×

zbMATH — the first resource for mathematics

Optimal large-time behavior of the Vlasov-Maxwell-Boltzmann system in the whole space. (English) Zbl 1244.35010
Summary: We study the large-time behavior of classical solutions to the two-species Vlasov-Maxwell-Boltzmann system in the whole space \(\mathbb R^3 \). The existence of global-in-time nearby Maxwellian solutions is known from [R. M. Strain and Y. Guo, Commun. Partial Differ. Equations 31, No. 1–3, 417–429 (2006; Zbl 1096.82010)]. However, the asymptotic behavior of these solutions has been a challenging open problem. Building on our previous work on time decay for the simpler Vlasov-Poisson-Boltzmann system, we prove that these solutions converge to the global Maxwellian with the optimal decay rate of \(O(t^{-3/2 + 3/(2r)})\) in the \(L^2_\xi (L^r_x)\)-norm for any \(2 \leq r \leq \infty \) if initial perturbation is smooth enough and decays in space velocity fast enough at infinity. Moreover, some explicit rates for the electromagnetic field tending to zero are also provided.

MSC:
35B40 Asymptotic behavior of solutions to PDEs
35Q20 Boltzmann equations
35Q83 Vlasov equations
35Q61 Maxwell equations
PDF BibTeX XML Cite
Full Text: DOI arXiv
References:
[1] Alexandre , R. Morimoto , Y. Ukai , S. Xu , C.-J. Yang , T. The Boltzmann equation without angular cutoff in the whole space: III, Qualitative properties of solutions 2010 · Zbl 1426.76660
[2] Alexandre, Global existence and full regularity of the Boltzmann equation without angular cutoff, J. Amer. Math. Soc 24 pp 771– (2011) · Zbl 1248.35140
[3] Cercignani, The mathematical theory of dilute gases (1994) · Zbl 0813.76001
[4] Desvillettes, On the trend to global equilibrium for spatially inhomogeneous kinetic systems: the Boltzmann equation, Invent. Math 159 (2) pp 245– (2005) · Zbl 1162.82316
[5] DiPerna, Global weak solution of Vlasov-Maxwell systems, Comm. Pure Appl. Math 42 (6) pp 729– (1989) · Zbl 0698.35128
[6] DiPerna, On the Cauchy problem for Boltzmann equation: global existence and weak stability, Ann. Math 130 (2) pp 321– (1989) · Zbl 0698.45010
[7] Dolbeault, Hypocoercivity for kinetic equations with linear relaxation terms, C. R. Acad. Sci. Paris 347 (9-10) pp 511– (2009) · Zbl 1177.35054
[8] Duan, On the Cauchy problem for the Boltzmann equation in the whole space: global existence and uniform stability in L{\(\xi\)}2 (HxN), J. Differential Equations 244 (12) pp 3204– (2008) · Zbl 1150.82019
[9] Duan, Hypocoercivity of linear degenerately dissipative kinetic equations, Nonlinearity 24 (8) pp 2165– (2011) · Zbl 1237.35125
[10] Duan, Stability of the Boltzmann equation with potential forces on torus, Phys. D 238 (17) pp 1808– (2009) · Zbl 1173.35638
[11] Duan, Global smooth flows for the compressible Euler-Maxwell system: Relaxation case, J. Hyperbolic Differ. Equ. · Zbl 1292.76080
[12] Duan, Optimal time decay of the Vlasov-Poisson-Boltzmann system in \input amssym \({\mathbb R}^3\), Arch. Ration. Mech. Anal 199 (1) pp 291– (2010) · Zbl 1232.35169
[13] Duan, combination of energy method and spectral analysis for study of equations of gas motion, Front. Math. China 4 (2) pp 253– (2009) · Zbl 1180.35103
[14] Duan, Optimal decay estimates on the linearized Boltzmann equation with time-dependent forces and their applications, Comm. Math. Phys 277 (1) pp 189– (2008) · Zbl 1175.82047
[15] Duan, Stability of the one-species Vlasov-Poisson-Boltzmann system, SIAM J. Math. Anal 41 (6) pp 2353– (2009) · Zbl 1323.82036
[16] Duan, Boltzmann equation with external force and Vlasov-Poisson-Boltzmann system in infinite vacuum, Discrete Contin. Dyn. Syst 16 (1) pp 253– (2006) · Zbl 1185.35035
[17] Glassey, Decay of the linearized Boltzmann-Vlasov system, Transport Theory Statist. Phys 28 (2) pp 135– (1999) · Zbl 0983.82018
[18] Glassey, Perturbation of essential spectra of evolution operators and the Vlasov-Poisson-Boltzmann system, Discrete Contin. Dynam. Systems 5 pp 457– (1999) · Zbl 0951.35102
