On global solutions to the Vlasov-Poisson system with radiation damping

Meixia Xiao; Xianwen Zhang

doi:10.3934/krm.2018046

Article Contents

2018, Volume 11, Issue 5: 1183-1209. Doi: 10.3934/krm.2018046

This issue Previous Article Uniform stability of the Cucker-Smale model and its application to the Mean-Field limit Next Article A deterministic-stochastic method for computing the Boltzmann collision integral in $\mathcal{O}(MN)$ operations

On global solutions to the Vlasov-Poisson system with radiation damping

Meixia Xiao and
Xianwen Zhang^,

School of Mathematics and Statistics, Huazhong University of Science and Technology, Wuhan, Hubei 430074, China

* Corresponding author: Xianwen Zhang
* Corresponding author: Xianwen Zhang

Received: May 2017

Revised: September 2017

Published: May 2018

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Abstract

In this paper, the dynamics of three dimensional Vlasov-Poisson system with radiation damping is investigated. We prove global existence of a classical as well as weak solution that propagates boundedness of velocity-space support or velocity-space moment of order two respectively. This kind of solutions possess finite mass but need not necessarily have finite kinetic energy. Moreover, uniqueness of the classical solution is also shown.

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Mathematics Subject Classification: Primary: 35Q83, 35L60, 82C21; Secondary: 82D10.

