메뉴 건너뛰기




Volumn 54, Issue 1, 2001, Pages 1-42

On the trend to global equilibrium in spatially inhomogeneous entropy-dissipating systems: The linear Fokker-Planck equation

Author keywords

[No Author keywords available]

Indexed keywords


EID: 0035526707     PISSN: 00103640     EISSN: None     Source Type: Journal    
DOI: 10.1002/1097-0312(200101)54:1<1::AID-CPA1>3.0.CO;2-Q     Document Type: Article
Times cited : (238)

References (40)
  • 1
    • 0000224209 scopus 로고
    • 1 for the spatially homogeneous Boltzmann equation
    • 1 for the spatially homogeneous Boltzmann equation. Arch. Rational Mech. Anal. 103 (1988), no. 2, 151-167.
    • (1988) Arch. Rational Mech. Anal. , vol.103 , Issue.2 , pp. 151-167
    • Arkeryd, L.1
  • 2
    • 84943150390 scopus 로고
    • 1 trend to equilibrium for the Boltzmann equation
    • 1 trend to equilibrium for the Boltzmann equation. Stud. Appl. Math. 87 (1992), no. 3, 283-288.
    • (1992) Stud. Appl. Math. , vol.87 , Issue.3 , pp. 283-288
    • Arkeryd, L.1
  • 3
    • 0007406044 scopus 로고    scopus 로고
    • On logarithmic Sobolev inequalities, and the rate of convergence to equilibrium for Fokker-Planck type equations
    • in press
    • Arnold, A.; Markowich, P.; Toscani, G.; Unterreiter, A. On logarithmic Sobolev inequalities, and the rate of convergence to equilibrium for Fokker-Planck type equations. J. Funct. Anal., in press.
    • J. Funct. Anal.
    • Arnold, A.1    Markowich, P.2    Toscani, G.3    Unterreiter, A.4
  • 4
    • 0001119849 scopus 로고
    • Diffusions hypercontractives. Séminaire de probabilités, XIX, 1983/84, 177-206
    • Springer, Berlin-New York
    • Bakry, D.; Emery, M. Diffusions hypercontractives. Séminaire de probabilités, XIX, 1983/84, 177-206. Lecture Notes in Mathematics, 1123. Springer, Berlin-New York, 1985.
    • (1985) Lecture Notes in Mathematics, 1123
    • Bakry, D.1    Emery, M.2
  • 6
    • 84972497473 scopus 로고
    • On long time asymptotics of the Vlasov-Fokker-Planck equation and of the Vlasov-Poisson-Fokker-Planck system with Coulombic and Newtonian potentials
    • Bouchut, F.; Dolbeault, J. On long time asymptotics of the Vlasov-Fokker-Planck equation and of the Vlasov-Poisson-Fokker-Planck system with Coulombic and Newtonian potentials. Differential Integral Equations 8 (1995), no. 3, 487-514.
    • (1995) Differential Integral Equations , vol.8 , Issue.3 , pp. 487-514
    • Bouchut, F.1    Dolbeault, J.2
  • 7
    • 0002356751 scopus 로고
    • Strict entropy production bounds and stability of the rate of convergence to equilibrium for the Boltzmann equation
    • Carlen, E. A.; Carvalho, M. C. Strict entropy production bounds and stability of the rate of convergence to equilibrium for the Boltzmann equation. J. Statist. Phys. 67 (1992), no. 3-4, 575-608.
    • (1992) J. Statist. Phys. , vol.67 , Issue.3-4 , pp. 575-608
    • Carlen, E.A.1    Carvalho, M.C.2
  • 8
    • 21344493381 scopus 로고
    • Entropy production estimates for Boltzmann equations with physically realistic collision kernels
    • Carlen, E. A.; Carvalho, M. C. Entropy production estimates for Boltzmann equations with physically realistic collision kernels. J. Statist. Phys. 74 (1994), no. 3-4, 743-782.
    • (1994) J. Statist. Phys. , vol.74 , Issue.3-4 , pp. 743-782
    • Carlen, E.A.1    Carvalho, M.C.2
  • 10
    • 4243219284 scopus 로고
    • La condition d'hypoellipticité d'hörmander
    • Chaleyat-Maurel, M. La condition d'hypoellipticité d'Hörmander. Astérisque 84-85 (1981), 189-202.
    • (1981) Astérisque , vol.84-85 , pp. 189-202