[19] Gressman, Global classical solutions of the Boltzmann equation with longrange interactions, Proc. Natl. Acad. Sci. USA 107 (13) pp 5744– (2010) · Zbl 1205.82120
[20] Gressman, Global classical solutions of the Boltzmann equation without angular cut-off, J. Amer. Math. Soc 24 (3) pp 771– (2011) · Zbl 1248.35140
[21] Guo, The Vlasov-Poisson-Boltzmann system near vacuum, Comm. Math. Phys 218 (2) pp 293– (2001) · Zbl 0981.35057
[22] Guo, The Vlasov-Poisson-Boltzmann system near Maxwellians, Comm. Pure Appl. Math 55 (9) pp 1104– (2002) · Zbl 1027.82035
[23] Guo, The Vlasov-Maxwell-Boltzmann system near Maxwellians, Invent. Math 153 (3) pp 593– (2003) · Zbl 1029.82034
[24] Guo, The Boltzmann equation in the whole space, Indiana Univ. Math. J 53 (4) pp 1081– (2004) · Zbl 1065.35090
[25] Ide, Decay property of regularity-loss type for dissipative Timoshenko system, Math. Models Meth. Appl. Sci 18 (5) pp 647– (2008) · Zbl 1153.35013
[26] Ide, Decay property of regularity-loss type and nonlinear effects for dissipative Timoshenko system, Math. Models Meth. Appl. Sci 18 (7) pp 1001– (2008) · Zbl 1160.35346
[27] Jang, Vlasov-Maxwell-Boltzmann diffusive limit, Arch. Ration. Mech. Anal 194 (2) pp 531– (2009) · Zbl 1347.76062
[28] Kawashima , S.
[29] Kawashima, The Boltzmann equation and thirteen moments, Japan J. Appl. Math 7 pp 301– (1990) · Zbl 0702.76090
[30] Liu, Energy method for the Boltzmann equation, Phys. D 188 (3-4) pp 178– (2004) · Zbl 1098.82618
[31] Liu, Boltzmann equation: micro-macro decompositions and positivity of shock profiles, Comm. Math. Phys 246 (1) pp 133– (2004) · Zbl 1092.82034
[32] Liu, The Green’s function and large-time behavior of solutions for the one-dimensional Boltzmann equation, Comm. Pure Appl. Math 57 (12) pp 1543– (2004) · Zbl 1111.76047
[33] Markowich, Semiconductor equations (1990)
[34] Mischler, On the initial boundary value problem for the Vlasov-Poisson-Boltzmann system, Comm. Math. Phys 210 (2) pp 447– (2000) · Zbl 0983.45007
[35] Mouhot, Quantitative perturbative study of convergence to equilibrium for collisional kinetic models in the torus, Nonlinearity 19 (4) pp 969– (2006) · Zbl 1169.82306
[36] Mouhot, Landau damping, J. Math. Phys 51 (1) pp 015204– (2010) · Zbl 1247.82081
[37] Strain, The Vlasov-Maxwell-Boltzmann system in the whole space, Comm. Math. Phys 268 (2) pp 543– (2006) · Zbl 1129.35022
[38] Strain, Asymptotic stability of the relativistic Boltzmann equation for the soft potentials, Comm. Math. Phys 300 (2) pp 529– (2010) · Zbl 1214.35072
[39] Strain , R. M.
[40] Strain, Stability of the relativistic Maxwellian in a collisional plasma, Comm. Math. Phys 251 (2) pp 263– (2004) · Zbl 1113.82070
[41] Strain, Almost exponential decay near Maxwellian, Comm. Partial Differential Equations 31 (1-3) pp 417– (2006) · Zbl 1096.82010
[42] Strain, Exponential decay for soft potentials near Maxwellian, Arch. Ration. Mech. Anal 187 (2) pp 287– (2008) · Zbl 1130.76069
[43] Taylor, Partial differential equations III. Nonlinear equations (2011) · Zbl 1206.35004
[44] Ueda, Dissipative structure of the regularity-loss type and time asymptotic decay of solutions for the Euler-Maxwell system, Preprint (2010)
[45] Ukai, On the existence of global solutions of mixed problem for non-linear Boltzmann equation, Proc. Japan Acad 50 pp 179– (1974) · Zbl 0312.35061
[46] Ukai, The Boltzmann equation in the space L2 L{\(\beta\)}: global and time-periodic solutions, Anal. Appl. (Singap.) 4 (3) pp 263– (2006) · Zbl 1096.35012
[47] Villani, Hypocoercivity, Mem. Amer. Math. Soc 202 (950) pp iv– (2009)
[48] Yang, Cauchy problem for the Vlasov-Poisson-Boltzmann system, Arch. Rational Mech. Anal 182 (3) pp 415– (2006) · Zbl 1104.76086
[49] Yang, Global existence of classical solutions to the Vlasov-Poisson-Boltzmann system, Comm. Math. Phys 268 (3) pp 569– (2006) · Zbl 1129.35023
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.