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The Vlasov-Maxwell-Boltzmann system near Maxwellians in the whole space with very soft potentials. (English) Zbl 1414.82027
This paper solves the problem of the global existence of perturbative classical solutions around a global Maxwellian to Vlasov-Maxwell-Boltzmann (VMB) systems in the whole space, provided that the initial perturbation has sufficient regularity and velocity-integrability. This is done thanks to the proposed approach where the global classical solutions to the VBM system for the whole range of soft potentials are given. The obtained result shows that, as long as the initial data is small with enough regularity, one can establish the global existence of small amplitude classical solutions for the full range of cutoff intermolecular interactions with \(-3 < \gamma\le 1\).
The paper is divided into four main sections. In the introduction, we have the scope with the information about VMB systems like dilute ionized plasmas consisting of two-species particles. Section 2 gives necessary theorems and explanations about difficulties in the case \(-3 < \gamma\le 1\). It shows the main results; they are given in the following steps:
explanations about existing approaches with the list of main difficulties: (i) how to control the possible velocity-growth, (ii) how to control the convection term \(\nu \cdot \nabla_x f\) in the weighted energy estimates.
the case of soft potentials, \(-1 \le\gamma \le 1\).
presentation of the authors’ approach.
Section 3 shows all necessary proofs for the main results. Section 4 gives proofs of lemmas and estimates used in Section 3.

82C40 Kinetic theory of gases in time-dependent statistical mechanics
35Q83 Vlasov equations
35Q61 Maxwell equations
35Q20 Boltzmann equations
35L60 First-order nonlinear hyperbolic equations
35A01 Existence problems for PDEs: global existence, local existence, non-existence
Full Text: DOI arXiv
[1] Adams R.: Sobolev Spaces. Academic Press, New York (1985)
[2] Alexandre, R.; Morimoto, Y.; Ukai, S.; Yang, C.-J.; Yang, T., The Boltzmann equation without angular cutoff in the whole space: I, global existence for soft potential, J. Funct. Anal., 263, 915-1010, (2012) · Zbl 1232.35110
[3] Chapman, S., Cowling, T.G.: The Mathematical Theory of Non-uniform Gases. An account of the Kinetic Theory of Viscosity, Thermal Conduction and Diffusion in Gases, 3rd edn, prepared in co-operation with D. Burnett. Cambridge University Press, London (1970) · Zbl 0726.76084
[4] Duan, R.-J., Global smooth dynamics of a fully ionized plasma with long-range collisions, Ann. Inst. Henri Poincar Anal. Non Linaire, 31, 751-778, (2014) · Zbl 1305.82057
[5] Duan, R.-J.; Liu, S.-Q., The Vlasov-Poisson-Boltzmann system without angular cutoff, Commun. Math. Phys., 324, 1-45, (2013) · Zbl 1285.35115
[6] Duan, R.-J.; Liu, S.-Q.; Yang, T.; Zhao, H.-J., Stabilty of the nonrelativistic Vlasov-Maxwell-Boltzmann system for angular non-cutoff potentials, Kinet. Relat. Models, 6, 159-204, (2013) · Zbl 1288.35476
[7] Duan, R.-J.; Strain, R.M., Optimal large-time behavior of the Vlasov-Maxwell-Boltzmann system in the whole space, Commun. Pure Appl. Math., 24, 1497-1546, (2011) · Zbl 1244.35010
[8] Duan, R.-J.; Yang, T.; Zhao, H.-J., The Vlasov-Poisson-Boltzmann system for soft potentials, Math. Methods Models Appl. Sci., 23, 979-1028, (2013) · Zbl 1437.76056
[9] Grad, H.; Laurmann, J.A. (ed.), Asymptotic theory of the Boltzmann equation II, No. 1, 26-59, (1963), New York
