# American Institute of Mathematical Sciences

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Global classical solutions for the "One and one-half'' dimensional relativistic Vlasov-Maxwell-Fokker-Planck system
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March  2015, 8(1): 153-168. doi: 10.3934/krm.2015.8.153

## Global magnetic confinement for the 1.5D Vlasov-Maxwell system

 1 Department of Mathematics, Pennsylvania State University, State College, PA 16802, United States 2 Department of Mathematics, The University of Akron, Akron, OH 44325, United States 3 Brown University, Department of Mathematics and Lefschetz Center for Dynamical Systems, Providence, RI 02912

Received  August 2014 Revised  September 2014 Published  December 2014

We establish the global-in-time existence and uniqueness of classical solutions to the one and one-half'' dimensional relativistic Vlasov--Maxwell systems in a bounded interval, subject to an external magnetic field which is infinitely large at the spatial boundary. We prove that the large external magnetic field confines the particles to a compact set away from the boundary. This excludes the known singularities that typically occur due to particles that repeatedly bounce off the boundary. In addition to the confinement, we follow the techniques introduced by Glassey and Schaeffer, who studied the Cauchy problem without boundaries.
Citation: Toan T. Nguyen, Truyen V. Nguyen, Walter A. Strauss. Global magnetic confinement for the 1.5D Vlasov-Maxwell system. Kinetic and Related Models, 2015, 8 (1) : 153-168. doi: 10.3934/krm.2015.8.153
##### References:
 [1] F. Bouchut, F. Golse and C. Pallard, Classical solutions and the Glassey-Strauss theorem for the 3D Vlasov-Maxwell system, Arch. Ration. Mech. Anal., 170 (2003), 1-15. doi: 10.1007/s00205-003-0265-6. [2] S. Caprino, G. Cavallaro and C. Marchioro, Time evolution of a Vlasov-Poisson plasma with magnetic confinement, Kinet. Relat. Models, 5 (2012), 729-742. doi: 10.3934/krm.2012.5.729. [3] S. Caprino, G. Cavallaro and C. Marchioro, On a magnetically confined plasma with infinite charge, SIAM J. Math. Anal., 46 (2014), 133-164. doi: 10.1137/130916527. [4] R. DiPerna and P.-L. Lions, Global weak solutions of Vlasov-Maxwell systems, Comm. Pure Appl. Math., 42 (1989), 729-757. doi: 10.1002/cpa.3160420603. [5] P. Garabedian, A unified theory of tokamaks and stellarators, Comm. Pure Appl. Math., 47 (1994), 281-292. doi: 10.1002/cpa.3160470303. [6] R. Glassey, The Cauchy Problem in Kinetic Theory, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 1996. doi: 10.1137/1.9781611971477. [7] R. Glassey and J. Schaeffer, On the "one and one-half dimensional'' relativistic Vlasov-Maxwell system, Math. Methods Appl. Sci., 13 (1990), 169-179. doi: 10.1002/mma.1670130207. [8] R. Glassey and J. Schaeffer, The "two and one-half-dimensional'' relativistic Vlasov Maxwell system, Comm. Math. Phys., 185 (1997), 257-284. doi: 10.1007/s002200050090. [9] R. Glassey and J. Schaeffer, The relativistic Vlasov-Maxwell system in two space dimensions, I, II, Arch. Rational Mech. Anal., 141 (1998), 331-354, 355-374. doi: 10.1007/s002050050079. [10] R. Glassey and W. Strauss, Singularity formation in a collisionless plasma could occur only at high velocities, Arch. Rational Mech. Anal., 92 (1986), 59-90. doi: 10.1007/BF00250732. [11] Y. Guo, Global weak solutions of the Vlasov-Maxwell system with boundary conditions, Comm. Math. Phys., 154 (1993), 245-263. doi: 10.1007/BF02096997. [12] Y. Guo, Singular solutions of the Vlasov-Maxwell system on a half line, Arch. Rational Mech. Anal., 131 (1995), 241-304. doi: 10.1007/BF00382888. [13] D. Han-Kwan, On the confinement of a tokamak plasma, SIAM J. Math. Anal., 42 (2010), 2337-2367. doi: 10.1137/090774574. [14] S. Klainerman and G. Staffilani, A new approach to study the Vlasov-Maxwell system, Commun. Pure Appl. Anal., 1 (2002), 103-125. [15] S. Mischler, Kinetic equations with Maxwell boundary conditions, Ann. Sci. Éc. Norm. Supér., (4) 43 (2010), 719-760. [16] R. B. White, The Theory of Toroidally Confined Plasmas, Second edition. Imperial College Press, London, 2001. doi: 10.1142/p237.

