zbMATH — the first resource for mathematics

Geometry Search for the term Geometry in any field. Queries are case-independent.
Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact.
"Topological group" Phrases (multi-words) should be set in "straight quotation marks".
au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted.
Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff.
"Quasi* map*" py: 1989 The resulting documents have publication year 1989.
so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14.
"Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic.
dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles.
py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses).
la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

a & b logic and
a | b logic or
!ab logic not
abc* right wildcard
"ab c" phrase
(ab c) parentheses
any anywhere an internal document identifier
au author, editor ai internal author identifier
ti title la language
so source ab review, abstract
py publication year rv reviewer
cc MSC code ut uncontrolled term
dt document type (j: journal article; b: book; a: book article)
On the trend to equilibrium for the Vlasov-Poisson-Boltzmann equation. (English) Zbl 1151.35077
Summary: The dynamics of dilute electrons and plasma can be modeled by Vlasov-Poisson-Boltzmann equation, for which the equilibrium state can be a global Maxwellian. In this paper, we show that the rate of convergence to equilibrium is $O(t^{ - \infty })$, by using a method developed for the Boltzmann equation without external force in [{\it L. Desvillettes} and {\it C. Villani}, Invent. Math. 159, No. 2, 245--316 (2005; Zbl 1162.82316)]. In particular, the idea of this method is to show that the solution $f$ cannot stay near any local Maxwellians for long. The improvement in this paper is to handle the effect from the external force governed by the Poisson equation. Moreover, by using the macro-micro decomposition, we simplify the estimation on the time derivatives of the deviation of the solution from the local Maxwellian with same macroscopic components.

35Q35PDEs in connection with fluid mechanics
82C40Kinetic theory of gases (time-dependent statistical mechanics)
82C70Transport processes (time-dependent statistical mechanics)
78A35Motion of charged particles
Full Text: DOI
[1] Caflisch, R.: The Boltzmann equation with a soft potential, Comm. math. Phys. 74, 71-109 (1980) · Zbl 0434.76066
[2] Desvillettes, L.; Villani, C.: On the trend to global equilibrium in spatially inhomogeneous entropy-dissipating systems: the linear Fokker -- Planck equation, Comm. pure appl. Math. 54, 1-42 (2001) · Zbl 1029.82032 · doi:10.1002/1097-0312(200101)54:1<1::AID-CPA1>3.0.CO;2-Q
[3] Desvillettes, L.; Villani, C.: On the trend to global equilibrium for spatially inhomogeneous kinetic systems: the Boltzmann equation, Invent. math. 159, 245-316 (2005) · Zbl 1162.82316 · doi:10.1007/s00222-004-0389-9
[4] Guo, Y.: The Vlasov -- Poisson -- Boltzmann system near maxwellians, Comm. pure appl. Math. 55, 1104-1135 (2002) · Zbl 1027.82035
[5] Guo, Y.: The Vlasov -- Maxwell -- Boltzmann system near maxwellians, Invent. math. 153, 593-630 (2003) · Zbl 1029.82034
[6] Liu, T. P.; Yang, T.; Yu, S. H.: Energy method for the Boltzmann equation, Phys. D 188, 178-192 (2004) · Zbl 1098.82618 · doi:10.1016/j.physd.2003.07.011
[7] Strain, R. M.: The Vlasov -- Maxwell -- Boltzmann system in the whole space, Comm. math. Phys. 268, 543-567 (2006) · Zbl 1129.35022 · doi:10.1007/s00220-006-0109-y
[8] R.M. Strain, Y. Guo, Exponential decay for soft potentials near Maxwellian, preprint · Zbl 1130.76069 · doi:10.1007/s00205-007-0067-3
[9] Ukai, S.: On the existence of global solutions of mixed problem for non-linear Boltzmann equation, Proc. Japan acad. 50, 179-184 (1974) · Zbl 0312.35061 · doi:10.3792/pja/1195519027
[10] Villani, C.: A review of mathematical topics in collisional kinetic theory, Handb. math. Fluid dynam., 71-305 (2002) · Zbl 1170.82369
[11] Villani, C.: Cercignani’s conjecture is sometimes true and always almost true, Comm. math. Phys. 234, 455-490 (2003) · Zbl 1041.82018 · doi:10.1007/s00220-002-0777-1
[12] Yang, T.; Yu, H. J.; Zhao, H. J.: Cauchy problem for the Vlasov -- Poisson -- Boltzmann system, Arch. ration. Mech. anal. 182, 415-470 (2006) · Zbl 1104.76086
[13] Yang, T.; Zhao, H. J.: Global existence of classical solutions to the Vlasov -- Poisson -- Boltzmann system, Comm. math. Phys. 268, 569-605 (2006) · Zbl 1129.35023 · doi:10.1007/s00220-006-0103-4
[14] Yang, T.; Zhao, H. J.: A new energy method for the Boltzmann equation, J. math. Phys. 47, 053301 (2006) · Zbl 1111.82048 · doi:10.1063/1.2195528