Bulletin of the Korean Mathematical Society (대한수학회보)

Korean Mathematical Society (대한수학회)
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GLOBAL WEAK SOLUTIONS FOR THE RELATIVISTIC VLASOV-KLEIN-GORDON SYSTEM IN TWO DIMENSIONS

  • Xiao, Meixia (School of Mathematics and Statistics Huazhong University of Science and Technology) ;
  • Zhang, Xianwen (School of Mathematics and Statistics Huazhong University of Science and Technology)
  • Received : 2017.02.26
  • Accepted : 2017.08.08
  • Published : 2018.03.31
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Abstract

This paper is concerned with global existence of weak solutions to the relativistic Vlasov-Klein-Gordon system. The energy of this system is conserved, but the interaction term ${\int}_{{\mathbb{R}}^n}\;{\rho}{\varphi}dx$ in it need not be positive. So far existence of global weak solutions has been established only for small initial data [9, 14]. In two dimensions, this paper shows that the interaction term can be estimated by the kinetic energy to the power of ${\frac{4q-4}{3q-2}}$ for 1 < q < 2. As a consequence, global existence of weak solutions for general initial data is obtained.

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References

  1. 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
  2. R. J. Diperna and P. L. Lions, Global weak solutions of Vlasov-Maxwell system, Comm. Pure Appl. Math. 42 (1989), no. 6, 729-757. https://doi.org/10.1002/cpa.3160420603
  3. R. T. Glassey, The Cauchy Problem in Kinetic Theory, SIAM, Philadelphia, 1996.
  4. R. T. Glassey and J. 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
  5. 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
  6. F. Golse, P. L. Lions, B. Perthame, and R. Sentis, Regularity of the moments of the solution of a transport equation, J. Funct. Anal. 76 (1988), no. 1, 110-125. https://doi.org/10.1016/0022-1236(88)90051-1
  7. S.-Y. Ha and H. Lee, Global existence of classical solutions to the damped Vlasov-Klein-Gordon equations with small data, J. Math. Phys. 50 (2009), no. 5, 053302, 33 pp.
  8. L. Hormander, Lectures on Nonlinear Hyperbolic Differential Equations, New York: Springer Verlag, 1997.
  9. M. Kunzinger, G. Rein, R. Steinbauer, and G. Teschl, Global weak solutions of the relativistic Vlasov-Klein-Gordon system, Comm. Math. Phys. 238 (2003), no. 1-2, 367-378. https://doi.org/10.1007/s00220-003-0861-1
  10. M. Kunzinger, G. Rein, R. Steinbauer, and G. Teschl, On classical solutions of the relativistic Vlasov-Klein-Gordon system, Electron. J. Differential Equations 1 (2005), no. 1, 1-17.
  11. E. H. Lieb and M. Loss, Analysis, AMS, Providence, RI, 2001.
  12. 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
  13. G. Rein, Global weak solutions to the relativistic Vlasov-Maxwell system revisited, Commun. Math. Sci. 2 (2004), no. 2, 145-158. https://doi.org/10.4310/CMS.2004.v2.n2.a1
  14. M. Wei and W. Zhu, Global weak solutions of the relativistic Vlasov-Klein-Gordon system in two dimensions, Ann. Differential Equations 4 (2007), no. 4, 511-518.