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Mean-field limit of a spatially-extended FitzHugh-Nagumo neural network
A note on two species collisional plasma in bounded domains
Department of Mathematics, University of Wisconsin-Madison, Madison, WI 53706, USA |
We construct a unique global-in-time solution to the two species Vlasov-Poisson-Boltzmann system in convex domains with the diffuse boundary condition, which can be viewed as one of the ideal scattering boundary model. The construction follows a new $ L^{2} $-$ L^{\infty} $ framework in [
References:
[1] |
L. Bernis and L. Desvillettes,
Propagation of singularities for classical solutions of the Vlasov-Poisson-Boltzmann equation, Discrete Contin. Dyn. Syst., 24 (2009), 13-33.
doi: 10.3934/dcds.2009.24.13. |
[2] |
Y. B. Cao, Regularity of boltzmann equation with external fields in convex domains of diffuse reflection, SIAM J. Math. Anal., 51 (2019), 3195–3275, arXiv: 1812.09388.
doi: 10.1137/18M1234928. |
[3] |
Y. B. Cao, C. Kim and D. Lee, Global strong solutions of the Vlasov-Poisson-Boltzmann system in bounded domains, Arch. Ration. Mech. Anal., 233 (2019), 1027–1130, https://doi.org/10.1007/s00205-019-01374-9.
doi: 10.1007/s00205-019-01374-9. |
[4] |
C. Cercignani, R. Illner and M. Pulvirenti, The Mathematical Theory of Dilute Gases, Applied Mathematical Sciences, 106. Springer-Verlag, New York, 1994.
doi: 10.1007/978-1-4419-8524-8. |
[5] |
L. Desvillettes and J. Dolbeault,
On long time asymptotics of the Vlasov-Poisson-Boltzmann equation, Comm. Partial Differential Equations, 16 (1991), 451-489.
doi: 10.1080/03605309108820765. |
[6] |
R. J. Duan and R. M. Strain,
Optimal large-time behavior of the Vlasov-Maxwell-Boltzmann system in the whole space, Comm. Pure Appl. Math., 64 (2011), 1497-1546.
doi: 10.1002/cpa.20381. |
[7] |
R. Esposito, Y. Guo, C. Kim and R. Marra,
Non-isothermal boundary in the Boltzmann theory and Fourier law, Comm. Math. Phys., 323 (2003), 177-239.
doi: 10.1007/s00220-013-1766-2. |
[8] |
R. Esposito, Y. Guo, C. Kim and R. Marra, Stationary solutions to the Boltzmann equation in the hydrodynamic limit, Ann. PDE, 4 (2018), Art. 1,119 pp.
doi: 10.1007/s40818-017-0037-5. |
[9] |
D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, Reprint of the 1998 edition, Classics in Mathematics, Springer-Verlag, Berlin, 2001. |
[10] |
R. T. Glassey, The Cauchy Problems in Kinetic Theory, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 1996.
doi: 10.1137/1.9781611971477. |
[11] |
R. T. Glassey and W. A. Strauss,
Decay of the linearized Boltzmann-Vlasov system, Transport Theory Statist. Phys., 28 (1999), 135-156.
doi: 10.1080/00411459908205653. |
[12] |
Y. Guo,
Regularity of the Vlasov equations in a half space, Indiana. Math. J., 43 (1994), 255-320.
doi: 10.1512/iumj.1994.43.43013. |
[13] |
Y. Guo,
Decay and continuity of Boltzmann equation in bounded domains, Arch. Rational Mech. Anal., 197 (2010), 713-809.
doi: 10.1007/s00205-009-0285-y. |
[14] |
Y. Guo,
The Vlasov-Poisson-Boltzmann system near Maxwellians, Comm. Pure Appl. Math., 55 (2002), 1104-1135.
doi: 10.1002/cpa.10040. |
[15] |
Y. Guo,
The Vlasov-Maxwell-Boltzmann system near Maxwellians, Invent. Math., 153 (2003), 593-630.
