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October  2018, 11(5): 1183-1209. doi: 10.3934/krm.2018046

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

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

* Corresponding author: Xianwen Zhang

Received  May 2017 Revised  September 2017 Published  May 2018

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.

Citation: Meixia Xiao, Xianwen Zhang. On global solutions to the Vlasov-Poisson system with radiation damping. Kinetic & Related Models, 2018, 11 (5) : 1183-1209. doi: 10.3934/krm.2018046
References:
[1]

J. P. Aubin, Un théorème de compacité, C. R. Acad. Sci. Paris, 256 (1963), 5042-5044. Google Scholar

[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. Google Scholar

[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. Google Scholar

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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. Google Scholar

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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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[8]

J. ChenX. 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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[12]

R. J. DiPernaP. L. Lions and Y. Meyer, Lp regularity of velocity averages, Ann. Inst. H. Poincaré Anal. Non Linéaire, 8 (1991), 271-287. doi: 10.1016/S0294-1449(16)30264-5. Google Scholar

[13]

R. T. Glassey, The Cauchy Problem in Kinetic Theory, SIAM, Philadelphia, 1996. doi: 10.1137/1.9781611971477. Google Scholar

[14]

F. GolseB. 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. Google Scholar

[15]

F. GolseP. L. LionsB. 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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[28]

C. Pallard, Large velocities in a collisionless plasma, J. Differential Equations, 252 (2012), 2864-2876. doi: 10.1016/j.jde.2011.09.020. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[37]

J. Schaeffer, Asymptotic growth bounds for the Vlasov-Poisson system, Math. Methods Appl. Sci., 34 (2011), 262-277. doi: 10.1002/mma.1354. Google Scholar

[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. Google Scholar

show all references

References:
[1]

J. P. Aubin, Un théorème de compacité, C. R. Acad. Sci. Paris, 256 (1963), 5042-5044. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[8]

J. ChenX. 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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[12]

R. J. DiPernaP. L. Lions and Y. Meyer, Lp regularity of velocity averages, Ann. Inst. H. Poincaré Anal. Non Linéaire, 8 (1991), 271-287. doi: 10.1016/S0294-1449(16)30264-5. Google Scholar

[13]

R. T. Glassey, The Cauchy Problem in Kinetic Theory, SIAM, Philadelphia, 1996. doi: 10.1137/1.9781611971477. Google Scholar

[14]

F. GolseB. 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. Google Scholar

[15]

F. GolseP. L. LionsB. 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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[28]

C. Pallard, Large velocities in a collisionless plasma, J. Differential Equations, 252 (2012), 2864-2876. doi: 10.1016/j.jde.2011.09.020. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[37]

J. Schaeffer, Asymptotic growth bounds for the Vlasov-Poisson system, Math. Methods Appl. Sci., 34 (2011), 262-277. doi: 10.1002/mma.1354. Google Scholar

[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. Google Scholar

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