American Institute of Mathematical Sciences

Advanced
February  2013, 33(2): 723-737. doi: 10.3934/dcds.2013.33.723

On global existence of classical solutions for the Vlasov-Poisson system in convex bounded domains

Hyung Ju Hwang 1, , Jaewoo Jung 2, and  Juan J. L. Velázquez 3,

1. 

Department of Mathematics, Pohang University of Science and Technology, Pohang, Gyeongbuk, South Korea
2. 

Department of Mathematics, Pohang University of Science and Technology, Pohang, 790-784, South Korea
3. 

Institute for Applied Mathematics, University of Bonn, Endenicher Allee 60, 53115 Bonn, Germany

Received  July 2011 Revised  March 2012 Published  September 2012

We prove global existence of strong solutions for the Vlasov-Poisson system in a convex bounded domain in the plasma physics case assuming homogeneous Dirichlet boundary conditions for the electric potential and the specular reflection boundary conditions for the distribution density.
Keywords: boundary value problem, global existence, convex domain., Vlasov-Poisson.
Mathematics Subject Classification: Primary: 82D10; Secondary: 35Q83, 76X0.
Citation: 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
References:
[1]

C. Bardos and P. Degond, Global existence for the Vlasov-Poisson system in 3 space variables with small initial data,, Ann. Inst. H. Poincaré Anal. Non Linéaire, 2 (1985), 101.   Google Scholar

[2]

J. Batt, Global symmetric solutions of the initial value problem of stellar dynamics,, J. Differential Equations, 25 (1977), 342.  doi: 10.1016/0022-0396(77)90049-3.  Google Scholar

[3]

J. W. Connor, An analytic solution for the distribution of neutral particles in a Maxwellian plasma using the method of singular eigenfunctions,, Plasma Physics, 19 (1977), 853.  doi: 10.1088/0032-1028/19/9/006.  Google Scholar

[4]

J. W. Gadzuk, Theory of dielectric screening of an impurity at the surface of an electron gas,, J. Phys. Chem. Solids, 30 (1969), 2307.  doi: 10.1016/0022-3697(69)90157-7.  Google Scholar

[5]

R. Glassey, "The Cauchy Problem in Kinetic Theory,", Society for Industrial and Applied Mathematics (SIAM), (1996).   Google Scholar

[6]

Y. Guo, Singular solutions of Vlasov-Maxwell system on a half line,, Arch. Ration. Mech. Anal., 131 (1995), 241.  doi: 10.1007/BF00382888.  Google Scholar

[7]

Y. Guo, Regularity for the Vlasov equations in a half space,, Indiana Univ. Math. J., 43 (1994), 255.  doi: 10.1512/iumj.1994.43.43013.  Google Scholar

[8]

J. H. Hopps and W. L. Waldron, Surface modes in electron plasmas,, Physical Review A, 15 (1977), 1721.  doi: 10.1103/PhysRevA.15.1721.  Google Scholar

[9]

E. Horst, On the classical solutions of the initial value problem for the unmodified non-linear Vlasov equation, Parts I,, Math. Methods Appl. Sci., 3 (1981), 229.   Google Scholar

[10]

E. Horst, On the classical solutions of the initial value problem for the unmodified non-linear Vlasov equation, Parts II,, Math. Methods Appl. Sci., 4 (1982), 19.   Google Scholar

[11]

H. J. Hwang, Regularity for the Vlasov-Poisson system in a convex domain,, SIAM J. Math. Anal., 36 (2004), 121.  doi: 10.1137/S0036141003422278.  Google Scholar

[12]

H. J. Hwang and J.J . L. Velázquez, On global existence for the Vlasov-Poisson system in a half space,, J. Differential Equations, 247 (2009), 1915.  doi: 10.1016/j.jde.2009.06.004.  Google Scholar

[13]

H. J. Hwang and J. J. L. Velázquez, Global existence for the Vlasov-Poisson system in bounded domains,, Arch. Ration. Mech. Anal., 195 (2010), 763.  doi: 10.1007/s00205-009-0239-4.  Google Scholar

