA

    Charles Fefferman

Authors
  1. Diego Córdoba
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  2. Charles Fefferman
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Correspondence to Diego Córdoba .

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Editors and Affiliations

  1. Tokyo, Tokyo, Japan

    Yoshikazu Giga

  2. Toulon, Var, France

    Antonin Novotny

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Córdoba, D., Fefferman, C. (2016). Water Waves With or Without Surface Tension. In: Giga, Y., Novotny, A. (eds) Handbook of Mathematical Analysis in Mechanics of Viscous Fluids. Springer, Cham. https://doi.org/10.1007/978-3-319-10151-4_30-1

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  • DOI: https://doi.org/10.1007/978-3-319-10151-4_30-1

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