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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 & 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.  doi: 10.1007/s00205-003-0265-6.  Google Scholar

[2]

S. Caprino, G. Cavallaro and C. Marchioro, Time evolution of a Vlasov-Poisson plasma with magnetic confinement,, Kinet. Relat. Models, 5 (2012), 729.  doi: 10.3934/krm.2012.5.729.  Google Scholar

[3]

S. Caprino, G. Cavallaro and C. Marchioro, On a magnetically confined plasma with infinite charge,, SIAM J. Math. Anal., 46 (2014), 133.  doi: 10.1137/130916527.  Google Scholar

[4]

R. DiPerna and P.-L. Lions, Global weak solutions of Vlasov-Maxwell systems,, Comm. Pure Appl. Math., 42 (1989), 729.  doi: 10.1002/cpa.3160420603.  Google Scholar

[5]

P. Garabedian, A unified theory of tokamaks and stellarators,, Comm. Pure Appl. Math., 47 (1994), 281.  doi: 10.1002/cpa.3160470303.  Google Scholar

[6]

R. Glassey, The Cauchy Problem in Kinetic Theory,, Society for Industrial and Applied Mathematics (SIAM), (1996).  doi: 10.1137/1.9781611971477.  Google Scholar

[7]

R. Glassey and J. Schaeffer, On the "one and one-half dimensional'' relativistic Vlasov-Maxwell system,, Math. Methods Appl. Sci., 13 (1990), 169.  doi: 10.1002/mma.1670130207.  Google Scholar

[8]

R. Glassey and J. Schaeffer, The "two and one-half-dimensional'' relativistic Vlasov Maxwell system,, Comm. Math. Phys., 185 (1997), 257.  doi: 10.1007/s002200050090.  Google Scholar

[9]

R. Glassey and J. Schaeffer, The relativistic Vlasov-Maxwell system in two space dimensions, I, II,, Arch. Rational Mech. Anal., 141 (1998), 331.  doi: 10.1007/s002050050079.  Google Scholar

[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.  doi: 10.1007/BF00250732.  Google Scholar

[11]

Y. Guo, Global weak solutions of the Vlasov-Maxwell system with boundary conditions,, Comm. Math. Phys., 154 (1993), 245.  doi: 10.1007/BF02096997.  Google Scholar

[12]

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

[13]

D. Han-Kwan, On the confinement of a tokamak plasma,, SIAM J. Math. Anal., 42 (2010), 2337.  doi: 10.1137/090774574.  Google Scholar

[14]

S. Klainerman and G. Staffilani, A new approach to study the Vlasov-Maxwell system,, Commun. Pure Appl. Anal., 1 (2002), 103.   Google Scholar

[15]

S. Mischler, Kinetic equations with Maxwell boundary conditions,, Ann. Sci. Éc. Norm. Supér., 43 (2010), 719.   Google Scholar

[16]

R. B. White, The Theory of Toroidally Confined Plasmas,, Second edition. Imperial College Press, (2001).  doi: 10.1142/p237.  Google Scholar

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.  doi: 10.1007/s00205-003-0265-6.  Google Scholar

[2]

S. Caprino, G. Cavallaro and C. Marchioro, Time evolution of a Vlasov-Poisson plasma with magnetic confinement,, Kinet. Relat. Models, 5 (2012), 729.  doi: 10.3934/krm.2012.5.729.  Google Scholar

[3]

S. Caprino, G. Cavallaro and C. Marchioro, On a magnetically confined plasma with infinite charge,, SIAM J. Math. Anal., 46 (2014), 133.  doi: 10.1137/130916527.  Google Scholar

[4]

R. DiPerna and P.-L. Lions, Global weak solutions of Vlasov-Maxwell systems,, Comm. Pure Appl. Math., 42 (1989), 729.  doi: 10.1002/cpa.3160420603.  Google Scholar

[5]

P. Garabedian, A unified theory of tokamaks and stellarators,, Comm. Pure Appl. Math., 47 (1994), 281.  doi: 10.1002/cpa.3160470303.  Google Scholar

[6]

R. Glassey, The Cauchy Problem in Kinetic Theory,, Society for Industrial and Applied Mathematics (SIAM), (1996).  doi: 10.1137/1.9781611971477.  Google Scholar

[7]

R. Glassey and J. Schaeffer, On the "one and one-half dimensional'' relativistic Vlasov-Maxwell system,, Math. Methods Appl. Sci., 13 (1990), 169.  doi: 10.1002/mma.1670130207.  Google Scholar

[8]

R. Glassey and J. Schaeffer, The "two and one-half-dimensional'' relativistic Vlasov Maxwell system,, Comm. Math. Phys., 185 (1997), 257.  doi: 10.1007/s002200050090.  Google Scholar

[9]

R. Glassey and J. Schaeffer, The relativistic Vlasov-Maxwell system in two space dimensions, I, II,, Arch. Rational Mech. Anal., 141 (1998), 331.  doi: 10.1007/s002050050079.  Google Scholar

[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.  doi: 10.1007/BF00250732.  Google Scholar

[11]

Y. Guo, Global weak solutions of the Vlasov-Maxwell system with boundary conditions,, Comm. Math. Phys., 154 (1993), 245.  doi: 10.1007/BF02096997.  Google Scholar

[12]

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

[13]

D. Han-Kwan, On the confinement of a tokamak plasma,, SIAM J. Math. Anal., 42 (2010), 2337.  doi: 10.1137/090774574.  Google Scholar

[14]

S. Klainerman and G. Staffilani, A new approach to study the Vlasov-Maxwell system,, Commun. Pure Appl. Anal., 1 (2002), 103.   Google Scholar

[15]

S. Mischler, Kinetic equations with Maxwell boundary conditions,, Ann. Sci. Éc. Norm. Supér., 43 (2010), 719.   Google Scholar

[16]

R. B. White, The Theory of Toroidally Confined Plasmas,, Second edition. Imperial College Press, (2001).  doi: 10.1142/p237.  Google Scholar

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