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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 & Related Models, 2019, 12 (6) : 1359-1429. doi: 10.3934/krm.2019053
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. Google Scholar
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. Google Scholar
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. Google Scholar
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. Google Scholar
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. Google Scholar
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. Google Scholar
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. Google Scholar
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. Google Scholar
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. Google Scholar
R. T. Glassey, The Cauchy Problems in Kinetic Theory, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 1996. doi: 10.1137/1.9781611971477. Google Scholar
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. Google Scholar
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. Google Scholar
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. Google Scholar
Y. Guo, The Vlasov-Poisson-Boltzmann system near Maxwellians, Comm. Pure Appl. Math., 55 (2002), 1104-1135. doi: 10.1002/cpa.10040. Google Scholar
Y. Guo, The Vlasov-Maxwell-Boltzmann system near Maxwellians, Invent. Math., 153 (2003), 593-630. doi: 10.1007/s00222-003-0301-z. Google Scholar
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. Google Scholar
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. Google Scholar
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. Google Scholar
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. Google Scholar
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. Google Scholar
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. Google Scholar
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. Google Scholar
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. Google Scholar
J. C. Maxwell, On the dynamical theory of gases, Phil. Trans. Roy. Soc. London, 157 (1866), 49-88. Google Scholar
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. Google Scholar
Zhendong Fang, Hao Wang. Convergence from two-species Vlasov-Poisson-Boltzmann system to two-fluid incompressible Navier-Stokes-Fourier-Poisson system. Discrete & Continuous Dynamical Systems - B, 2021 doi: 10.3934/dcdsb.2021231
Guochun Wu, Yinghui Zhang. Global analysis of strong solutions for the viscous liquid-gas two-phase flow model in a bounded domain. Discrete & Continuous Dynamical Systems - B, 2018, 23 (4) : 1411-1429. doi: 10.3934/dcdsb.2018157
Laurent Bernis, Laurent Desvillettes. Propagation of singularities for classical solutions of the Vlasov-Poisson-Boltzmann equation. Discrete & Continuous Dynamical Systems, 2009, 24 (1) : 13-33. doi: 10.3934/dcds.2009.24.13
Jean Dolbeault. An introduction to kinetic equations: the Vlasov-Poisson system and the Boltzmann equation. Discrete & Continuous Dynamical Systems, 2002, 8 (2) : 361-380. doi: 10.3934/dcds.2002.8.361
Mehdi Badsi. Collisional sheath solutions of a bi-species Vlasov-Poisson-Boltzmann boundary value problem. Kinetic & Related Models, 2021, 14 (1) : 149-174. doi: 10.3934/krm.2020052
Renjun Duan, Tong Yang, Changjiang Zhu. Boltzmann equation with external force and Vlasov-Poisson-Boltzmann system in infinite vacuum. Discrete & Continuous Dynamical Systems, 2006, 16 (1) : 253-277. doi: 10.3934/dcds.2006.16.253
Hyung Ju Hwang, Jaewoo Jung, Juan J. L. Velázquez. On global existence of classical solutions for the Vlasov-Poisson system in convex bounded domains. Discrete & Continuous Dynamical Systems, 2013, 33 (2) : 723-737. doi: 10.3934/dcds.2013.33.723
Megan Griffin-Pickering, Mikaela Iacobelli. Global strong solutions in $ {\mathbb{R}}^3 $ for ionic Vlasov-Poisson systems. Kinetic & Related Models, 2021, 14 (4) : 571-597. doi: 10.3934/krm.2021016
Hai-Liang Li, Tong Yang, Mingying Zhong. Diffusion limit of the Vlasov-Poisson-Boltzmann system. Kinetic & Related Models, 2021, 14 (2) : 211-255. doi: 10.3934/krm.2021003
Gleb G. Doronin, Nikolai A. Larkin. Kawahara equation in a bounded domain. Discrete & Continuous Dynamical Systems - B, 2008, 10 (4) : 783-799. doi: 10.3934/dcdsb.2008.10.783
Xiaoli Li. Global strong solution for the incompressible flow of liquid crystals with vacuum in dimension two. Discrete & Continuous Dynamical Systems, 2017, 37 (9) : 4907-4922. doi: 10.3934/dcds.2017211
Chuntian Wang. The existence of strong solutions to the $3D$ Zakharov-Kuznestov equation in a bounded domain. Discrete & Continuous Dynamical Systems, 2014, 34 (11) : 4897-4910. doi: 10.3934/dcds.2014.34.4897
Ke Lin, Chunlai Mu. Convergence of global and bounded solutions of a two-species chemotaxis model with a logistic source. Discrete & Continuous Dynamical Systems - B, 2017, 22 (6) : 2233-2260. doi: 10.3934/dcdsb.2017094
Robert T. Glassey, Walter A. Strauss. Perturbation of essential spectra of evolution operators and the Vlasov-Poisson-Boltzmann system. Discrete & Continuous Dynamical Systems, 1999, 5 (3) : 457-472. doi: 10.3934/dcds.1999.5.457
Hao Wang. Uniform stability estimate for the Vlasov-Poisson-Boltzmann system. Discrete & Continuous Dynamical Systems, 2021, 41 (2) : 657-680. doi: 10.3934/dcds.2020292
Marc Briant. Stability of global equilibrium for the multi-species Boltzmann equation in $L^\infty$ settings. Discrete & Continuous Dynamical Systems, 2016, 36 (12) : 6669-6688. doi: 10.3934/dcds.2016090
Xue-Li Song, Yan-Ren Hou. Pullback $\mathcal{D}$-attractors for the non-autonomous Newton-Boussinesq equation in two-dimensional bounded domain. Discrete & Continuous Dynamical Systems, 2012, 32 (3) : 991-1009. doi: 10.3934/dcds.2012.32.991
Yongfu Wang. Global strong solution to the two dimensional nonhomogeneous incompressible heat conducting Navier-Stokes flows with vacuum. Discrete & Continuous Dynamical Systems - B, 2020, 25 (11) : 4317-4333. doi: 10.3934/dcdsb.2020099
Jishan Fan, Fucai Li, Gen Nakamura. Global strong solution to the two-dimensional density-dependent magnetohydrodynamic equations with vaccum. Communications on Pure & Applied Analysis, 2014, 13 (4) : 1481-1490. doi: 10.3934/cpaa.2014.13.1481
Radjesvarane Alexandre, Yoshinori Morimoto, Seiji Ukai, Chao-Jiang Xu, Tong Yang. Bounded solutions of the Boltzmann equation in the whole space. Kinetic & Related Models, 2011, 4 (1) : 17-40. doi: 10.3934/krm.2011.4.17