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http://dx.doi.org/10.4134/BKMS.b170175

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)
Publication Information
Bulletin of the Korean Mathematical Society / v.55, no.2, 2018 , pp. 591-598 More about this Journal
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.
Keywords
Klein-Gordon field; Vlasov equation; weak solution; global existence;
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1 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.   DOI
2 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.
3 L. Hormander, Lectures on Nonlinear Hyperbolic Differential Equations, New York: Springer Verlag, 1997.
4 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.   DOI
5 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.
6 E. H. Lieb and M. Loss, Analysis, AMS, Providence, RI, 2001.
7 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.   DOI
8 G. Rein, Global weak solutions to the relativistic Vlasov-Maxwell system revisited, Commun. Math. Sci. 2 (2004), no. 2, 145-158.   DOI
9 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.
10 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.   DOI
11 R. J. Diperna and P. L. Lions, Global weak solutions of Vlasov-Maxwell system, Comm. Pure Appl. Math. 42 (1989), no. 6, 729-757.   DOI
12 R. T. Glassey, The Cauchy Problem in Kinetic Theory, SIAM, Philadelphia, 1996.
13 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.   DOI
14 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.   DOI