doi: 10.1007/s00222-003-0301-z. |
[16] |
Y. Guo, C. Kim, D. Tonon and A. Trescases,
Regularity of the Boltzmann equation in convex domains, Invent. Math., 207 (2017), 115-290.
doi: 10.1007/s00222-016-0670-8. |
[17] |
Y. Guo, C. Kim, D. Tonon and A. Trescases,
A. BV-regularity of the Boltzmann equation in non-convex domains, Arch. Rational Mech. Anal., 220 (2016), 1045-1093.
doi: 10.1007/s00205-015-0948-9. |
[18] |
H. J. Hwang and J. Velázquez,
Global existence for the Vlasov-Poisson system in bounded domains, Arch. Rat. Mech. Anal., 195 (2010), 763-796.
doi: 10.1007/s00205-009-0239-4. |
[19] |
H. J. Hwang, J. H. Jang and J. Jung,
The Fokker-Planck equation with absorbing boundary conditions in bounded domains, SIAM J. Math. Anal., 50 (2018), 2194-2232.
doi: 10.1137/16M1109928. |
[20] |
C. Kim,
Boltzmann equation with a large potential in a periodic box, Comm. Partial Differential Equations, 39 (2014), 1393-1423.
doi: 10.1080/03605302.2014.903278. |
[21] |
C. Kim and D. Lee,
The Boltzmann equation with specular boundary condition in convex domains, Comm. Pure Appl. Math., 71 (2018), 411-504.
doi: 10.1002/cpa.21705. |
[22] |
C. Kim and D. Lee,
Decay of the Boltzmann equation with the specular boundary condition in non-convex cylindrical domains, Arch. Ration. Mech. Anal., 230 (2018), 49-123.
doi: 10.1007/s00205-018-1241-5. |
[23] |
C. Kim,
Formation and propagation of discontinuity for Boltzmann equation in non-convex domains, Comm. Math. Phys., 308 (2011), 641-701.
doi: 10.1007/s00220-011-1355-1. |
[24] |
J. C. Maxwell,
On the dynamical theory of gases, Phil. Trans. Roy. Soc. London, 157 (1866), 49-88.
|
[25] |
S. Mischler,
On the initial boundary value problem for the Vlasov-Poisson-Boltzmann system, Commun. Math. Phys., 210 (2000), 447-466.
doi: 10.1007/s002200050787. |
show all references
References:
[1] |
L. Bernis and L. Desvillettes,
Propagation of singularities for classical solutions of the Vlasov-Poisson-Boltzmann equation, Discrete Contin. Dyn. Syst., 24 (2009), 13-33.
doi: 10.3934/dcds.2009.24.13. |
[2] |
Y. B. Cao, Regularity of boltzmann equation with external fields in convex domains of diffuse reflection, SIAM J. Math. Anal., 51 (2019), 3195–3275, arXiv: 1812.09388.
doi: 10.1137/18M1234928. |
[3] |
Y. B. Cao, C. Kim and D. Lee, Global strong solutions of the Vlasov-Poisson-Boltzmann system in bounded domains, Arch. Ration. Mech. Anal., 233 (2019), 1027–1130, https://doi.org/10.1007/s00205-019-01374-9.
doi: 10.1007/s00205-019-01374-9. |
[4] |
C. Cercignani, R. Illner and M. Pulvirenti, The Mathematical Theory of Dilute Gases, Applied Mathematical Sciences, 106. Springer-Verlag, New York, 1994.
doi: 10.1007/978-1-4419-8524-8. |
[5] |
L. Desvillettes and J. Dolbeault,
On long time asymptotics of the Vlasov-Poisson-Boltzmann equation, Comm. Partial Differential Equations, 16 (1991), 451-489.
doi: 10.1080/03605309108820765. |
[6] |
R. J. Duan and R. M. Strain,
Optimal large-time behavior of the Vlasov-Maxwell-Boltzmann system in the whole space, Comm. Pure Appl. Math., 64 (2011), 1497-1546.
doi: 10.1002/cpa.20381. |
[7] |
R. Esposito, Y. Guo, C. Kim and R. Marra,
Non-isothermal boundary in the Boltzmann theory and Fourier law, Comm. Math. Phys., 323 (2003), 177-239.
