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December  2019, 12(6): 1359-1429. doi: 10.3934/krm.2019053

A note on two species collisional plasma in bounded domains

Yunbai Cao

Department of Mathematics, University of Wisconsin-Madison, Madison, WI 53706, USA

Received  March 2019 Published  September 2019

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 [3]. In our knowledge this result is the first construction of strong solutions for two species plasma models with self-consistent field in general bounded domains.

Keywords: Boltzmann equation, bounded domain, Vlasov Poisson, diffuse reflection, global strong solution, two species.
Mathematics Subject Classification: Primary: 35Q20, 35Q83; Secondary: 35Q70.
Citation: Yunbai Cao. A note on two species collisional plasma in bounded domains. Kinetic and Related Models, 2019, 12 (6) : 1359-1429. doi: 10.3934/krm.2019053
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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