Abstract
We prove convergence to equilibrium with explicit rates for various kinetic equations with relatively bad control of the distribution tails: in particular, Boltzmann-type equations with (smoothed) soft potentials. We compensate the lack of uniform-in-time estimates by the use of precise logarithmic Sobolev-type inequalities, and the assumption that the initial datum decays rapidly at large velocities. Our method not only gives explicit results on the times of convergence, but is also able to cover situations in which compactness arguments apparently do not apply (even mere convergence to equilibrium was an open problem for soft potentials).
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REFERENCES
R. Alexandre and C. Villani, On the Boltzmann equation for long-range interactions and the Landau approximation in plasma physics. Work in preparation.
L. Arkeryd, L ∞ estimates for the space-homogeneous Boltzmann equation, J. Stat. Phys. 31(2):347–361 (1983).
A. Arnold, P. Markowich, T. Toscani, and A. Unterreiter, On logarithmic Sobolev inequalities, Csiszár–Kullback inequalities, and the rate of convergence to equilibrium for Fokker–Planck type equations. Preprint, 1998.
D. Bakry and M. Emery, Diffusions hypercontractives. In Sém. Proba. XIX, LNM, No. 1123 (Springer, 1985), pp. 177–206.
S. Bobkov and F. Götze, Exponential integrability and transportation cost related to logarithmic sobolev inequalities, J. Funct. Anal. 163(1):1–28 (1999).
E. Carlen and M. Carvalho, Strict entropy production bounds and stability of the rate of convergence to equilibrium for the Boltzmann equation, J. Stat. Phys. 67(3–4):575–608 (1992).
E. Carlen and M. Carvalho, Entropy production estimates for Boltzmann equations with physically realistic collision kernels, J. Stat. Phys. 74(3–4):743–782 (1994).
C. Cercignani, The Boltzmann Equation and its Applications(Springer, 1988).
C. Cercignani, R. Illner, and M. Pulvirenti, The Mathematical Theory of Dilute Gases(Springer, 1994).
L. Desvillettes, Some applications of the method of moments for the homogeneous Boltzmann equation. Arch. Rat. Mech. Anal. 123(4):387–395 (1993).
L. Desvillettes and C. Villani, On the spatially homogeneous Landau equation for hard potentials. Part II: H-theorem and applications (Preprint Univ. Orléans, 1997). To appear in Comm. P.D.E.
L. Desvillettes and C. Villani, On the spatially homogeneous Landau equation for hard potentials. Part I: Existence, uniqueness and smoothness (Preprint ENS Cachan, 1998). To appear in Comm. P.D.E.
R. Holley and D. Stroock, Logarithmic Sobolev inequalities and stochastic Ising models, J. Stat. Phys. 46(5–6):1159–1194 (1987).
P. Lions and C. Villani, Régularité optimale de racines carrées, C.R. Acad. Sci. Paris Série I 321:1537–1541 (1995).
P. Lizorkin, Interpolation of weighted Lp spaces, Dokl. Akad. Nauk. SSSR 222:1 (1975). Trad. Sov. Math. Dokl. 16(3):577–581 (1975).
H. McKean, Entropy is the only increasing functional of Kac's 1-dimensional caricature of a Maxwellian gas, Z. für Wahrscheinlichkeitstheorie 2:167–172 (1963).
H. Risken, The Fokker_Planck Equation, Methods of Solution and Applications, second ed. (Springer-Verlag, Berlin, 1989). 1308 Toscani and Villani
G. Toscani, Entropy production and the rate of convergence to equilibrium for the Fokker–Planck equation. Quarterly of Appl. Math. LVII:521–541 (1999).
G. Toscani and C. Villani, Sharp entropy dissipation bounds and explicit rate of trend to equilibrium for the spatially homogeneous Boltzmann equation, Comm. Math. Phys. 203(3):667–706 (1999).
C. Villani, Contribution ø l'étude mathé matique des équations de Boltzmann et de Landau en théorie cinétique des gaz et des plasmas. PhD thesis (Univ. Paris-Dauphine, 1998).
C. Villani, On a new class of weak solutions for the spatially homogeneous Boltzmann and Landau equations. Arch. Rat. Mech. Anal. 143(3):273–307 (1998).
Wennberg, B. Regularity in the Boltzmann equation and the Radon transform, Comm. P.D.E. 19(11–12):2057–2074 (1994).
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Toscani, G., Villani, C. On the Trend to Equilibrium for Some Dissipative Systems with Slowly Increasing a Priori Bounds. Journal of Statistical Physics 98, 1279–1309 (2000). https://doi.org/10.1023/A:1018623930325
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DOI: https://doi.org/10.1023/A:1018623930325