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 & 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]

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

[2]

Ahmad Z. Fino, Wenhui Chen. A global existence result for two-dimensional semilinear strongly damped wave equation with mixed nonlinearity in an exterior domain. Communications on Pure & Applied Analysis, 2020, 19 (12) : 5387-5411. doi: 10.3934/cpaa.2020243

[3]

Antoine Benoit. Weak well-posedness of hyperbolic boundary value problems in a strip: when instabilities do not reflect the geometry. Communications on Pure & Applied Analysis, 2020, 19 (12) : 5475-5486. doi: 10.3934/cpaa.2020248

[4]

Mokhtar Bouloudene, Manar A. Alqudah, Fahd Jarad, Yassine Adjabi, Thabet Abdeljawad. Nonlinear singular $ p $ -Laplacian boundary value problems in the frame of conformable derivative. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020442

[5]

Marco Ghimenti, Anna Maria Micheletti. Compactness results for linearly perturbed Yamabe problem on manifolds with boundary. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020453

[6]

Lingfeng Li, Shousheng Luo, Xue-Cheng Tai, Jiang Yang. A new variational approach based on level-set function for convex hull problem with outliers. Inverse Problems & Imaging, , () : -. doi: 10.3934/ipi.2020070

[7]

Mengni Li. Global regularity for a class of Monge-Ampère type equations with nonzero boundary conditions. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2020267

[8]

Shenglan Xie, Maoan Han, Peng Zhu. A posteriori error estimate of weak Galerkin fem for second order elliptic problem with mixed boundary condition. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020340

[9]

Zhilei Liang, Jiangyu Shuai. Existence of strong solution for the Cauchy problem of fully compressible Navier-Stokes equations in two dimensions. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020348

[10]

Jie Zhang, Yuping Duan, Yue Lu, Michael K. Ng, Huibin Chang. Bilinear constraint based ADMM for mixed Poisson-Gaussian noise removal. Inverse Problems & Imaging, , () : -. doi: 10.3934/ipi.2020071

[11]

Haiyu Liu, Rongmin Zhu, Yuxian Geng. Gorenstein global dimensions relative to balanced pairs. Electronic Research Archive, 2020, 28 (4) : 1563-1571. doi: 10.3934/era.2020082

[12]

Jianhua Huang, Yanbin Tang, Ming Wang. Singular support of the global attractor for a damped BBM equation. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020345

[13]

Wenbin Li, Jianliang Qian. Simultaneously recovering both domain and varying density in inverse gravimetry by efficient level-set methods. Inverse Problems & Imaging, , () : -. doi: 10.3934/ipi.2020073

[14]

Denis Bonheure, Silvia Cingolani, Simone Secchi. Concentration phenomena for the Schrödinger-Poisson system in $ \mathbb{R}^2 $. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020447

[15]

Dan Zhu, Rosemary A. Renaut, Hongwei Li, Tianyou Liu. Fast non-convex low-rank matrix decomposition for separation of potential field data using minimal memory. Inverse Problems & Imaging, , () : -. doi: 10.3934/ipi.2020076

[16]

Min Chen, Olivier Goubet, Shenghao Li. Mathematical analysis of bump to bucket problem. Communications on Pure & Applied Analysis, 2020, 19 (12) : 5567-5580. doi: 10.3934/cpaa.2020251

[17]

Qingfang Wang, Hua Yang. Solutions of nonlocal problem with critical exponent. Communications on Pure & Applied Analysis, 2020, 19 (12) : 5591-5608. doi: 10.3934/cpaa.2020253

[18]

Cheng He, Changzheng Qu. Global weak solutions for the two-component Novikov equation. Electronic Research Archive, 2020, 28 (4) : 1545-1562. doi: 10.3934/era.2020081

[19]

Christian Beck, Lukas Gonon, Martin Hutzenthaler, Arnulf Jentzen. On existence and uniqueness properties for solutions of stochastic fixed point equations. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020320

[20]

Stefano Bianchini, Paolo Bonicatto. Forward untangling and applications to the uniqueness problem for the continuity equation. Discrete & Continuous Dynamical Systems - A, 2020  doi: 10.3934/dcds.2020384

2019 Impact Factor: 1.338

Metrics

  • PDF downloads (36)
  • HTML views (0)
  • Cited by (1)

[Back to Top]