American Institute of Mathematical Sciences

Advanced
March  2011, 4(1): 215-226. doi: 10.3934/krm.2011.4.215

On a charge interacting with a plasma of unbounded mass

Silvia Caprino 1, and  Carlo Marchioro 2,

1. 

Dipartimento di Matematica, Università di Tor Vergata, Via della Ricerca Scientifica 1, 00133 Roma
2. 

Dipartimento di Matematica, Università La Sapienza, Piazzale Aldo Moro 2, 00185 Roma

Received  September 2010 Revised  November 2010 Published  January 2011

We consider a positive Vlasov-Helmholtz plasma in interaction with a positive point charge in $\R^2$ and we prove an existence and uniqueness theorem for this system without any assumption on the decay at infinity of the spatial density.
Keywords: infinitely extended systems., Vlasov-Helmholtz plasma.
Mathematics Subject Classification: Primary: 82D10, 35Q99; Secondary: 35L6.
Citation: Silvia Caprino, Carlo Marchioro. On a charge interacting with a plasma of unbounded mass. Kinetic & Related Models, 2011, 4 (1) : 215-226. doi: 10.3934/krm.2011.4.215
References:
[1]

P. Butta', G. Ferrari and C. Marchioro, Speedy motions of a body immersed in an infinitely extended medium,, Jour. Stat. Phys., 140 (2010), 1182.   Google Scholar

[2]

E. Caglioti, S. Caprino, C. Marchioro and M. Pulvirenti, The Vlasov equation with infinite mass,, Arch. Rational Mech. Anal., 159 (2001), 85.  doi: 10.1007/s002050100150.  Google Scholar

[3]

S. Caprino and C. Marchioro, On the plasma-charge model,, Kinetic and Related Problems, 3 (2010), 241.   Google Scholar

[4]

S. Caprino, C. Marchioro and M. Pulvirenti, On the two dimensional Vlasov-Helmholtz equation with infinite mass,, Commun. PDE, 27 (2002), 791.  doi: 10.1081/PDE-120002874.  Google Scholar

[5]

E. Horst, On the classical solutions of the initial value problem for the unmodified non-linear Vlasov equation I,, Math. Meth. Appl. Sci., 3 (1981), 229.  doi: 10.1002/mma.1670030117.  Google Scholar

[6]

E. Horst, On the classical solutions of the initial value problem for the unmodified non-linear Vlasov equation II,, Math. Meth. Appl. Sci., 4 (1982), 19.  doi: 10.1002/mma.1670040104.  Google Scholar

[7]

P. L. Lions and B. Perthame, Propagation of moments and regularity for the 3-dimensional Vlasov-Poisson system,, Invent. Math., 105 (1996), 415.  doi: 10.1007/BF01232273.  Google Scholar

[8]

G. Loeper, Uniqueness of the solution to the Vlasov-Poisson system with bounded density,, Jour. de Math. Pure Appl., 86 (2006), 68.   Google Scholar

[9]

A. Majda, G. Majda and Y. Zheng, Concentrations in the one-dimensional Vlasov-Poisson equations. I. Temporal development and non-unique weak solutions in the single component case,, Phys. D, 74 (1994), 268.  doi: 10.1016/0167-2789(94)90198-8.  Google Scholar

[10]

A. Majda, G. Majda and Y. Zheng, Concentrations in the one-dimensional Vlasov-Poisson equations. II. Screening and the necessity for measure-valued solutions in the two component case,, Phys. D, 79 (1994), 41.  doi: 10.1016/0167-2789(94)90037-X.  Google Scholar

[11]

C. Marchioro, E. Miot and M. Pulvirenti, The Cauchy problem for the 3-D Vlasov-Poisson system with point charges,, Arch. Rational Mech. Anal. (2010) in press., (2010).   Google Scholar

[12]

S. Okabe and T. Ukai, On classical solutions in the large in time for the two-dimensional Vlasov's equation,, Osaka Jour. Math., 15 (1978), 245.   Google Scholar

[13]

S. Pankavich, Global existence for the three dimensional Vlasov-Poisson system with steady spatial asymptotics,, Comm. PDE, 31 (2006), 349.  doi: 10.1080/03605300500358004.  Google Scholar

[14]

K. Pfaffelmoser, Global classical solutions of the Vlasov-Poisson system in three dimensions for general initial data,, Jour. Diff. Eq., 95 (1992), 281.  doi: 10.1016/0022-0396(92)90033-J.  Google Scholar

[15]

D. Salort, Transport equations with unbounded force fields and application to the Vlasov-Poisson equation,, Math. Mod. Meth. Appl. Sci., 19 (2009), 199.  doi: 10.1142/S0218202509003401.  Google Scholar

[16]

J. Schaeffer, Global existence of smooth solutions to the Vlasov-Poisson system in three dimensions,, Commun. PDE., 16 (1991), 1313.   Google Scholar

[17]

S. Wollman, Global in time solutions to the two-dimensional Vlasov-Poisson system,, Commun. Pure Appl. Math., 33 (1980), 173.  doi: 10.1002/cpa.3160330205.  Google Scholar