    • Chaleyat-Maurel, M.1
  • 11
    • 0000489740 scopus 로고
    • Information-type measures of difference of probability distributions and indirect observations
    • Csiszár, I. Information-type measures of difference of probability distributions and indirect observations. Studia Sci. Math. Hungar. 2 (1967), 299-318.
    • (1967) Studia Sci. Math. Hungar. , vol.2 , pp. 299-318
    • Csiszár, I.1
  • 12
    • 0002931582 scopus 로고
    • Convergence to equilibrium in large time for Boltzmann and B.G.K. Equations
    • Desvillettes, L. Convergence to equilibrium in large time for Boltzmann and B.G.K. equations. Arch. Rational Mech. Anal. 110 (1990), no. 1, 73-91.
    • (1990) Arch. Rational Mech. Anal. , vol.110 , Issue.1 , pp. 73-91
    • Desvillettes, L.1
  • 13
    • 67649350730 scopus 로고
    • Convergence to equilibrium in various situations for the solution of the Boltzmann equation
    • Nonlinear kinetic theory and mathematical aspects of hyperbolic systems (Rapallo, 1992), 101-114. World Scientific, River Edge, N.J.
    • Desvillettes, L. Convergence to equilibrium in various situations for the solution of the Boltzmann equation. Nonlinear kinetic theory and mathematical aspects of hyperbolic systems (Rapallo, 1992), 101-114. Series on Advances in Mathematics for Applied Sciences, 9. World Scientific, River Edge, N.J., 1992.
    • (1992) Series on Advances in Mathematics for Applied Sciences , pp. 9
    • Desvillettes, L.1
  • 14
    • 21844483733 scopus 로고
    • About the regularizing properties of the non-cut-off Kac equation
    • Desvillettes, L. About the regularizing properties of the non-cut-off Kac equation. Comm. Math. Phys. 168 (1995), no. 2, 417-440.
    • (1995) Comm. Math. Phys. , vol.168 , Issue.2 , pp. 417-440
    • Desvillettes, L.1
  • 15
    • 0000278898 scopus 로고    scopus 로고
    • On the spatially homogeneous Landau equation for hard potentials. I. Existence, uniqueness and smoothness
    • Desvillettes, L.; Villani, C. On the spatially homogeneous Landau equation for hard potentials. I. Existence, uniqueness and smoothness. Comm. Partial Differential Equations 25 (2000), no. 1-2, 179-259.
    • (2000) Comm. Partial Differential Equations , vol.25 , Issue.1-2 , pp. 179-259
    • Desvillettes, L.1    Villani, C.2
  • 16
    • 0000278899 scopus 로고    scopus 로고
    • On the spatially homogeneous Landau equation for hard potentials. II. H-theorem and applications
    • Desvillettes, L., and Villani, C. On the spatially homogeneous Landau equation for hard potentials. II. H-theorem and applications. Comm. Partial Differential Equations 25 (2000), no. 1-2, 261-298.
    • (2000) Comm. Partial Differential Equations , vol.25 , Issue.1-2 , pp. 261-298
    • Desvillettes, L.1    Villani, C.2
  • 18
    • 0007362455 scopus 로고
    • On Boltzmann's H-theorem
    • Grad, H. On Boltzmann's H-theorem. J. Soc. Indust. Appl. Math. 13 (1965), 259-277.
    • (1965) J. Soc. Indust. Appl. Math. , vol.13 , pp. 259-277
    • Grad, H.1
  • 19
    • 0001429116 scopus 로고
    • Logarithmic Sobolev inequalities
    • Gross, L. Logarithmic Sobolev inequalities. Amer. J. Math. 97 (1975), no. 4, 1061-1083.
    • (1975) Amer. J. Math. , vol.97 , Issue.4 , pp. 1061-1083
    • Gross, L.1
  • 20
    • 0002236459 scopus 로고
    • Logarithmic Sobolev inequalities and contractivity properties of semigroups
    • Dirichlet forms (Varenna, 1992), 54-88. Springer, Berlin
    • Gross, L. Logarithmic Sobolev inequalities and contractivity properties of semigroups. Dirichlet forms (Varenna, 1992), 54-88. Lecture Notes in Mathematics, 1563. Springer, Berlin, 1993.
