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On the limit behavior of the magnetic Zakharov system

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Abstract

In this paper, we prove that the solutions of magnetic Zakharov system converge to those of generalized Zakharov system in Sobolev space H s, s > 3/2, when parameter β → ∞. Further, when parameter (α, β) → ∞ together, we prove that the solutions of magnetic Zakharov system converge to those of Schrödinger equation with magnetic effect in Sobolev space H s, s > 3/2. Moreover, the convergence rate is also obtained.

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References

  1. Added H, Added S. Existence globle de solutions fortes pour les équations de la turbulence de Langmuir en dimension 2. C R Acad Sci Paris, 1986, 299: 551–554

    MathSciNet  Google Scholar 

  2. Added H, Added S. Equations of Langmuir turbulence and nonlinear Schrödinger equation: smoothness and approximation. J Funct Anal, 1988, 79: 183–210

    Article  MathSciNet  MATH  Google Scholar 

  3. Bourgain J, Colliander J. On wellposedness of the Zakharov system. Int Math Res Not, 1996, 11: 515–546

    Article  MathSciNet  Google Scholar 

  4. Bejenaru I, Herr S, Holmer J, et al. On the 2D Zakharov system with L 2 Schrödinger data. Nonlinearity, 2009, 22: 1063–1089

    Article  MathSciNet  MATH  Google Scholar 

  5. Colliander J, Holmer J, Tzirakis N. Low regularity global well-posedness for the Zakharov and Klein-Gordon-Schrödinger systems. Trans Amer Math Soc, 2008, 360: 4619–4638

    Article  MathSciNet  MATH  Google Scholar 

  6. Chase J B. Role of spontaneous magnetic fields in a laser-created deuterium plasma. Phys Fluids, 1973, 16: 1142–1148

    Article  Google Scholar 

  7. Cazenave T, Weissler F B. The Cauchy problem for the critical nonlinear Schrödinger equation in H s. Nonlinear Anal, 1990, 14: 807–836

    Article  MathSciNet  MATH  Google Scholar 

  8. Fang D Y, Pecher H, Zhong S J. Low regularity global well-posedness for the two-dimensional Zakharov system. Analysis (Munich), 2009, 29: 265–282

    MathSciNet  MATH  Google Scholar 

  9. Glangetas L, Merle F. Existence of self-similar blow-up solutions for Zakharov equation in dimension two. Part I. Comm Math Phys, 1994, 160: 173–215

    Article  MathSciNet  MATH  Google Scholar 

  10. Glangetas L, Merle F. Concentration properties of blow-up solutions and instability results for Zakharov equation in dimension two. Part II. Comm Math Phys, 1994, 160: 349–389

    Article  MathSciNet  MATH  Google Scholar 

  11. Ginibre J, Tsutsumi Y, Velo G. On the Cauchy problem for the Zakharov system. J Funct Anal, 1997, 151: 384–436

    Article  MathSciNet  MATH  Google Scholar 

  12. Guo B L, Zhang J J, Pu X K. On the existence and uniqueness of smooth solution for a generalized Zakharov equation. J Math Anal Appl, 2010, 365: 238–253

    Article  MathSciNet  MATH  Google Scholar 

  13. Guo B L, Zhang J J. Well-posedness of the Cauchy problem for the magnetic Zakharov type system. Nonlinearity, 2011, 24: 2191–2210

    Article  MathSciNet  MATH  Google Scholar 

  14. Han L J, Wang B X. Global wellposedness and limit behavior for the generalized finite-depth-fluid equation with small critical data. J Differential Equations, 2008, 245: 2103–2144

    Article  MathSciNet  MATH  Google Scholar 

  15. He X T. The pondermotive force and magnetic field generation effects resulting from the non-linear interaction between plasma-wave and particles (in Chinese). Acta Phys Sinica, 1983, 32: 325–337

