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The inviscid limit for the 2D Navier-Stokes equations in bounded domains
A toy model for the relativistic Vlasov-Maxwell system
1. | School of Mathematics, Cardiff University, Cardiff CF24 4AG, Wales, United Kingdom |
2. | Department of Applied Mathematics and Statistics, Colorado School of Mines, Golden, CO, USA |
3. | School of Mathematics, Beijing Institute of Technology, Beijing, China |
The global-in-time existence of classical solutions to the relativistic Vlasov-Maxwell (RVM) system in three space dimensions remains elusive after nearly four decades of mathematical research. In this note, a simplified "toy model" is presented and studied. This toy model retains one crucial aspect of the RVM system: the phase-space evolution of the distribution function is governed by a transport equation whose forcing term satisfies a wave equation with finite speed of propagation.
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] |
P. Gérard and C. Pallard,
A mean-field toy model for resonant transport, Kinet. Relat. Models, 3 (2010), 299-309.
doi: 10.3934/krm.2010.3.299. |
[3] |
R. T. Glassey and J. W. Schaeffer,
Global existence for the relativistic Vlasov-Maxwell system with nearly neutral initial data, Comm. Math. Phys., 119 (1988), 353-384.
doi: 10.1007/BF01218078. |
[4] |
R. T. Glassey and J. W. 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. |
[5] |
R. T. Glassey and J. W. Schaeffer,
The "two and one-half dimensional" relativistic vlasov maxwell system, Comm. Math. Phys., 185 (1997), 257-284.
doi: 10.1007/s002200050090. |
[6] |
R. T. Glassey and W. A. 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. |
[7] |
R. T. Glassey and W. A. Strauss,
Absence of shocks in an initially dilute collisionless plasma, Comm. Math. Phys., 113 (1987), 191-208.
doi: 10.1007/BF01223511. |
[8] |
S. Klainerman and G. Staffilani,
A new approach to study the Vlasov-Maxwell system, Commun. Pure Appl. Anal., 1 (2002), 103-125.
doi: 10.3934/cpaa.2002.1.103. |
[9] |
J. Luk and R. M. Strain,
A new continuation criterion for the relativistic Vlasov-Maxwell system, Comm. Math. Phys., 331 (2014)), 1005-1027.
doi: 10.1007/s00220-014-2108-8. |
[10] |
C. Nguyen and S. Pankavich,
A one-dimensional kinetic model of plasma dynamics with a transport field, Evol. Equ. Control Theory, 3 (2014), 681-698.
doi: 10.3934/eect.2014.3.681. |
[11] |
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. |
[12] |
J. W. Schaeffer,
The classical limit of the relativistic Vlasov-Maxwell system, Comm. Math. Phys., 104 (1986), 403-421.
doi: 10.1007/BF01210948. |
[13] |
J. W. Schaeffer,
Global existence of smooth solutions to the vlasov poisson system in three dimensions, Comm. Partial Differential Equations, 16 (1991), 1313-1335.
doi: 10.1080/03605309108820801. |
[14] |
S. Wollman,
An existence and uniqueness theorem for the Vlasov-Maxwell system, Comm. Pure Appl. Math., 37 (1984), 457-462.
doi: 10.1002/cpa.3160370404. |
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] |
P. Gérard and C. Pallard,
A mean-field toy model for resonant transport, Kinet. Relat. Models, 3 (2010), 299-309.
doi: 10.3934/krm.2010.3.299. |
[3] |
R. T. Glassey and J. W. Schaeffer,
Global existence for the relativistic Vlasov-Maxwell system with nearly neutral initial data, Comm. Math. Phys., 119 (1988), 353-384.
doi: 10.1007/BF01218078. |
[4] |
R. T. Glassey and J. W. 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. |
[5] |
R. T. Glassey and J. W. Schaeffer,
The "two and one-half dimensional" relativistic vlasov maxwell system, Comm. Math. Phys., 185 (1997), 257-284.
doi: 10.1007/s002200050090. |
[6] |
R. T. Glassey and W. A. 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. |
[7] |
R. T. Glassey and W. A. Strauss,
Absence of shocks in an initially dilute collisionless plasma, Comm. Math. Phys., 113 (1987), 191-208.
doi: 10.1007/BF01223511. |
[8] |
S. Klainerman and G. Staffilani,
A new approach to study the Vlasov-Maxwell system, Commun. Pure Appl. Anal., 1 (2002), 103-125.
doi: 10.3934/cpaa.2002.1.103. |
[9] |
J. Luk and R. M. Strain,
A new continuation criterion for the relativistic Vlasov-Maxwell system, Comm. Math. Phys., 331 (2014)), 1005-1027.
doi: 10.1007/s00220-014-2108-8. |
[10] |
C. Nguyen and S. Pankavich,
A one-dimensional kinetic model of plasma dynamics with a transport field, Evol. Equ. Control Theory, 3 (2014), 681-698.
doi: 10.3934/eect.2014.3.681. |
[11] |
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. |
[12] |
J. W. Schaeffer,
The classical limit of the relativistic Vlasov-Maxwell system, Comm. Math. Phys., 104 (1986), 403-421.
doi: 10.1007/BF01210948. |
[13] |
J. W. Schaeffer,
Global existence of smooth solutions to the vlasov poisson system in three dimensions, Comm. Partial Differential Equations, 16 (1991), 1313-1335.
doi: 10.1080/03605309108820801. |
[14] |
S. Wollman,
An existence and uniqueness theorem for the Vlasov-Maxwell system, Comm. Pure Appl. Math., 37 (1984), 457-462.
doi: 10.1002/cpa.3160370404. |
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