    • (1993) Lecture Notes in Mathematics , pp. 1563
    • Gross, L.1
  • 21
    • 15244346736 scopus 로고
    • Hypoelliptic second order differential equations
    • Hörmander, L. Hypoelliptic second order differential equations. Acta Math. 119 (1967), 147-171.
    • (1967) Acta Math. , vol.119 , pp. 147-171
    • Hörmander, L.1
  • 22
    • 0000170568 scopus 로고
    • On a class of ultraparabolic equations
    • Il'in, A. M. On a class of ultraparabolic equations. Dokl. Akad. Nauk SSSR 159 (1964), 1214-1217; Sov. Math. Dokl. 5 (1964), 1673-1676.
    • (1964) Dokl. Akad. Nauk SSSR , vol.159 , pp. 1214-1217
    • Il'in, A.M.1
  • 23
    • 0002389336 scopus 로고
    • Il'in, A. M. On a class of ultraparabolic equations. Dokl. Akad. Nauk SSSR 159 (1964), 1214-1217; Sov. Math. Dokl. 5 (1964), 1673-1676.
    • (1964) Sov. Math. Dokl. , vol.5 , pp. 1673-1676
  • 24
    • 0002460456 scopus 로고
    • Pseudo-differential operators and hypoellipticity. Partial differential equations
    • Univ. California, Berkeley, Calif., 1971, Amererican Mathematical Society, Providence, R.I.
    • Kohn, J. J. Pseudo-differential operators and hypoellipticity. Partial differential equations (Proc. Sympos. Pure Math., Vol. XXIII, Univ. California, Berkeley, Calif., 1971), 61-69. Amererican Mathematical Society, Providence, R.I., 1973.
    • (1973) Proc. Sympos. Pure Math. , vol.23 , pp. 61-69
    • Kohn, J.J.1
  • 25
    • 84941465845 scopus 로고
    • A lower bound for discrimination information in terms of variation
    • Kullback, S. A lower bound for discrimination information in terms of variation. IEEE Trans. Information Theory 4 (1967), 126-127.
    • (1967) IEEE Trans. Information Theory , vol.4 , pp. 126-127
    • Kullback, S.1
  • 26
    • 0001156542 scopus 로고
    • L'algèbre de lie des gradients itérés d'un générateur markovien - Développements de moyennes et d'entropies
    • Ledoux, M. L'algèbre de Lie des gradients itérés d'un générateur markovien - développements de moyennes et d'entropies. Ann. Sci. École Norm. Sup. (4) 28 (1995), no. 4, 435-460.
    • (1995) Ann. Sci. École Norm. Sup. (4) , vol.28 , Issue.4 , pp. 435-460
    • Ledoux, M.1
  • 27
    • 0000834860 scopus 로고
    • Compactness in Boltzmann's equation via Fourier integral operators and applications, I, II, III
    • Lions, P-L. Compactness in Boltzmann's equation via Fourier integral operators and applications, I, II, III. J. Math. Kyoto Univ. 34 (1994), 391-427, 429-461, no. 2, 539-584.
    • (1994) J. Math. Kyoto Univ. , vol.34 , pp. 391-427
    • Lions, P.-L.1
  • 28
    • 0033408149 scopus 로고    scopus 로고
    • Spatial decay solutions of the Boltzmann equation: Converse properties of long time limiting behavior
    • Lu, X. Spatial decay solutions of the Boltzmann equation: converse properties of long time limiting behavior. SIAM J. Math. Anal. 30 (1999), no. 5, 1151-1174.
    • (1999) SIAM J. Math. Anal. , vol.30 , Issue.5 , pp. 1151-1174
    • Lu, X.1
  • 29
    • 0031488084 scopus 로고    scopus 로고
    • Boltzmann equation with infinite energy: Renormalized solutions and distributional solutions for small initial data and initial data close to a Maxwellian
    • Mischler, S.; Perthame, B. Boltzmann equation with infinite energy: renormalized solutions and distributional solutions for small initial data and initial data close to a Maxwellian. SIAM J. Math. Anal. 28 (1997), no. 5, 1015-1027.
    • (1997) SIAM J. Math. Anal. , vol.28 , Issue.5 , pp. 1015-1027
    • Mischler, S.1    Perthame, B.2