[18]

S. Wollman, Global in time solution to the three-dimensional Vlasov-Poisson system,, Jour. Math. Anal. Appl., 176 (1993), 76.  doi: 10.1006/jmaa.1993.1200.  Google Scholar

[19]

Y. Zheng and A. Majda, Existence of global weak solutions to one-component Vlasov-Poisson and Fokker-Planck-Poisson systems in one space dimension with measures as initial data,, Comm. Pure Appl. Math., 47 (1994), 1365.  doi: 10.1002/cpa.3160471004.  Google Scholar

show all references

References:
[1]

P. Butta', G. Ferrari and C. Marchioro, Speedy motions of a body immersed in an infinitely extended medium,, Jour. Stat. Phys., 140 (2010), 1182.   Google Scholar

[2]

E. Caglioti, S. Caprino, C. Marchioro and M. Pulvirenti, The Vlasov equation with infinite mass,, Arch. Rational Mech. Anal., 159 (2001), 85.  doi: 10.1007/s002050100150.  Google Scholar

[3]

S. Caprino and C. Marchioro, On the plasma-charge model,, Kinetic and Related Problems, 3 (2010), 241.   Google Scholar

[4]

S. Caprino, C. Marchioro and M. Pulvirenti, On the two dimensional Vlasov-Helmholtz equation with infinite mass,, Commun. PDE, 27 (2002), 791.  doi: 10.1081/PDE-120002874.  Google Scholar

[5]

E. Horst, On the classical solutions of the initial value problem for the unmodified non-linear Vlasov equation I,, Math. Meth. Appl. Sci., 3 (1981), 229.  doi: 10.1002/mma.1670030117.  Google Scholar

[6]

E. Horst, On the classical solutions of the initial value problem for the unmodified non-linear Vlasov equation II,, Math. Meth. Appl. Sci., 4 (1982), 19.  doi: 10.1002/mma.1670040104.  Google Scholar

[7]

P. L. Lions and B. Perthame, Propagation of moments and regularity for the 3-dimensional Vlasov-Poisson system,, Invent. Math., 105 (1996), 415.  doi: 10.1007/BF01232273.  Google Scholar

[8]

G. Loeper, Uniqueness of the solution to the Vlasov-Poisson system with bounded density,, Jour. de Math. Pure Appl., 86 (2006), 68.   Google Scholar

[9]

A. Majda, G. Majda and Y. Zheng, Concentrations in the one-dimensional Vlasov-Poisson equations. I. Temporal development and non-unique weak solutions in the single component case,, Phys. D, 74 (1994), 268.  doi: 10.1016/0167-2789(94)90198-8.  Google Scholar

[10]

A. Majda, G. Majda and Y. Zheng, Concentrations in the one-dimensional Vlasov-Poisson equations. II. Screening and the necessity for measure-valued solutions in the two component case,, Phys. D, 79 (1994), 41.  doi: 10.1016/0167-2789(94)90037-X.  Google Scholar

[11]

C. Marchioro, E. Miot and M. Pulvirenti, The Cauchy problem for the 3-D Vlasov-Poisson system with point charges,, Arch. Rational Mech. Anal. (2010) in press., (2010).   Google Scholar

[12]

S. Okabe and T. Ukai, On classical solutions in the large in time for the two-dimensional Vlasov's equation,, Osaka Jour. Math., 15 (1978), 245.   Google Scholar

[13]

S. Pankavich, Global existence for the three dimensional Vlasov-Poisson system with steady spatial asymptotics,, Comm. PDE, 31 (2006), 349.  doi: 10.1080/03605300500358004.  Google Scholar

[14]

K. Pfaffelmoser, Global classical solutions of the Vlasov-Poisson system in three dimensions for general initial data,, Jour. Diff. Eq., 95 (1992), 281.  doi: 10.1016/0022-0396(92)90033-J.  Google Scholar

[15]

D. Salort, Transport equations with unbounded force fields and application to the Vlasov-Poisson equation,, Math. Mod. Meth. Appl. Sci., 19 (2009), 199.  doi: 10.1142/S0218202509003401.  Google Scholar

[16]

J. Schaeffer, Global existence of smooth solutions to the Vlasov-Poisson system in three dimensions,, Commun. PDE., 16 (1991), 1313.   Google Scholar

[17]

S. Wollman, Global in time solutions to the two-dimensional Vlasov-Poisson system,, Commun. Pure Appl. Math., 33 (1980), 173.  doi: 10.1002/cpa.3160330205.  Google Scholar

[18]

S. Wollman, Global in time solution to the three-dimensional Vlasov-Poisson system,, Jour. Math. Anal. Appl., 176 (1993), 76.  doi: 10.1006/jmaa.1993.1200.  Google Scholar

[19]

Y. Zheng and A. Majda, Existence of global weak solutions to one-component Vlasov-Poisson and Fokker-Planck-Poisson systems in one space dimension with measures as initial data,, Comm. Pure Appl. Math., 47 (1994), 1365.  doi: 10.1002/cpa.3160471004.  Google Scholar

