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

[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. 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.

[13]

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

[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.

[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.

[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.

show all references

References:
[1]

J. P. Aubin, Un théoréme 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. 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.

[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. 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.

[13]

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

[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.

[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.

[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.

[1]

Jack Schaeffer. Global existence for the Vlasov-Poisson system with steady spatial asymptotic behavior. Kinetic & Related Models, 2012, 5 (1) : 129-153. doi: 10.3934/krm.2012.5.129

[2]

Zili Chen, Xiuting Li, Xianwen Zhang. The two dimensional Vlasov-Poisson system with steady spatial asymptotics. Kinetic & Related Models, 2017, 10 (4) : 977-1009. doi: 10.3934/krm.2017039

[3]

Hyung Ju Hwang, Jaewoo Jung, Juan J. L. Velázquez. On global existence of classical solutions for the Vlasov-Poisson system in convex bounded domains. Discrete & Continuous Dynamical Systems - A, 2013, 33 (2) : 723-737. doi: 10.3934/dcds.2013.33.723

[4]

Blanca Ayuso, José A. Carrillo, Chi-Wang Shu. Discontinuous Galerkin methods for the one-dimensional Vlasov-Poisson system. Kinetic & Related Models, 2011, 4 (4) : 955-989. doi: 10.3934/krm.2011.4.955

[5]

Jean Dolbeault. An introduction to kinetic equations: the Vlasov-Poisson system and the Boltzmann equation. Discrete & Continuous Dynamical Systems - A, 2002, 8 (2) : 361-380. doi: 10.3934/dcds.2002.8.361

[6]

Francis Filbet, Roland Duclous, Bruno Dubroca. Analysis of a high order finite volume scheme for the 1D Vlasov-Poisson system. Discrete & Continuous Dynamical Systems - S, 2012, 5 (2) : 283-305. doi: 10.3934/dcdss.2012.5.283

[7]

Lan Luo, Hongjun Yu. Global solutions to the relativistic Vlasov-Poisson-Fokker-Planck system. Kinetic & Related Models, 2016, 9 (2) : 393-405. doi: 10.3934/krm.2016.9.393

[8]

Silvia Caprino, Guido Cavallaro, Carlo Marchioro. Time evolution of a Vlasov-Poisson plasma with magnetic confinement. Kinetic & Related Models, 2012, 5 (4) : 729-742. doi: 10.3934/krm.2012.5.729

[9]

Gang Li, Xianwen Zhang. A Vlasov-Poisson plasma of infinite mass with a point charge. Kinetic & Related Models, 2018, 11 (2) : 303-336. doi: 10.3934/krm.2018015

[10]

Dongming Wei. 1D Vlasov-Poisson equations with electron sheet initial data. Kinetic & Related Models, 2010, 3 (4) : 729-754. doi: 10.3934/krm.2010.3.729

[11]

Silvia Caprino, Guido Cavallaro, Carlo Marchioro. A Vlasov-Poisson plasma with unbounded mass and velocities confined in a cylinder by a magnetic mirror. Kinetic & Related Models, 2016, 9 (4) : 657-686. doi: 10.3934/krm.2016011

[12]

Ling Hsiao, Fucai Li, Shu Wang. Combined quasineutral and inviscid limit of the Vlasov-Poisson-Fokker-Planck system. Communications on Pure & Applied Analysis, 2008, 7 (3) : 579-589. doi: 10.3934/cpaa.2008.7.579

[13]

Yemin Chen. Smoothness of classical solutions to the Vlasov-Poisson-Landau system. Kinetic & Related Models, 2008, 1 (3) : 369-386. doi: 10.3934/krm.2008.1.369

[14]

Kosuke Ono, Walter A. Strauss. Regular solutions of the Vlasov-Poisson-Fokker-Planck system. Discrete & Continuous Dynamical Systems - A, 2000, 6 (4) : 751-772. doi: 10.3934/dcds.2000.6.751

[15]

Robert T. Glassey, Walter A. Strauss. Perturbation of essential spectra of evolution operators and the Vlasov-Poisson-Boltzmann system. Discrete & Continuous Dynamical Systems - A, 1999, 5 (3) : 457-472. doi: 10.3934/dcds.1999.5.457

[16]

Peng Jiang. Unique global solution of an initial-boundary value problem to a diffusion approximation model in radiation hydrodynamics. Discrete & Continuous Dynamical Systems - A, 2015, 35 (7) : 3015-3037. doi: 10.3934/dcds.2015.35.3015

[17]

Miroslav Grmela, Michal Pavelka. Landau damping in the multiscale Vlasov theory. Kinetic & Related Models, 2018, 11 (3) : 521-545. doi: 10.3934/krm.2018023

[18]

Toan T. Nguyen, Truyen V. Nguyen, Walter A. Strauss. Global magnetic confinement for the 1.5D Vlasov-Maxwell system. Kinetic & Related Models, 2015, 8 (1) : 153-168. doi: 10.3934/krm.2015.8.153

[19]

Renjun Duan, Tong Yang, Changjiang Zhu. Boltzmann equation with external force and Vlasov-Poisson-Boltzmann system in infinite vacuum. Discrete & Continuous Dynamical Systems - A, 2006, 16 (1) : 253-277. doi: 10.3934/dcds.2006.16.253

[20]

Yuanjie Lei, Linjie Xiong, Huijiang Zhao. One-species Vlasov-Poisson-Landau system near Maxwellians in the whole space. Kinetic & Related Models, 2014, 7 (3) : 551-590. doi: 10.3934/krm.2014.7.551

2016 Impact Factor: 1.261

Metrics

  • PDF downloads (3)
  • HTML views (20)
  • Cited by (0)

Other articles
by authors

[Back to Top]