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

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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

Abstract
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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:

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[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
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[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

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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

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