Browse > Article
http://dx.doi.org/10.4134/BKMS.b181077

MILD SOLUTIONS FOR THE RELATIVISTIC VLASOV-KLEIN-GORDON SYSTEM  

Xiao, Meixia (School of Mathematics and Computer Science Wuhan Textile University)
Zhang, Xianwen (School of Mathematics and Statistics Huazhong University of Science and Technology)
Publication Information
Bulletin of the Korean Mathematical Society / v.56, no.6, 2019 , pp. 1447-1465 More about this Journal
Abstract
In this paper, the relativistic Vlasov-Klein-Gordon system in one dimension is investigated. This non-linear dynamics system consists of a transport equation for the distribution function combined with Klein-Gordon equation. Without any assumption of continuity or compact support of any initial particle density $f_0$, we prove the existence and uniqueness of the mild solution via the iteration method.
Keywords
Klein-Gordon field; mild solutions; Bessel function; characteristics;
Citations & Related Records
Times Cited By KSCI : 1  (Citation Analysis)
연도 인용수 순위
1 R. J. DiPerna, P.-L. Lions, and Y. Meyer, $L^p$ regularity of velocity averages, Ann. Inst. H. Poincare Anal. Non Lineaire 8 (1991), no. 3-4, 271-287. https://doi.org/10.1016/S0294-1449(16)30264-5   DOI
2 R. T. Glassey, The Cauchy problem in kinetic theory, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 1996. https://doi.org/10.1137/1.9781611971477
3 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   DOI
4 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. https://doi.org/10.1063/1.3131613
5 E. Horst, R. Hunze, and H. Neunzert, 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   DOI
6 E. Horst and H. Neunzert, On the classical solutions of the initial value problem for the unmodied nonlinear Vlasov equation. I. General theory, Math. Methods Appl. Sci. 3 (1981), no. 2, 229-248. https://doi.org/10.1002/mma.1670030117   DOI
7 E. Horst and H. Neunzert, On the classical solutions of the initial value problem for the unmodied nonlinear Vlasov equation. II. Special cases, Math. Methods Appl. Sci. 4 (1982), no. 1, 19-32. https://doi.org/10.1002/mma.1670040104   DOI
8 M. Kunzinger, G. Rein, R. Steinbauer, and G. Teschl, On classical solutions of the relativistic Vlasov-Klein-Gordon system, Electron. J. Differential Equations 2005 (2005), no. 01, 17 pp.
9 M. Bostan, Mild solutions for the relativistic Vlasov-Maxwell system for laser-plasma interaction, Quart. Appl. Math. 65 (2007), no. 1, 163-187. https://doi.org/10.1090/S0033-569X-07-01047-4   DOI
10 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   DOI
11 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   DOI
12 K. Pfaelmoser, Global classical solutions of the Vlasov-Poisson system in three dimensions for general initial data, J. Dierential Equations 95 (1992), no. 2, 281-303. https://doi.org/10.1016/0022-0396(92)90033-J   DOI
13 G. Rein, Collisionless kinetic equations from astrophysics-the Vlasov-Poisson system, in Handbook of dierential equations: evolutionary equations. Vol. III, 383-476, Handb. Dier. Equ, Elsevier/North-Holland, Amsterdam, 2007. https://doi.org/10.1016/S1874-5717(07)80008-9
14 J. Schaeer, Global existence of smooth solutions to the Vlasov-Poisson system in three dimensions, Comm. Partial Dierential Equations 16 (1991), no. 8-9, 1313-1335. https://doi.org/10.1080/03605309108820801   DOI
15 M. Xiao and X. Zhang, Global weak solutions for the relativistic Vlasov-Klein-Gordon system in two dimensions, Bull. Korean Math. Soc. 55 (2018), no. 2, 591-598. https://doi.org/10.4134/BKMS.b170175   DOI
16 M. Wei and W. Zhu, Global weak solutions of the relativistic Vlasov-Klein-Gordon system in two dimensions, Ann. Dierential Equations 23 (2007), no. 4, 511-518.
17 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   DOI
18 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   DOI
19 A. A. Arsen'ev, Global existence of a weak solution of Vlasov's system of equations, U.S.S.R. Comp. Math. and Math. Phys. 15 (1975), 131-143.   DOI
20 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
21 M. Bostan, Mild solutions for the one-dimensional Nordstrom-Vlasov system, Nonlinearity 20 (2007), no. 5, 1257-1281.   DOI