    Google Scholar 

  16. Jiang S, Ju Q C, Li H L, et al. Quasi-neutral limit of the full bipolar Euler-Poisson system. Sci China Math, 2010, 53: 3099–3114

    Article  MathSciNet  MATH  Google Scholar 

  17. Kono M, Skoric M M, Haar D T. Spontaneous excitation of magnetic fields and collapse dynamics in a Langmuir plasma. J Plasma Phys, 1981, 26: 123–146

    Article  Google Scholar 

  18. Laurey C. The Cauchy problem for a generalized Zakharov system. Differential Integral Equations, 1995, 8: 105–130

    MathSciNet  MATH  Google Scholar 

  19. Masmoudi N, Nakanishi K. Nonrelativistic limit from Maxwell-Klein-Gordon and Maxwell-Dirac to Poisson-Schrödinger. Int Math Res Not, 2003, 13: 697–734

    Article  MathSciNet  Google Scholar 

  20. Masmoudi N, Nakanishi K. From the Klein-Gordon-Zakharov system to the nonlinear Schrödinger equation. J Hyperbolic Differ Equ, 2005, 2: 975–1008

    Article  MathSciNet  MATH  Google Scholar 

  21. Masmoudi N, Nakanishi K. Energy convergence for singular limits of Zakharov type systems. Invent Math, 2008, 172: 535–583

    Article  MathSciNet  MATH  Google Scholar 

  22. Ozawa T, Tsutsumi Y. The nonlinear Schrödinger limit and the initial layer of the Zakharov equations. Differential Integral Equations, 1992, 5: 721–745

    MathSciNet  MATH  Google Scholar 

  23. Ozawa T, Tsutsumi Y. Existence and smooth effect of solutions for the Zakharov equations. Pub RIMS Kyoto Univ, 1992, 28: 329–361

    Article  MathSciNet  MATH  Google Scholar 

  24. Sulem C, Sulem P L. Quelques résulatats de régularité pour les équation de la turbulence de Langmuir. C R Acad Sci Paris, 1979, 289: 173–176

    MathSciNet  MATH  Google Scholar 

  25. Schochet S H, Weinstein M I. The nonlinear Schrödinger limit of the Zakharov equations governing Langmuir turbulence. Comm Math Phys, 1986, 106: 569–580

    Article  MathSciNet  MATH  Google Scholar 

  26. Stamper J A. Spontanous magnetic fields in Laser-Produced plasmas. Phys Rev Lett, 1971, 26: 1012–1015

    Article  Google Scholar 

  27. Stamper J A, Tidman D A. Magnetic field generation due to radiation pressure in a laser-produced plasma. Phys Fluids, 1973, 16: 2024–2025

    Article  Google Scholar 

  28. Tribel H. Theory of Function Spaces. Boston: Birkhäuser, 1983

    Book  Google Scholar 

  29. Zakharov V E. Collapse of Langmuir waves. Sov Phys JETP, 1972, 35: 908–914

    Google Scholar 

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Authors and Affiliations

  1. Department of of Mathematics and Physics, North China Electric Power University, Beijing, 102206, China

    LiJia Han

  2. College of Mathematics, Physics and Information Engineering, Jiaxing University, Jiaxing, 314001, China

    JingJun Zhang

  3. College of Mathematics and Software Science, Sichuan Normal University, Chengdu, 610068, China

    ZaiHui Gan

  4. Institute of Applied Physics and Computational Mathematics, Beijing, 100088, China

    BoLing Guo

Authors
  1. LiJia Han
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  2. JingJun Zhang
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  3. ZaiHui Gan
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  4. BoLing Guo
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Correspondence to LiJia Han.

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Han, L., Zhang, J., Gan, Z. et al. On the limit behavior of the magnetic Zakharov system. Sci. China Math. 55, 509–540 (2012). https://doi.org/10.1007/s11425-011-4325-3

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  • DOI: https://doi.org/10.1007/s11425-011-4325-3

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