Advertisement

Communications in Mathematical Physics

, Volume 331, Issue 3, pp 1005–1027 | Cite as

A New Continuation Criterion for the Relativistic Vlasov–Maxwell System

  • Jonathan Luk
  • Robert M. StrainEmail author
Article

Abstract

The global existence of solutions to the relativistic Vlasov–Maxwell system given sufficiently regular finite energy initial data is a longstanding open problem. The main result of Glassey and Strauss (Arch Ration Mech Anal 92:59–90, 1986) shows that a solution (f, E, B) remains C 1 as long as the momentum support of f remains bounded. Alternate proofs were later given by Bouchut et al. (Arch Ration Mech Anal 170:1–15, 2003) and Klainerman and Staffilani (Commun Pure Appl Anal 1:103–125, 2002). We show that only the boundedness of the momentum support of f after projecting to any two dimensional plane is needed for (f, E, B) to remain C 1.

Keywords

Global Existence Singular Term Vlasov Equation Collisionless Plasma Global Regularity 
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.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Bouchut F., Golse F., Pallard C.: Classical solutions and the Glassey–Strauss theorem for the 3D Vlasov–Maxwell system. Arch. Ration. Mech. Anal. 170, 1–15 (2003)zbMATHMathSciNetCrossRefGoogle Scholar
  2. 2.
    Glassey R., Schaeffer J.: On symmetric solutions of the relativistic Vlasov–Poisson system. Commun. Math. Phys. 101, 459–473 (1985)ADSzbMATHMathSciNetCrossRefGoogle Scholar
  3. 3.
    Glassey R., Schaeffer J.: Global existence for the relativistic Vlasov–Maxwell system with nearly neutral initial data. Commun. Math. Phys. 119, 353–384 (1988)ADSzbMATHMathSciNetCrossRefGoogle Scholar
  4. 4.
    Glassey R., Schaeffer J.: The two and one half dimensional relativistic Vlasov–Maxwell system. Commun. Math. Phys. 185, 257–284 (1997)ADSzbMATHMathSciNetCrossRefGoogle Scholar
  5. 5.
    Glassey R., Schaeffer J.: The relativistic Vlasov–Maxwell system in two space dimensions: part I. Arch. Ration. Mech. Anal. 141, 331–354 (1998)zbMATHMathSciNetCrossRefGoogle Scholar
  6. 6.
    Glassey R., Schaeffer J.: The relativistic Vlasov–Maxwell system in two space dimensions: part II. Arch. Ration. Mech. Anal. 141, 355–374 (1998)MathSciNetCrossRefGoogle Scholar
  7. 7.
    Glassey R., Strauss W.: Singularity formation in a collisionless plasma could only occur at large velocities. Arch. Ration. Mech. Anal. 92, 59–90 (1986)zbMATHMathSciNetCrossRefGoogle Scholar
  8. 8.
    Glassey R., Strauss W.: Absence of shocks in an initially dilute collisionless plasma. Commun. Math. Phys. 113, 191–208 (1987)ADSzbMATHMathSciNetCrossRefGoogle Scholar
  9. 9.
    Glassey R., Strauss W.: High velocity particles in a collisionless plasma. Math. Methods Appl. Sci. 9, 46–52 (1987)ADSzbMATHMathSciNetCrossRefGoogle Scholar
  10. 10.
    Glassey R., Strauss W.: Large velocities in the relativistic Vlasov–Maxwell equations. J. Fac. Sci. Univ. Tokyo Sect. IA Math. 36, 615–527 (1989)Google Scholar
  11. 11.
    Klainerman S., Staffilani G.: A new approach to study the Vlasov–Maxwell system. Commun. Pure Appl. Anal. 1, 103–125 (2002)zbMATHMathSciNetGoogle Scholar
  12. 12.
    Luk, J., Strain, R.: Strichartz estimates and moment bounds for the Vlasov–Maxwell system II. Continuation criterion in the 3D case, preprint (2014). arXiv:1406.0169
  13. 13.
    Lions P.-L., Perthame B.: Propagation of moments and regularity for the 3-dimensional Vlasov–Poisson system. Invent. Math 105, 415–430 (1991)ADSzbMATHMathSciNetCrossRefGoogle Scholar
  14. 14.
    Pallard C.: On the boundedness of the momentum support of solutions to the relativistic Vlasov–Maxwell system. Indiana Univ. Math. J. 54(5), 1395–1409 (2005)zbMATHMathSciNetCrossRefGoogle Scholar
  15. 15.
    Pfaffermoser K.: Global classical solutions of the Vlasov–Poisson system in three dimensions for general initial data. J. Differ. Equ. 95, 281–303 (1992)ADSCrossRefGoogle Scholar
  16. 16.
    Rein G.: Generic global solutions of the relativistic Vlasov–Maxwell system of plasma physics. Commun. Math. Phys. 135, 41–78 (1990)ADSzbMATHMathSciNetCrossRefGoogle Scholar
  17. 17.
    Schaeffer J.: Global existence of smooth solutions to the Vlasov–Poisson system in three dimensions. Commun. P.D.E. 16(8–9), 1313–1335 (1991)zbMATHMathSciNetCrossRefGoogle Scholar
  18. 18.
    Schaeffer J.: The classical limit of the relativistic Vlasov–Maxwell system. Commun. Math. Phys. 104(3), 403–421 (1986)ADSzbMATHMathSciNetCrossRefGoogle Scholar
  19. 19.
    Schaeffer J.: A small data theorem for collisionless plasma that includes high velocity particles. Indiana Univ. Math. J. 53(1), 1–34 (2004)zbMATHMathSciNetCrossRefGoogle Scholar
  20. 20.
    Sospedra-Alfonso R., Illner R.: Classical solvability of the relativistic Vlasov–Maxwell system with bounded spatial density. Math. Methods Appl. Sci. 33(6), 751–757 (2010)zbMATHMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2014

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of PennsylvaniaPhiladelphiaUSA

Personalised recommendations