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[1]	J. P. Aubin, Un théorème de compacité, C. R. Acad. Sci. Paris, 256 (1963), 5042-5044.
[2]	J. Batt, Ein Existenzbeweis für die Vlasov-Gleichung der Stellar-dyamik bei gemittelter Dichte, Arch. Rational Mech. Anal., 13 (1963), 296-308. doi: 10.1007/BF01262698.
[3]	J. Batt, Global symmetric solutions of the initial value problem of stellar dynamics, J. Differential Equations, 25 (1977), 342-364. doi: 10.1016/0022-0396(77)90049-3.
[4]	S. Bauer, A non-relativistic model of plasma physics containing a radiation reaction term, Kinet. Relat. Models, 11 (2018), 25-42. doi: 10.3934/krm.2018002.
[5]	F. Bouchut and L. Desvillettes, Averaging lemmas without time Fourier transform and application to discretized kinetic equations, Proc. Roy. Soc. Edinburgh Sect. A, 129 (1999), 19-36. doi: 10.1017/S030821050002744X.
[6]	F. Castella, Propagation of space moments in the Vlasov-Poisson equation and further results, Ann. Inst. H. Poincaré Anal. Non Linéaire, 16 (1999), 503-533. doi: 10.1016/S0294-1449(99)80026-2.
[7]	J. Chen and X. Zhang, Global existence of small amplitude solutions to the Vlasov-Poisson system with radiation damping, Internat. J. Math., 26 (2015), 1550098(19 pages). doi: 10.1142/S0129167X15500986.
[8]	J. Chen, X. Zhang and R. Gao, Existence, uniqueness and asymptotic behavior for the Vlasov-Poisson system with radiation damping, Acta Math. Sin., English Series, 33 (2017), 635-656. doi: 10.1007/s10114-016-6310-9.
[9]	Z. Chen and X. Zhang, Global existence to the Vlasov-Poisson system and propagation of moments without assumption of finite kinetic energy, Comm. Math. Phys., 343 (2016), 851-879. doi: 10.1007/s00220-016-2616-9.
[10]	Z. Chen and X. Zhang, Sub-linear estimate of large velocities in a collisionless plasma, Comm. Math. Sci., 12 (2014), 279-291. doi: 10.4310/CMS.2014.v12.n2.a4.
[11]	R. J. DiPerna and P. L. Lions, Global weak solutions of Vlasov-Maxwell systems, Comm. Pure Appl. Math., 42 (1989), 729-757. doi: 10.1002/cpa.3160420603.
[12]	R. J. DiPerna, P. L. Lions and Y. Meyer, L_p regularity of velocity averages, Ann. Inst. H. Poincaré Anal. Non Linéaire, 8 (1991), 271-287. doi: 10.1016/S0294-1449(16)30264-5.
[13]	R. T. Glassey, The Cauchy Problem in Kinetic Theory, SIAM, Philadelphia, 1996. doi: 10.1137/1.9781611971477.
[14]	F. Golse, B. Perthame and R. Sentis, Un résultat de compacité pour les équations de transport et application au calcul de la limite de la valeur propre principale d'un opérateur de transport, C. R. Acad. Sci. Paris Sér. I Math., 301 (1985), 341-344.
[15]	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), 110-125. doi: 10.1016/0022-1236(88)90051-1.
[16]	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), 229-248. doi: 10.1002/mma.1670030117.
[17]	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), 19-32. doi: 10.1002/mma.1670040104.
[18]	E. Horst and R. Hunze, Weak solutions of the initial value problem for the unmodified nonlinear Vlasov equation, Math. Methods Appl. Sci., 6 (1984), 262-279. doi: 10.1002/mma.1670060118.
[19]	E. Horst, On the asymptotic growth of the solutions of the Vlasov-Poisson system, Math. Methods Appl. Sci., 16 (1993), 75-85. doi: 10.1002/mma.1670160202.
[20]	R. Illner and G. Rein, Time decay of the solutions of the Vlasov-Poisson system in the plasma physical case, Math. Methods Appl. Sci., 19 (1996), 1409-1413. doi: 10.1002/(SICI)1099-1476(19961125)19:17<1409::AID-MMA836>3.0.CO;2-2.
[21]	M. Kunze and A. D. Rendall, The Vlasov-Poisson system with radiation damping, Ann. Henri Poincaré, 2 (2001), 857-886. doi: 10.1007/s00023-001-8596-z.
[22]	P. L. Lions, A kinetic formulation of multidimensional scalar conservation laws and related equations, J. Amer. Math. Soc., 7 (1994), 169-191. doi: 10.1090/S0894-0347-1994-1201239-3.
[23]	P. L. Lions and B. Perthame, Propagation of moments and regularity for the 3-dimensional Vlasov-Poisson system, Invent. Math., 105 (1991), 415-430. doi: 10.1007/BF01232273.
[24]	G. Loeper, Uniqueness of the solution to the Vlasov-Poisson system with bounded density, J. Math. Pures Appl., 86 (2006), 68-79. doi: 10.1016/j.matpur.2006.01.005.
[25]	E. Miot, A uniqueness criterion for unbounded solutions to the Vlasov-Poisson system, Comm. Math. Phys., 346 (2016), 469-482. doi: 10.1007/s00220-016-2707-7.
[26]	C. Pallard, A note on the growth of velocities in a collisionless plasma, Math. Methods Appl. Sci., 34 (2011), 803-806. doi: 10.1002/mma.1402.
[27]	C. Pallard, Growth estimates and uniform decay for a collisionless plasma, Kinet. Relat. Models, 4 (2011), 549-567. doi: 10.3934/krm.2011.4.549.
[28]	C. Pallard, Large velocities in a collisionless plasma, J. Differential Equations, 252 (2012), 2864-2876. doi: 10.1016/j.jde.2011.09.020.
[29]	C. Pallard, Moment propagation for weak solutions to the Vlasov-Poisson system, Comm. Partial Differential Equations, 37 (2012), 1273-1285. doi: 10.1080/03605302.2011.606863.
[30]	C. Pallard, Space moments of the Vlasov-Poisson system: Propagation and regularity, SIAM J. Math. Anal., 46 (2014), 1754-1770. doi: 10.1137/120881178.
[31]	B. Perthame, Time decay, propagation of low moments and dispersive effects for kinetic equations, Comm. Partial Differential Equations, 21 (1996), 659-686. doi: 10.1080/03605309608821201.
[32]	B. Perthame and P. E. Souganidis, A limiting case for velocity averaging, Ann. Sci. École Norm. Sup., 31 (1998), 591-598. doi: 10.1016/S0012-9593(98)80108-0.
[33]	K. Pfaffelmoser, Global classical solutions of the Vlasov-Poisson system in three dimensions for general initial data, J. Differential Equations, 95 (1992), 281-303. doi: 10.1016/0022-0396(92)90033-J.
[34]	G. Rein, Growth estimates for the solutions of the Vlasov-Poisson system in the plasma physics case, Math. Nachr., 191 (1998), 269-278. doi: 10.1002/mana.19981910114.
[35]	G. Rein, Collisionless kinetic equation from astrophysics-the Vlasov-Poisson system, in Handbook of Differential Equations: Evolutionary Equations (eds. C. M. Dafermos and E. Feireisl), Elsevier, 3 (2007), 383–476. doi: 10.1016/S1874-5717(07)80008-9.
[36]	D. Salort, Transport equations with unbounded force fields and application to the Vlasov-Poisson equation, Math. Models Methods Appl. Sci., 19 (2009), 199-228. doi: 10.1142/S0218202509003401.
[37]	J. Schaeffer, Asymptotic growth bounds for the Vlasov-Poisson system, Math. Methods Appl. Sci., 34 (2011), 262-277. doi: 10.1002/mma.1354.
[38]	X. Zhang and J. Wei, The Vlasov-Poisson system with infinite kinetic energy and initial data in $L^{p}(\mathbb{R}^{6})$, J. Math. Anal. Appl., 341 (2008), 548-558. doi: 10.1016/j.jmaa.2007.10.038.