[14]

S. V. Iordanskii, The Cauchy problem for the kinetic equation of plasma,, Trudy Mat. Inst. Steklov., 60 (1961), 181.   Google Scholar

[15]

P. L. Lions and B. Perthame, Propagation of moments and regularity of solutions for the 3-dimensional Vlasov-Poisson system,, Invent. Math., 105 (1991), 415.  doi: 10.1007/BF01232273.  Google Scholar

[16]

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

[17]

K. U. Riemann, The Bohm criterion and sheath formation,, J. Phys. D: Appl. Phys., 24 (1991), 492.  doi: 10.1088/0022-3727/24/4/001.  Google Scholar

[18]

A. Shivarova and I. Zhelyazkov, Surface waves in a homogeneous plasma sharply bounded by a dielectric,, Plasma Physics, 20 (1978), 1049.  doi: 10.1088/0032-1028/20/10/007.  Google Scholar

[19]

D. J. Struik, "Lectures on Classical Differential Geometry,", Dover Publications, (1988).   Google Scholar

[20]

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

show all references

References:
[1]

C. Bardos and P. Degond, Global existence for the Vlasov-Poisson system in 3 space variables with small initial data,, Ann. Inst. H. Poincaré Anal. Non Linéaire, 2 (1985), 101.   Google Scholar

[2]

J. Batt, Global symmetric solutions of the initial value problem of stellar dynamics,, J. Differential Equations, 25 (1977), 342.  doi: 10.1016/0022-0396(77)90049-3.  Google Scholar

[3]

J. W. Connor, An analytic solution for the distribution of neutral particles in a Maxwellian plasma using the method of singular eigenfunctions,, Plasma Physics, 19 (1977), 853.  doi: 10.1088/0032-1028/19/9/006.  Google Scholar

[4]

J. W. Gadzuk, Theory of dielectric screening of an impurity at the surface of an electron gas,, J. Phys. Chem. Solids, 30 (1969), 2307.  doi: 10.1016/0022-3697(69)90157-7.  Google Scholar

[5]

R. Glassey, "The Cauchy Problem in Kinetic Theory,", Society for Industrial and Applied Mathematics (SIAM), (1996).   Google Scholar

[6]

Y. Guo, Singular solutions of Vlasov-Maxwell system on a half line,, Arch. Ration. Mech. Anal., 131 (1995), 241.  doi: 10.1007/BF00382888.  Google Scholar

[7]

Y. Guo, Regularity for the Vlasov equations in a half space,, Indiana Univ. Math. J., 43 (1994), 255.  doi: 10.1512/iumj.1994.43.43013.  Google Scholar

[8]

J. H. Hopps and W. L. Waldron, Surface modes in electron plasmas,, Physical Review A, 15 (1977), 1721.  doi: 10.1103/PhysRevA.15.1721.  Google Scholar

[9]

E. Horst, On the classical solutions of the initial value problem for the unmodified non-linear Vlasov equation, Parts I,, Math. Methods Appl. Sci., 3 (1981), 229.   Google Scholar

[10]

E. Horst, On the classical solutions of the initial value problem for the unmodified non-linear Vlasov equation, Parts II,, Math. Methods Appl. Sci., 4 (1982), 19.   Google Scholar

[11]

H. J. Hwang, Regularity for the Vlasov-Poisson system in a convex domain,, SIAM J. Math. Anal., 36 (2004), 121.  doi: 10.1137/S0036141003422278.  Google Scholar

[12]

H. J. Hwang and J.J . L. Velázquez, On global existence for the Vlasov-Poisson system in a half space,, J. Differential Equations, 247 (2009), 1915.  doi: 10.1016/j.jde.2009.06.004.  Google Scholar

[13]

H. J. Hwang and J. J. L. Velázquez, Global existence for the Vlasov-Poisson system in bounded domains,, Arch. Ration. Mech. Anal., 195 (2010), 763.  doi: 10.1007/s00205-009-0239-4.  Google Scholar