[1]

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
[2]

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
[3]

Baptiste Fedele, Claudia Negulescu. Numerical study of an anisotropic Vlasov equation arising in plasma physics. Kinetic & Related Models, 2018, 11 (6) : 1395-1426. doi: 10.3934/krm.2018055
[4]

P. Adda, J. L. Dimi, A. Iggidir, J. C. Kamgang, G. Sallet, J. J. Tewa. General models of host-parasite systems. Global analysis. Discrete & Continuous Dynamical Systems - B, 2007, 8 (1) : 1-17. doi: 10.3934/dcdsb.2007.8.1
[5]

Siniša Slijepčević. Extended gradient systems: Dimension one. Discrete & Continuous Dynamical Systems - A, 2000, 6 (3) : 503-518. doi: 10.3934/dcds.2000.6.503
[6]

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
[7]

Matteo Petrera, Yuri B. Suris. Geometry of the Kahan discretizations of planar quadratic Hamiltonian systems. Ⅱ. Systems with a linear Poisson tensor. Journal of Computational Dynamics, 2019, 6 (2) : 401-408. doi: 10.3934/jcd.2019020
[8]

Denis de Carvalho Braga, Luis Fernando Mello, Carmen Rocşoreanu, Mihaela Sterpu. Lyapunov coefficients for non-symmetrically coupled identical dynamical systems. Application to coupled advertising models. Discrete & Continuous Dynamical Systems - B, 2009, 11 (3) : 785-803. doi: 10.3934/dcdsb.2009.11.785
[9]

Susanna Terracini, Juncheng Wei. DCDS-A Special Volume Qualitative properties of solutions of nonlinear elliptic equations and systems. Preface. Discrete & Continuous Dynamical Systems - A, 2014, 34 (6) : i-ii. doi: 10.3934/dcds.2014.34.6i
[10]

Anna Kostianko, Sergey Zelik. Inertial manifolds for 1D reaction-diffusion-advection systems. Part Ⅰ: Dirichlet and Neumann boundary conditions. Communications on Pure & Applied Analysis, 2017, 16 (6) : 2357-2376. doi: 10.3934/cpaa.2017116
[11]

Anna Kostianko, Sergey Zelik. Inertial manifolds for 1D reaction-diffusion-advection systems. Part Ⅱ: periodic boundary conditions. Communications on Pure & Applied Analysis, 2018, 17 (1) : 285-317. doi: 10.3934/cpaa.2018017
[12]

Jacques Demongeot, Dan Istrate, Hajer Khlaifi, Lucile Mégret, Carla Taramasco, René Thomas. From conservative to dissipative non-linear differential systems. An application to the cardio-respiratory regulation. Discrete & Continuous Dynamical Systems - S, 2018, 0 (0) : 0-0. doi: 10.3934/dcdss.2020181
[13]

Alberto Boscaggin, Anna Capietto. Infinitely many solutions to superquadratic planar Dirac-type systems. Conference Publications, 2009, 2009 (Special) : 72-81. doi: 10.3934/proc.2009.2009.72
[14]

Helmut Rüssmann. KAM iteration with nearly infinitely small steps in dynamical systems of polynomial character. Discrete & Continuous Dynamical Systems - S, 2010, 3 (4) : 683-718. doi: 10.3934/dcdss.2010.3.683
[15]

Yinbin Deng, Shuangjie Peng, Li Wang. Infinitely many radial solutions to elliptic systems involving critical exponents. Discrete & Continuous Dynamical Systems - A, 2014, 34 (2) : 461-475. doi: 10.3934/dcds.2014.34.461
[16]

Petru Jebelean. Infinitely many solutions for ordinary $p$-Laplacian systems with nonlinear boundary conditions. Communications on Pure & Applied Analysis, 2008, 7 (2) : 267-275. doi: 10.3934/cpaa.2008.7.267
[17]

Lushun Wang, Minbo Yang, Yu Zheng. Infinitely many segregated solutions for coupled nonlinear Schrödinger systems. Discrete & Continuous Dynamical Systems - A, 2019, 39 (10) : 6069-6102. doi: 10.3934/dcds.2019265
[18]

S. L. Ma'u, P. Ramankutty. An averaging method for the Helmholtz equation. Conference Publications, 2003, 2003 (Special) : 604-609. doi: 10.3934/proc.2003.2003.604
[19]

Silvia Caprino, Carlo Marchioro. On the plasma-charge model. Kinetic & Related Models, 2010, 3 (2) : 241-254. doi: 10.3934/krm.2010.3.241
[20]

John Sylvester. An estimate for the free Helmholtz equation that scales. Inverse Problems & Imaging, 2009, 3 (2) : 333-351. doi: 10.3934/ipi.2009.3.333

2018 Impact Factor: 1.38

Tools

Metrics

  • PDF downloads (22)
  • HTML views (0)
  • Cited by (4)

Other articles
by authors

[Back to Top]

Copyright © 2020 American Institute of Mathematical Sciences