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

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.
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 and Continuous Dynamical Systems, 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), 101-118.

[2]

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.

[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), 853-873. doi: 10.1088/0032-1028/19/9/006.

[4]

J. W. Gadzuk, Theory of dielectric screening of an impurity at the surface of an electron gas, J. Phys. Chem. Solids, 30 (1969), 2307-2319. doi: 10.1016/0022-3697(69)90157-7.

[5]

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

[6]

Y. Guo, Singular solutions of Vlasov-Maxwell system on a half line, Arch. Ration. Mech. Anal., 131 (1995), 241-304. doi: 10.1007/BF00382888.

[7]

Y. Guo, Regularity for the Vlasov equations in a half space, Indiana Univ. Math. J., 43 (1994), 255-320. doi: 10.1512/iumj.1994.43.43013.

[8]

J. H. Hopps and W. L. Waldron, Surface modes in electron plasmas, Physical Review A, 15 (1977), 1721-1729. doi: 10.1103/PhysRevA.15.1721.

[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), 229-248.

[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), 19-32.

[11]

H. J. Hwang, Regularity for the Vlasov-Poisson system in a convex domain, SIAM J. Math. Anal., 36 (2004), 121-171. doi: 10.1137/S0036141003422278.

[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), 1915-1948. doi: 10.1016/j.jde.2009.06.004.

[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), 763-796. doi: 10.1007/s00205-009-0239-4.

[14]

S. V. Iordanskii, The Cauchy problem for the kinetic equation of plasma, Trudy Mat. Inst. Steklov., 60 (1961), 181-194.

[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), 415-430. doi: 10.1007/BF01232273.

[16]

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.

[17]

K. U. Riemann, The Bohm criterion and sheath formation, J. Phys. D: Appl. Phys., 24 (1991), 492-518. doi: 10.1088/0022-3727/24/4/001.

[18]

A. Shivarova and I. Zhelyazkov, Surface waves in a homogeneous plasma sharply bounded by a dielectric, Plasma Physics, 20 (1978), 1049-1073. doi: 10.1088/0032-1028/20/10/007.

[19]

D. J. Struik, "Lectures on Classical Differential Geometry," Dover Publications, Inc., New York, 1988.

[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), 245-261.

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), 101-118.

[2]

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.

[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), 853-873. doi: 10.1088/0032-1028/19/9/006.

[4]

J. W. Gadzuk, Theory of dielectric screening of an impurity at the surface of an electron gas, J. Phys. Chem. Solids, 30 (1969), 2307-2319. doi: 10.1016/0022-3697(69)90157-7.

[5]

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

[6]

Y. Guo, Singular solutions of Vlasov-Maxwell system on a half line, Arch. Ration. Mech. Anal., 131 (1995), 241-304. doi: 10.1007/BF00382888.

[7]

Y. Guo, Regularity for the Vlasov equations in a half space, Indiana Univ. Math. J., 43 (1994), 255-320. doi: 10.1512/iumj.1994.43.43013.

[8]

J. H. Hopps and W. L. Waldron, Surface modes in electron plasmas, Physical Review A, 15 (1977), 1721-1729. doi: 10.1103/PhysRevA.15.1721.

[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), 229-248.

[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), 19-32.

[11]

H. J. Hwang, Regularity for the Vlasov-Poisson system in a convex domain, SIAM J. Math. Anal., 36 (2004), 121-171. doi: 10.1137/S0036141003422278.

[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), 1915-1948. doi: 10.1016/j.jde.2009.06.004.

[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), 763-796. doi: 10.1007/s00205-009-0239-4.

[14]

S. V. Iordanskii, The Cauchy problem for the kinetic equation of plasma, Trudy Mat. Inst. Steklov., 60 (1961), 181-194.

[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), 415-430. doi: 10.1007/BF01232273.

[16]

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.

[17]

K. U. Riemann, The Bohm criterion and sheath formation, J. Phys. D: Appl. Phys., 24 (1991), 492-518. doi: 10.1088/0022-3727/24/4/001.

[18]

A. Shivarova and I. Zhelyazkov, Surface waves in a homogeneous plasma sharply bounded by a dielectric, Plasma Physics, 20 (1978), 1049-1073. doi: 10.1088/0032-1028/20/10/007.

[19]

D. J. Struik, "Lectures on Classical Differential Geometry," Dover Publications, Inc., New York, 1988.

[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), 245-261.

[1]

Jack Schaeffer. Global existence for the Vlasov-Poisson system with steady spatial asymptotic behavior. Kinetic and 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 and Related Models, 2018, 11 (5) : 1183-1209. doi: 10.3934/krm.2018046

[3]

Mehdi Badsi. Collisional sheath solutions of a bi-species Vlasov-Poisson-Boltzmann boundary value problem. Kinetic and Related Models, 2021, 14 (1) : 149-174. doi: 10.3934/krm.2020052

[4]

Megan Griffin-Pickering, Mikaela Iacobelli. Global strong solutions in $ {\mathbb{R}}^3 $ for ionic Vlasov-Poisson systems. Kinetic and Related Models, 2021, 14 (4) : 571-597. doi: 10.3934/krm.2021016

[5]

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

[6]

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

[7]

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

[8]

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

[9]

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

[10]

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

[11]

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

[12]

Yulia O. Belyaeva, Björn Gebhard, Alexander L. Skubachevskii. A general way to confined stationary Vlasov-Poisson plasma configurations. Kinetic and Related Models, 2021, 14 (2) : 257-282. doi: 10.3934/krm.2021004

[13]

Jack Schaeffer. On time decay for the spherically symmetric Vlasov-Poisson system. Kinetic and Related Models, 2022, 15 (4) : 721-727. doi: 10.3934/krm.2021021

[14]

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 and Related Fields, 2011, 1 (1) : 61-81. doi: 10.3934/mcrf.2011.1.61

[15]

Gerhard Rein, Christopher Straub. On the transport operators arising from linearizing the Vlasov-Poisson or Einstein-Vlasov system about isotropic steady states. Kinetic and Related Models, 2020, 13 (5) : 933-949. doi: 10.3934/krm.2020032

[16]

Xianglong Duan. Sharp decay estimates for the Vlasov-Poisson and Vlasov-Yukawa systems with small data. Kinetic and Related Models, 2022, 15 (1) : 119-146. doi: 10.3934/krm.2021049

[17]

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

[18]

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

[19]

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

[20]

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

2021 Impact Factor: 1.588

Metrics

  • PDF downloads (88)
  • HTML views (0)
  • Cited by (1)

[Back to Top]