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Blowing Up Solutions to the Zakharov System for Langmuir Waves

  • Yuri Cher
  • Magdalena Czubak
  • Catherine SulemEmail author
Chapter
  • 784 Downloads
Part of the CRM Series in Mathematical Physics book series (CRM)

Abstract

Langmuir waves take place in a quasi-neutral plasma and are modeled by the Zakharov system. The phenomenon of collapse, described by blowing up solutions, plays a central role in their dynamics. We present in this article a review of the main mathematical properties of blowing up solutions. They include conditions for blowup in finite or infinite time, description of self-similar singular solutions and lower bounds for the rate of blowup of certain norms associated with the solutions.

Keywords

Energy Space Blowup Rate Langmuir Wave Blowup Solution Strichartz Estimate 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Notes

Acknowledgements

MC is partially supported by grant #246255 from the Simons Foundation. CS is partially supported by NSERC through grant number 46179–13 and Simons Foundation Fellowship #265059.

References

  1. 1.
    V.E. Zakharov, Sov. Phys. JETP 35(5), 908 (1972)ADSGoogle Scholar
  2. 2.
    V.E. Zakharov, A.F. Mastryukov, V.S. Synakh, Sov. J. Plasma Phys. 1, 339 (1975)ADSGoogle Scholar
  3. 3.
    P.A. Robinson, Rev. Mod. Phys. 69, 507 (1997)CrossRefADSGoogle Scholar
  4. 4.
    L. Bergé, Phys. Rep. 303(5–6), 259 (1998)MathSciNetCrossRefADSGoogle Scholar
  5. 5.
    C. Sulem, P.L. Sulem, The Nonlinear Schrödinger Equation. Self-focusing and Wave Collapse. Applied Mathematical Sciences, vol. 139 (Springer, New York, 1999)Google Scholar
  6. 6.
    B. Texier, Arch. Ration. Mech. Anal. 184(1), 121 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    C. Sulem, P.L. Sulem, C. R. Acad. Sci. Paris Sér. A-B 289(3), A173 (1979)MathSciNetADSGoogle Scholar
  8. 8.
    H. Added, S. Added, C. R. Acad. Sci. Paris Sér. I Math. 299(12), 551 (1984)MathSciNetzbMATHGoogle Scholar
  9. 9.
    S.H. Schochet, M.I. Weinstein, Commun. Math. Phys. 106(4), 569 (1986)MathSciNetCrossRefADSzbMATHGoogle Scholar
  10. 10.
    T. Ozawa, Y. Tsutsumi, Publ. Res. Inst. Math. Sci. 28(3), 329 (1992)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    L. Glangetas, F. Merle, Commun. Math. Phys. 160(1), 173 (1994)MathSciNetCrossRefADSzbMATHGoogle Scholar
  12. 12.
    L. Glangetas, F. Merle, Commun. Math. Phys. 160(2), 349 (1994)MathSciNetCrossRefADSzbMATHGoogle Scholar
  13. 13.
    J. Bourgain, J. Colliander, Int. Math. Res. Not. 1996(11), 515 (1996)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    J. Ginibre, Y. Tsutsumi, G. Velo, J. Funct. Anal. 151(2), 384 (1997)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    N. Tzvetkov, Differ. Integr. Equ. 13(4–6), 423 (2000)MathSciNetzbMATHGoogle Scholar
  16. 16.
    H. Pecher, Int. Math. Res. Not. 2001(19), 1027 (2001)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    J. Colliander, J. Holmer, N. Tzirakis, Trans. Am. Math. Soc. 360(9), 4619 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    I. Bejenaru, S. Herr, J. Holmer, D. Tataru, Nonlinearity 22(5), 1063 (2009)MathSciNetCrossRefADSzbMATHGoogle Scholar
  19. 19.
    I. Bejenaru, S. Herr, J. Funct. Anal. 261(2), 478 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    J. Ginibre, G. Velo, Hokkaido Math. J. 35(4), 865 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Z. Hani, F. Pusateri, J. Shatah, Commun. Math. Phys. 322(3), 731 (2013)MathSciNetCrossRefADSzbMATHGoogle Scholar
  22. 22.
    F. Merle, Commun. Pure Appl. Math. 49(8), 765 (1996)MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    W.A. Strauss, Commun. Math. Phys. 55(2), 149 (1977)CrossRefADSzbMATHGoogle Scholar
  24. 24.
    Z. Guo, K. Nakanishi, Int. Math. Res. Not. IMRN 2014(9), 2327 (2014)MathSciNetzbMATHGoogle Scholar
  25. 25.
    F. Merle, Commun. Math. Phys. 175(2), 433 (1996)MathSciNetCrossRefADSzbMATHGoogle Scholar
  26. 26.
    V.E. Zakharov, L.N. Shur, Sov. Phys. JETP 54(6), 1064 (1981)Google Scholar
  27. 27.
    H. Berestycki, P.L. Lions, Arch. Ration. Mech. Anal. 82(4), 313 (1983)MathSciNetzbMATHGoogle Scholar
  28. 28.
    H. Berestycki, P.L. Lions, Arch. Ration. Mech. Anal. 82(4), 347 (1983)MathSciNetzbMATHGoogle Scholar
  29. 29.
    L. Bergé, Luc, G. Pelletier, D. Pesme, Phys. Rev. A 42(8), 4962 (1990)Google Scholar
  30. 30.
    M. Landman, G.C. Papanicolaou, C. Sulem, P.L. Sulem, X.P. Wang, Phys. Rev. A 46(12), 7869 (1992)CrossRefADSGoogle Scholar
  31. 31.
    M.I. Weinstein, Commun. Math. Phys. 87(4), 567 (1982/1983)Google Scholar
  32. 32.
    O.B. Budneva, V.E. Zakharov, V.S. Synakh, Sov. J. Plasma Phys. 1, 335 (1975)ADSGoogle Scholar
  33. 33.
    V. Masselin, Adv. Differ. Equ. 6(10), 1153 (2001)MathSciNetzbMATHGoogle Scholar
  34. 34.
    J. Holmer, Electron. J. Differ. Equ. 24, (2007)Google Scholar
  35. 35.
    J. Colliander, M. Czubak, C. Sulem, J. Hyperbolic Differ. Equ. 10(3), 523 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  36. 36.
    F.B. Weissler, Isr. J. Math. 38(1–2), 29 (1981)MathSciNetCrossRefzbMATHGoogle Scholar
  37. 37.
    T. Cazenave, F.B. Weissler, Nonlinear Anal. 14(10), 807 (1990)MathSciNetCrossRefzbMATHGoogle Scholar
  38. 38.
    J. Bourgain, Geom. Funct. Anal. 3(2), 107 (1993)MathSciNetCrossRefzbMATHGoogle Scholar
  39. 39.
    F. Haas, P.K. Shukla, Phys. Rev. E 79(6), 066402 (2009)CrossRefADSGoogle Scholar
  40. 40.
    G. Simpson, C. Sulem, P.L. Sulem, Phys. Rev. E 80(5), 056405 (2009)CrossRefADSGoogle Scholar

Copyright information

© Springer International Publishing Switzerland 2016

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of TorontoTorontoCanada
  2. 2.Department of Mathematical SciencesBinghamton UniversityBinghamtonUSA

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