Bulletin of the Korean Mathematical Society (대한수학회보)

Korean Mathematical Society (대한수학회)
권호 표지
DOI QR코드

DOI QR Code

ON CLASSICAL SOLUTIONS AND THE CLASSICAL LIMIT OF THE VLASOV-DARWIN SYSTEM

  • Li, Xiuting (School of Automatica Huazhong University of Science and Technology) ;
  • Sun, Jiamu (School of Automatica Huazhong University of Science and Technology)
  • Received : 2017.10.30
  • Accepted : 2018.08.16
  • Published : 2018.09.30
Download PDF
⟨ Previous Next ⟩

Abstract

In this paper we study the initial value problem of the non-relativistic Vlasov-Darwin system with generalized variables (VDG). We first prove local existence and uniqueness of a nonnegative classical solution to VDG in three space variables, and establish the blow-up criterion. Then we show that it converges to the well-known Vlasov-Poisson system when the light velocity c tends to infinity in a pointwise sense.

Keywords

Acknowledgement

Supported by : China Postdoctoral Science Foundation

References

  1. K. Asano, On local solutions of the initial value problem for the Vlasov-Maxwell equation, Comm. Math. Phys. 106 (1986), no. 4, 551-568. https://doi.org/10.1007/BF01463395
  2. C. Bardos and P. Degond, Global existence for the Vlasov-Poisson equation in 3 space variables with small initial data, Ann. Inst. H. Poincare Anal. Non Lineaire 2 (1985), no. 2, 101-118. https://doi.org/10.1016/S0294-1449(16)30405-X
  3. J. Batt, Global symmetric solutions of the initial value problem of stellar dynamics, J. Differential Equations 25 (1977), no. 3, 342-364. https://doi.org/10.1016/0022-0396(77)90049-3
  4. S. Bauer and M. Kunze, The Darwin approximation of the relativistic Vlasov-Maxwell system, Ann. Henri Poincare 6 (2005), no. 2, 283-308. https://doi.org/10.1007/s00023-005-0207-y
  5. S. Benachour, F. Filbet, P. Laurencot, and E. Sonnendrucker, Global existence for the Vlasov-Darwin system in ${\mathbb{R}}^3$ for small initial data, Math. Methods Appl. Sci. 26 (2003), no. 4, 297-319. https://doi.org/10.1002/mma.355
  6. P. Degond and P.-A. Raviart, An analysis of the Darwin model of approximation to Maxwell's equations, Forum Math. 4 (1992), no. 1, 13-44.
  7. R. J. DiPerna and P.-L. Lions, Global weak solutions of Vlasov-Maxwell systems, Comm. Pure Appl. Math. 42 (1989), no. 6, 729-757. https://doi.org/10.1002/cpa.3160420603
  8. D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, reprint of the 1998 edition, Classics in Mathematics, Springer-Verlag, Berlin, 2001.
  9. R. T. Glassey, The Cauchy Problem in Kinetic Theory, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 1996.
  10. R. T. Glassey and J. W. Schaeffer, Global existence for the relativistic Vlasov-Maxwell system with nearly neutral initial data, Comm. Math. Phys. 119 (1988), no. 3, 353-384. https://doi.org/10.1007/BF01218078
  11. R. T. Glassey and J. W. Schaeffer, On the "one and one-half dimensional" relativistic Vlasov-Maxwell system, Math. Methods Appl. Sci. 13 (1990), no. 2, 169-179. https://doi.org/10.1002/mma.1670130207
  12. R. T. Glassey and J. W. Schaeffer, The "two and one-half-dimensional" relativistic Vlasov Maxwell system, Comm. Math. Phys. 185 (1997), no. 2, 257-284. https://doi.org/10.1007/s002200050090
  13. R. T. Glassey and J. W. Schaeffer, The relativistic Vlasov-Maxwell system in two space dimensions. I, Arch. Rational Mech. Anal. 141 (1998), no. 4, 331-354. https://doi.org/10.1007/s002050050079
  14. R. T. Glassey and J. W. Schaeffer, The relativistic Vlasov-Maxwell system in two space dimensions. II, Arch. Rational Mech. Anal. 141 (1998), no. 4, 355-374. https://doi.org/10.1007/s002050050080