[14]

S. V. Iordanskii, The Cauchy problem for the kinetic equation of plasma,, Trudy Mat. Inst. Steklov., 60 (1961), 181.   Google Scholar

[15]

P. L. Lions and B. Perthame, Propagation of moments and regularity of solutions for the 3-dimensional Vlasov-Poisson system,, Invent. Math., 105 (1991), 415.  doi: 10.1007/BF01232273.  Google Scholar

[16]

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

[17]

K. U. Riemann, The Bohm criterion and sheath formation,, J. Phys. D: Appl. Phys., 24 (1991), 492.  doi: 10.1088/0022-3727/24/4/001.  Google Scholar

[18]

A. Shivarova and I. Zhelyazkov, Surface waves in a homogeneous plasma sharply bounded by a dielectric,, Plasma Physics, 20 (1978), 1049.  doi: 10.1088/0032-1028/20/10/007.  Google Scholar

[19]

D. J. Struik, "Lectures on Classical Differential Geometry,", Dover Publications, (1988).   Google Scholar

[20]

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

[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]

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
[3]

Katherine Zhiyuan Zhang. Focusing solutions of the Vlasov-Poisson system. Kinetic & Related Models, 2019, 12 (6) : 1313-1327. doi: 10.3934/krm.2019051
[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]

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
[6]

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
[7]

Gianluca Crippa, Silvia Ligabue, Chiara Saffirio. Lagrangian solutions to the Vlasov-Poisson system with a point charge. Kinetic & Related Models, 2018, 11 (6) : 1277-1299. doi: 10.3934/krm.2018050
[8]

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
[9]

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
[10]

Ivonne Rivas, Muhammad Usman, Bing-Yu Zhang. Global well-posedness and asymptotic behavior of a class of initial-boundary-value problem of the Korteweg-De Vries equation on a finite domain. Mathematical Control & Related Fields, 2011, 1 (1) : 61-81. doi: 10.3934/mcrf.2011.1.61
[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]

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
[13]

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
[14]

Joackim Bernier, Michel Mehrenberger. Long-time behavior of second order linearized Vlasov-Poisson equations near a homogeneous equilibrium. Kinetic & Related Models, 2020, 13 (1) : 129-168. doi: 10.3934/krm.2020005
[15]

Piotr Kowalski. The existence of a solution for Dirichlet boundary value problem for a Duffing type differential inclusion. Discrete & Continuous Dynamical Systems - B, 2014, 19 (8) : 2569-2580. doi: 10.3934/dcdsb.2014.19.2569
[16]

Francesca Marcellini. Existence of solutions to a boundary value problem for a phase transition traffic model. Networks & Heterogeneous Media, 2017, 12 (2) : 259-275. doi: 10.3934/nhm.2017011
[17]

John R. Graef, Lingju Kong, Min Wang. Existence of multiple solutions to a discrete fourth order periodic boundary value problem. Conference Publications, 2013, 2013 (special) : 291-299. doi: 10.3934/proc.2013.2013.291
[18]

Feliz Minhós, T. Gyulov, A. I. Santos. Existence and location result for a fourth order boundary value problem. Conference Publications, 2005, 2005 (Special) : 662-671. doi: 10.3934/proc.2005.2005.662
[19]

Guglielmo Feltrin. Existence of positive solutions of a superlinear boundary value problem with indefinite weight. Conference Publications, 2015, 2015 (special) : 436-445. doi: 10.3934/proc.2015.0436
[20]

Eric R. Kaufmann. Existence and nonexistence of positive solutions for a nonlinear fractional boundary value problem. Conference Publications, 2009, 2009 (Special) : 416-423. doi: 10.3934/proc.2009.2009.416

2018 Impact Factor: 1.143

Tools

Metrics

  • PDF downloads (20)
  • HTML views (0)
  • Cited by (1)

Other articles
by authors

[Back to Top]

Copyright © 2019 American Institute of Mathematical Sciences