  • 30
    • 0001070817 scopus 로고
    • Scaling limit for interacting Ornstein-Uhlenbeck processes
    • Olla, S.; Varadhan, S. R. S. Scaling limit for interacting Ornstein-Uhlenbeck processes. Comm. Math. Phys. 135 (1991), no. 2, 355-378.
    • (1991) Comm. Math. Phys. , vol.135 , Issue.2 , pp. 355-378
    • Olla, S.1    Varadhan, S.R.S.2
  • 32
    • 0003299394 scopus 로고
    • The Fokker-Planck equation
    • Springer Series in Synergetics, 18. Springer, Berlin-New York
    • Risken, H. The Fokker-Planck equation. Methods of solution and applications. Second edition. Springer Series in Synergetics, 18. Springer, Berlin-New York, 1989.
    • (1989) Methods of Solution and Applications. Second Edition
    • Risken, H.1
  • 33
    • 12944295719 scopus 로고
    • Some inequalities satisfied by the quantities of information of Fisher and Shannon
    • Stam, A. Some inequalities satisfied by the quantities of information of Fisher and Shannon. Information and Control 2 (1959), 101-112.
    • (1959) Information and Control , vol.2 , pp. 101-112
    • Stam, A.1
  • 34
    • 0007365295 scopus 로고
    • H-theorem and asymptotic trend of the solution for a rarefied gas in the vacuum
    • Toscani, G. H-theorem and asymptotic trend of the solution for a rarefied gas in the vacuum. Arch. Rational Mech. Anal. 100 (1987), no. 1, 1-12.
    • (1987) Arch. Rational Mech. Anal. , vol.100 , Issue.1 , pp. 1-12
    • Toscani, G.1
  • 35
    • 0032632377 scopus 로고    scopus 로고
    • Entropy production and the rate of convergence to equilibrium for the Fokker-Planck equation
    • Toscani, G. Entropy production and the rate of convergence to equilibrium for the Fokker-Planck equation. Quart. Appl. Math. 57 (1999), no. 3, 521-541.
    • (1999) Quart. Appl. Math. , vol.57 , Issue.3 , pp. 521-541
    • Toscani, G.1
  • 36
    • 0033242961 scopus 로고    scopus 로고
    • Sharp entropy dissipation bounds and explicit rate of trend to equilibrium for the spatially homogeneous Boltzmann equation
    • Toscani, G.; Villani, C. Sharp entropy dissipation bounds and explicit rate of trend to equilibrium for the spatially homogeneous Boltzmann equation. Comm. Math. Phys. 203 (1999), no. 3, 667-706.
    • (1999) Comm. Math. Phys. , vol.203 , Issue.3 , pp. 667-706
    • Toscani, G.1    Villani, C.2
  • 37
    • 0034147246 scopus 로고    scopus 로고
    • On the trend to equilibrium for some dissipative systems with slowly increasing a priori bounds
    • Toscani, G.; Villani, C. On the trend to equilibrium for some dissipative systems with slowly increasing a priori bounds. J. Statist. Phys. 98 (2000), no. 5-6, 1279-1309.
    • (2000) J. Statist. Phys. , vol.98 , Issue.5-6 , pp. 1279-1309
    • Toscani, G.1    Villani, C.2
  • 38
    • 0003603398 scopus 로고
    • Fundamentals of Maxwell's kinetic theory of a simple monoatomic gas
    • Academic Press, New York-London
    • Truesdell, C.; Muncaster, R. G. Fundamentals of Maxwell's kinetic theory of a simple monoatomic gas. Pure and Applied Mathematics, 83. Academic Press, New York-London, 1980.
    • (1980) Pure and Applied Mathematics, 83.
    • Truesdell, C.1    Muncaster, R.G.2
  • 39
    • 0002952245 scopus 로고
    • The fundamental solution of a degenerate partial differential equation of parabolic type
    • Weber, M. The fundamental solution of a degenerate partial differential equation of parabolic type. Trans. Amer. Math. Soc. 71 (1951), 24-37.
    • (1951) Trans. Amer. Math. Soc. , vol.71 , pp. 24-37
    • Weber, M.1


* 이 정보는 Elsevier사의 SCOPUS DB에서 KISTI가 분석하여 추출한 것입니다.