  15. R. T. Glassey and W. A. Strauss, Singularity formation in a collisionless plasma could occur only at high velocities, Arch. Rational Mech. Anal. 92 (1986), no. 1, 59-90. https://doi.org/10.1007/BF00250732
  16. R. T. Glassey and W. A. Strauss, Absence of shocks in an initially dilute collisionless plasma, Comm. Math. Phys. 113 (1987), no. 2, 191-208. https://doi.org/10.1007/BF01223511
  17. F. Golse, Mean field kinetic equations, preprint, 2013.
  18. E. Horst, On the classical solutions of the initial value problem for the unmodified nonlinear Vlasov equation. I. General theory, Math. Methods Appl. Sci. 3 (1981), no. 2, 229-248. https://doi.org/10.1002/mma.1670030117
  19. E. Horst, On the classical solutions of the initial value problem for the unmodified nonlinear Vlasov equation. II. Special cases, Math. Methods Appl. Sci. 4 (1982), no. 1, 19-32. https://doi.org/10.1002/mma.1670040104
  20. E. Horst and R. Hunze, Weak solutions of the initial value problem for the unmodified nonlinear Vlasov equation, Math. Methods Appl. Sci. 6 (1984), no. 2, 262-279. https://doi.org/10.1002/mma.1670060118
  21. E. H. Lieb and M. Loss, Analysis, Graduate Studies in Mathematics, 14, American Mathematical Society, Providence, RI, 1997.
  22. P.-L. Lions and B. Perthame, Propagation of moments and regularity for the 3-dimensional Vlasov-Poisson system, Invent. Math. 105 (1991), no. 2, 415-430. https://doi.org/10.1007/BF01232273
  23. G. Loeper, Uniqueness of the solution to the Vlasov-Poisson system with bounded density, J. Math. Pures Appl. (9) 86 (2006), no. 1, 68-79. https://doi.org/10.1016/j.matpur.2006.01.005
  24. C. Pallard, The initial value problem for the relativistic Vlasov-Darwin system, Int. Math. Res. Not. 2006 (2006), Art. ID 57191, 31 pp.
  25. K. Pfaffelmoser, Global classical solutions of the Vlasov-Poisson system in three dimensions for general initial data, J. Differential Equations 95 (1992), no. 2, 281-303. https://doi.org/10.1016/0022-0396(92)90033-J
  26. G. Rein, Collisionless kinetic equations from astrophysics-the Vlasov-Poisson system, in Handbook of differential equations: evolutionary equations. Vol. III, 383-476, Handb. Differ. Equ, Elsevier/North-Holland, Amsterdam, 2007.
  27. J. Schaeffer, Asymptotic growth bounds for the Vlasov-Poisson system, Math. Methods Appl. Sci. 34 (2011), no. 3, 262-277. https://doi.org/10.1002/mma.1354
  28. J. Schaeffer, Global existence of smooth solutions to the Vlasov-Poisson system in three dimensions, Comm. Partial Differential Equations 16 (1991), no. 8-9, 1313-1335. https://doi.org/10.1080/03605309108820801
  29. J. Schaeffer, The classical limit of the relativistic Vlasov-Maxwell system, Comm. Math. Phys. 104 (1986), no. 3, 403-421. https://doi.org/10.1007/BF01210948
  30. M. Seehafer, Global classical solutions of the Vlasov-Darwin system for small initial data, Commun. Math. Sci. 6 (2008), no. 3, 749-764. https://doi.org/10.4310/CMS.2008.v6.n3.a11
  31. R. Sospedra-Alfonso and M. Agueh, Uniqueness of the compactly supported weak solutions of the relativistic Vlasov-Darwin system, Acta Appl. Math. 124 (2013), 207-227. https://doi.org/10.1007/s10440-012-9776-1
  32. R. Sospedra-Alfonso, M. Agueh, and R. Illner, Global classical solutions of the relativistic Vlasov-Darwin system with small Cauchy data: the generalized variables approach, Arch. Ration. Mech. Anal. 205 (2012), no. 3, 827-869. https://doi.org/10.1007/s00205-012-0518-3
  33. S. Wollman, An existence and uniqueness theorem for the Vlasov-Maxwell system, Comm. Pure Appl. Math. 37 (1984), no. 4, 457-462. https://doi.org/10.1002/cpa.3160370404