show all references

##### References:
 [1] F. Bouchut, F. Golse and C. Pallard, Classical solutions and the Glassey-Strauss theorem for the 3D Vlasov-Maxwell system, Arch. Ration. Mech. Anal., 170 (2003), 1-15. doi: 10.1007/s00205-003-0265-6. [2] S. Caprino, G. Cavallaro and C. Marchioro, Time evolution of a Vlasov-Poisson plasma with magnetic confinement, Kinet. Relat. Models, 5 (2012), 729-742. doi: 10.3934/krm.2012.5.729. [3] S. Caprino, G. Cavallaro and C. Marchioro, On a magnetically confined plasma with infinite charge, SIAM J. Math. Anal., 46 (2014), 133-164. doi: 10.1137/130916527. [4] R. DiPerna and P.-L. Lions, Global weak solutions of Vlasov-Maxwell systems, Comm. Pure Appl. Math., 42 (1989), 729-757. doi: 10.1002/cpa.3160420603. [5] P. Garabedian, A unified theory of tokamaks and stellarators, Comm. Pure Appl. Math., 47 (1994), 281-292. doi: 10.1002/cpa.3160470303. [6] R. Glassey, The Cauchy Problem in Kinetic Theory, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 1996. doi: 10.1137/1.9781611971477. [7] R. Glassey and J. Schaeffer, On the "one and one-half dimensional'' relativistic Vlasov-Maxwell system, Math. Methods Appl. Sci., 13 (1990), 169-179. doi: 10.1002/mma.1670130207. [8] R. Glassey and J. Schaeffer, The "two and one-half-dimensional'' relativistic Vlasov Maxwell system, Comm. Math. Phys., 185 (1997), 257-284. doi: 10.1007/s002200050090. [9] R. Glassey and J. Schaeffer, The relativistic Vlasov-Maxwell system in two space dimensions, I, II, Arch. Rational Mech. Anal., 141 (1998), 331-354, 355-374. doi: 10.1007/s002050050079. [10] R. Glassey and W. Strauss, Singularity formation in a collisionless plasma could occur only at high velocities, Arch. Rational Mech. Anal., 92 (1986), 59-90. doi: 10.1007/BF00250732. [11] Y. Guo, Global weak solutions of the Vlasov-Maxwell system with boundary conditions, Comm. Math. Phys., 154 (1993), 245-263. doi: 10.1007/BF02096997. [12] Y. Guo, Singular solutions of the Vlasov-Maxwell system on a half line, Arch. Rational Mech. Anal., 131 (1995), 241-304. doi: 10.1007/BF00382888. [13] D. Han-Kwan, On the confinement of a tokamak plasma, SIAM J. Math. Anal., 42 (2010), 2337-2367. doi: 10.1137/090774574. [14] S. Klainerman and G. Staffilani, A new approach to study the Vlasov-Maxwell system, Commun. Pure Appl. Anal., 1 (2002), 103-125. [15] S. Mischler, Kinetic equations with Maxwell boundary conditions, Ann. Sci. Éc. Norm. Supér., (4) 43 (2010), 719-760. [16] R. B. White, The Theory of Toroidally Confined Plasmas, Second edition. Imperial College Press, London, 2001. doi: 10.1142/p237.
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