doi: 10.1007/s00220-013-1766-2. |
[8] |
R. Esposito, Y. Guo, C. Kim and R. Marra, Stationary solutions to the Boltzmann equation in the hydrodynamic limit, Ann. PDE, 4 (2018), Art. 1,119 pp.
doi: 10.1007/s40818-017-0037-5. |
[9] |
D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, Reprint of the 1998 edition, Classics in Mathematics, Springer-Verlag, Berlin, 2001. |
[10] |
R. T. Glassey, The Cauchy Problems in Kinetic Theory, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 1996.
doi: 10.1137/1.9781611971477. |
[11] |
R. T. Glassey and W. A. Strauss,
Decay of the linearized Boltzmann-Vlasov system, Transport Theory Statist. Phys., 28 (1999), 135-156.
doi: 10.1080/00411459908205653. |
[12] |
Y. Guo,
Regularity of the Vlasov equations in a half space, Indiana. Math. J., 43 (1994), 255-320.
doi: 10.1512/iumj.1994.43.43013. |
[13] |
Y. Guo,
Decay and continuity of Boltzmann equation in bounded domains, Arch. Rational Mech. Anal., 197 (2010), 713-809.
doi: 10.1007/s00205-009-0285-y. |
[14] |
Y. Guo,
The Vlasov-Poisson-Boltzmann system near Maxwellians, Comm. Pure Appl. Math., 55 (2002), 1104-1135.
doi: 10.1002/cpa.10040. |
[15] |
Y. Guo,
The Vlasov-Maxwell-Boltzmann system near Maxwellians, Invent. Math., 153 (2003), 593-630.
doi: 10.1007/s00222-003-0301-z. |
[16] |
Y. Guo, C. Kim, D. Tonon and A. Trescases,
Regularity of the Boltzmann equation in convex domains, Invent. Math., 207 (2017), 115-290.
doi: 10.1007/s00222-016-0670-8. |
[17] |
Y. Guo, C. Kim, D. Tonon and A. Trescases,
A. BV-regularity of the Boltzmann equation in non-convex domains, Arch. Rational Mech. Anal., 220 (2016), 1045-1093.
doi: 10.1007/s00205-015-0948-9. |
[18] |
H. J. Hwang and J. Velázquez,
Global existence for the Vlasov-Poisson system in bounded domains, Arch. Rat. Mech. Anal., 195 (2010), 763-796.
doi: 10.1007/s00205-009-0239-4. |
[19] |
H. J. Hwang, J. H. Jang and J. Jung,
The Fokker-Planck equation with absorbing boundary conditions in bounded domains, SIAM J. Math. Anal., 50 (2018), 2194-2232.
doi: 10.1137/16M1109928. |
[20] |
C. Kim,
Boltzmann equation with a large potential in a periodic box, Comm. Partial Differential Equations, 39 (2014), 1393-1423.
doi: 10.1080/03605302.2014.903278. |
[21] |
C. Kim and D. Lee,
The Boltzmann equation with specular boundary condition in convex domains, Comm. Pure Appl. Math., 71 (2018), 411-504.
doi: 10.1002/cpa.21705. |
[22] |
C. Kim and D. Lee,
Decay of the Boltzmann equation with the specular boundary condition in non-convex cylindrical domains, Arch. Ration. Mech. Anal., 230 (2018), 49-123.
doi: 10.1007/s00205-018-1241-5. |
[23] |
C. Kim,
Formation and propagation of discontinuity for Boltzmann equation in non-convex domains, Comm. Math. Phys., 308 (2011), 641-701.
doi: 10.1007/s00220-011-1355-1. |
[24] |
J. C. Maxwell,
On the dynamical theory of gases, Phil. Trans. Roy. Soc. London, 157 (1866), 49-88.
|
[25] |
S. Mischler,
On the initial boundary value problem for the Vlasov-Poisson-Boltzmann system, Commun. Math. Phys., 210 (2000), 447-466.
doi: 10.1007/s002200050787. |
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