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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, 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.  Google Scholar

[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.  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), 853-873. 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), 2307-2319. 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), Philadelphia, PA, 1996.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  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), 229-248.  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), 19-32.  Google Scholar

[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.  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), 1915-1948. 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), 763-796. 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), 181-194.  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), 415-430. 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), 281-303. 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), 492-518. 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), 1049-1073. doi: 10.1088/0032-1028/20/10/007.  Google Scholar

[19]

D. J. Struik, "Lectures on Classical Differential Geometry," Dover Publications, Inc., New York, 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), 245-261.  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), 101-118.  Google Scholar

[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.  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), 853-873. 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), 2307-2319. 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), Philadelphia, PA, 1996.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  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), 229-248.  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), 19-32.  Google Scholar

[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.  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), 1915-1948. 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), 763-796. 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), 181-194.  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), 415-430. 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), 281-303. 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), 492-518. 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), 1049-1073. doi: 10.1088/0032-1028/20/10/007.  Google Scholar

[19]

D. J. Struik, "Lectures on Classical Differential Geometry," Dover Publications, Inc., New York, 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), 245-261.  Google Scholar

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