[10] Gressman, P.T.; Strain, R.M., Global classical solutions of the Boltzmann equation without angular cut-off, J. Am. Math. Soc., 24, 771-847, (2011) · Zbl 1248.35140
[11] Gualdani, M.P., Mischler, S., Mouhot, C.: Factorization for non-symmetric operators and exponential H-theorem. arXiv:1006.5523 · Zbl 06889665
[12] Guo, Y., The Landau equation in a periodic box, Commun. Math. Phys., 231, 391-434, (2002) · Zbl 1042.76053
[13] Guo, Y., Classical solutions to the Boltzmann equation formolecules with an angular cutoff, Arch. Ration. Mech. Anal., 169, 305-353, (2003) · Zbl 1044.76056
[14] Guo, Y., The Vlasov-Maxwell-Boltzmann system near maxwellians, Invent. Math., 153, 593-630, (2003) · Zbl 1029.82034
[15] Guo, Y., The Boltzmann equation in the whole space, Indiana Univ. Math. J., 53, 1081-1094, (2004) · Zbl 1065.35090
[16] Guo, Y., The Vlasov-Poisson-laudau system in a periodic box, J. Am. Math. Soc., 25, 759-812, (2012) · Zbl 1251.35167
[17] Guo, Y.; Strain, R., Momentum regularity and stability of the relativistic Vlasov-Maxwell-Boltzmann system, Commun. Math. Phys., 310, 649-673, (2012) · Zbl 1245.35130
[18] Guo, Y.; Wang, Y.J., Decay of dissipative equation and negative Sobolev spaces, Commun. Partial Differ. Equ., 37, 2165-2208, (2012) · Zbl 1258.35157
[19] Hosono, T.; Kawashima, S., Decay property of regularity-loss type and application to some nonlinear hyperbolic-elliptic system, Math. Models Methods Appl. Sci., 16, 1839-1859, (2006) · Zbl 1108.35014
[20] Jang, J., Vlasov-Maxwell-Boltzmann diffusive limit, Arch. Ration. Mech. Anal., 194, 531-584, (2009) · Zbl 1347.76062
[21] Krall N.A., Trivelpiece A.W.: Principles of Plasma Physics. McGraw-Hill, New York (1973)
[22] Lei, Y.-J.; Zhao, H.-J., Negative Sobolev spaces and the two-species Vlasov-Maxwell-Landau system in the whole space, J. Funct. Anal., 267, 3710-3757, (2014) · Zbl 1304.35694
[23] Liu, T.-P.; Yang, T.; Yu, S.-H., Energy method for the Boltzmann equation, Phys. D, 188, 178-192, (2004) · Zbl 1098.82618
[24] Liu, T.-P.; Yu, S.-H., Boltzmann equation: micro-macro decompositions and positivity of shock profiles, Commun. Math. Phys., 246, 133-179, (2004) · Zbl 1092.82034
[25] Strain, R.M., The Vlasov-Maxwell-Boltzmann system in the whole space, Commun. Math. Phys., 268, 543-567, (2006) · Zbl 1129.35022
[26] Strain, R.M.; Guo, Y., Stability of the relativistic Maxwellian in a collisional plasma, Commun. Math. Phys., 251, 263-320, (2004) · Zbl 1113.82070
[27] Strain, R.M.; Guo, Y., Exponential decay for soft potentials near Maxwellian, Arch. Ration. Mech. Anal., 187, 287-339, (2008) · Zbl 1130.76069
[28] Strain, R.M.; Zhu, K.-Y., The Vlasov-Poisson-Landau system in \({{\mathbb{R}}^{3}_{x}}\), Arch. Ration. Mech. Anal., 210, 615-671, (2013) · Zbl 1294.35168
[29] Villani, C.: A review of mathematical topics in collisional kinetic theory. In: Friedlander, S., Serre, D. (eds.) Handbook of Mathematical Fluid Dynamics, vol. I. North-Holland, Amsterdam, pp. 71-305 (2002) · Zbl 1170.82369
[30] Wang, Y.-J., Golobal solution and time decay of the Vlasov-Poisson-Landau system in \({{\mathbb{R}}^{3}_{x}}\), SIAM J. Math. Anal., 44, 3281-3323, (2012) · Zbl 1263.82044
[31] Xiao, Q.-H.; Xiong, L.-J.; Zhao, H.-J., The Vlasov-posson-Boltzmann system with angular cutoff for soft potential, J. Differ. Equ., 255, 1196-1232, (2013) · Zbl 1284.35442
[32] Xiao, Q.-H., Xiong, L.-J., Zhao, H.-J.: The Vlasov-Posson-Boltzmann system for the whole range of cutoff soft potentials. J. Funct. Anal. 272, 166-226 (2017) · Zbl 1457.76205
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