Abstract:
In this paper we investigate the large-time asymptotic of linearized very fast diffusion equations with and without potential confinements. These equations do not satisfy, in general, logarithmic Sobolev inequalities, but, as we show by using the 'Bakry-Emery reverse approach', in the confined case they have a positive spectral gap at the eigenvalue zero. We present estimates for this spectral gap and draw conclusions on the time decay of the solution, which we show to be exponential for the problem with confinement and algebraic for the pure diffusive case. These results hold for arbitrary algebraically large diffusion speeds, if the solutions have the mass-conservation property.
Registro:
Documento: |
Artículo
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Título: | Poincaré inequalities for linearizations of very fast diffusion equations |
Autor: | Carrillo, J.A.; Lederman, C.; Markowich, P.A.; Toscani, G. |
Filiación: | Depto. de Matemática Aplicada, Universidad de Granada, 18071 Granada, Spain Departamento de Matemática, Universidad de Buenos Aires, 1428 Buenos Aires, Argentina Institüt für Mathematik, Universität Wien, Boltzmanngasse 9, A-1090 Wien, Austria Dipartimento di Matematica, Universitá di Pavia, Via Ferrata 1, 27100 Pavia, Italy
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Año: | 2002
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Volumen: | 15
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Número: | 3
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Página de inicio: | 565
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Página de fin: | 580
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DOI: |
http://dx.doi.org/10.1088/0951-7715/15/3/303 |
Título revista: | Nonlinearity
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Título revista abreviado: | Nonlinearity
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ISSN: | 09517715
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Registro: | https://bibliotecadigital.exactas.uba.ar/collection/paper/document/paper_09517715_v15_n3_p565_Carrillo |
Referencias:
- Arnold, A., Markowich, P., Toscani, G., Unterreiter, A., On convex Sobolev inequalities and the rate of convergence to equilibrium for Fokker-Planck type equations (2001) Commun. Partial Diff. Eqns, 26, pp. 43-100. , [AMTU]
- Bakry, D., Emery, M., Diffusions hypercontractives (1985) Lecture Notes in Mathematics, 1123, pp. 177-206. , [BE] Sém. Proba. vol XIX (Berlin: Springer)
- Bobkov, S., Götze, F., Exponential integrability and transportation cost related to logarithmic Sobolev inequalities (1999) J. Funct. Anal., 163, pp. 1-28. , [BG]
- Carrillo, J.A., Tbscani, G., Exponential convergence toward equilibrium for homogeneous Fokker-Planck type equations (1998) Math. Meth. Appl. Sci., 21, pp. 1269-1286. , [CT1]
- Carrillo, J.A., Toscani, G., Asymptotic L1-decay of solutions of the porous medium equation to self-similarity (2000) Indiana Univ. Math. J., 49, pp. 113-1141. , [CT2]
- Carrillo, J.A., Juengel, A., Markowich, P., Toscani, G., Unterreiter, A., Entropy dissipation methods for degenerate parabolic problems and generalized Sobolev inequalities (2001) Monatsh. Math., 133, pp. 1-82. , [CJMTU]
- Dolbeault, J., Del Pino, M., (1999) Generalized Sobolev Inequalities and Asymptotic Behavior in Fast Diffusion and Porous Medium Problems, , [DP] Preprint CEREMADE no 9905
- Friedman, A., Kamin, S., The asymptotic behavior of gas in an N-dimensional porous medium (1980) Trans. Am. Math. Soc., 262, pp. 551-563. , [FK]
- Gross, L., Logarithmic Sobolev inequalities (1975) Am. J. Math., 97, pp. 1061-1083. , [Gr]
- Lederman, C., Markowich, P.A., (2001) On Fast-diffusion Equations with Infinite Equilibrium Entropy and Finite Equilibrium Mass, , [LM] Preprint
- Ledoux, M., Concentration of measure and logarithmic Sobolev inequalities (1999) Lecture Notes in Mathematics, 1709, pp. 120-216. , [Le] Sém. Proba. vol XXXIII (Berlin: Springer)
- McCann, R.J., A convexity principle for interacting gases (1997) Adv. Math., 128, pp. 153-179. , [McCa]
- Markowich, P.A., Villani, C., On the trend to equilibrium for the Fokker-Planck equation: An interplay between physics and functional analysis (2000) Mat. Contemp., 19, pp. 1-29. , [MV] VI Workshop on Partial Differential Equations, Part II (Rio de Janeiro 1999)
- Muckenhaupt, B., (1972) Hardy's Inequality with Weights Studio Mathematica, 44, pp. 31-38. , [Mu]
- Otto, F., The geometry of dissipative evolution equations: The porous medium equation (2001) Commun. Partial Diff. Eqns, 26, pp. 101-174. , [Ot]
- Stam, A., Some inequalities satisfied by the quantities of information of Fisher and Shannon (1959) Inform. Control, 2, pp. 101-112. , [St]
- Toscani, G., Sur l'inégalité logarithmique de Sobolev (1997) C. R. Acad. Sci., Paris A, 324, pp. 689-694. , [To] série 1
- Toscani, G., Villani, C., On the trend to equilibrium for some dissipative systems with slowly increasing a priori bounds (2000) J. Statist. Phys., 98, pp. 1279-1309. , [TV]
- Vázquez, J.L., An introduction to the mathematical theory of the porous medium equation Shape optimization and free boundaries (1990) NATO Adv. Sci. Inst. Ser. C Math. Phys. Sci., 380, pp. 347-389. , [Val] Montreal, PQ, Dordrecht: Kluwer
- Vázquez, J.L., Asymptotic behaviour for the porous medium equation in the whole space Notas del Curso de Doctorado Métodos Asintóticos en Ecuaciones de Evolución, , [Va2]
- Witelski, T.P., Bernoff, A.J., Self-similar asymptotics for linear and nonlinear diffusion equations (1998) Stud. Appl. Math., 100, pp. 153-193. , [WB]
- Zel'Dovich, I.B., Barenblatt, G.I., The asymptotic properties of self-modelling solutions of the nonstationary gas filtration equations (1958) Sov. Phys. Dokl., 3, pp. 44-47. , [ZB]
Citas:
---------- APA ----------
Carrillo, J.A., Lederman, C., Markowich, P.A. & Toscani, G.
(2002)
. Poincaré inequalities for linearizations of very fast diffusion equations. Nonlinearity, 15(3), 565-580.
http://dx.doi.org/10.1088/0951-7715/15/3/303---------- CHICAGO ----------
Carrillo, J.A., Lederman, C., Markowich, P.A., Toscani, G.
"Poincaré inequalities for linearizations of very fast diffusion equations"
. Nonlinearity 15, no. 3
(2002) : 565-580.
http://dx.doi.org/10.1088/0951-7715/15/3/303---------- MLA ----------
Carrillo, J.A., Lederman, C., Markowich, P.A., Toscani, G.
"Poincaré inequalities for linearizations of very fast diffusion equations"
. Nonlinearity, vol. 15, no. 3, 2002, pp. 565-580.
http://dx.doi.org/10.1088/0951-7715/15/3/303---------- VANCOUVER ----------
Carrillo, J.A., Lederman, C., Markowich, P.A., Toscani, G. Poincaré inequalities for linearizations of very fast diffusion equations. Nonlinearity. 2002;15(3):565-580.
http://dx.doi.org/10.1088/0951-7715/15/3/303