Markov property of Gaussian states of canonical commutation relation algebras
The Markov property of Gaussian states of canonical commutation relation algebras is studied. The detailed description is given by the representing block matrix. The proof is short and allows infinite...
The Markov property of Gaussian states of canonical commutation relation algebras is studied. The detailed description is given by the representing block matrix. The proof is short and allows infinite...
Hyers–Ulam stability on a generalized quadratic functional equation in distributions and hyperfunctions
We consider the Hyers–Ulam stability of a generalized quadratic functional equation in the spaces of distributions of Schwartz and hyperfunctions of Gelfand modulo bounded distributions and hype...
We consider the Hyers–Ulam stability of a generalized quadratic functional equation in the spaces of distributions of Schwartz and hyperfunctions of Gelfand modulo bounded distributions and hype...
Asymptotic behavior for logarithmic diffusion
J. Math. Phys. 50, 113518 (2009); doi:10.1063/1.3259207
Published 20 November 2009
You are not logged in to this journal. Log in
In this paper we prove, via the entropy dissipation method, that the solutions of the d-dimensional logarithmic diffusion equation, with nonhomogeneous Dirichlet boundary data, decay exponentially in time toward its own steady state. The result is valid not only in L1-norm (as customary when applying entropy dissipation methods) but also in any Lp-norm with p![[is-an-element-of]](http://scitation.aip.org/stockgif3/isin.gif)
[1,+![[infinity]](http://scitation.aip.org/stockgif3/infin.gif)
).
©2009 American Institute of Physics
[1,+).
©2009 American Institute of Physics
| History: | Received 12 April 2009; accepted 5 October 2009; published 20 November 2009 |
| Permalink: |
http://link.aip.org/link/?JMAPAQ/50/113518/1 |
| BUY THIS ARTICLE | (US$28) |
|
|
Connotea
CiteULike
del.icio.us
BibSonomy
REFERENCES (26)
For access to fully linked references, you need to log in.
For access to fully linked references, you need to Log in.
- Arnold, A., Markowich, P., Toscani, G., and Unterreiter, A., “On convex Sobolev inequalities and the rate of convergence to equilibrium for Fokker-Planck type equations,”
Commun. Partial Differ. Equ. 26, 43 (2001) . - Berryman, J. G. and Holland, C. J., “Asymptotic behavior of the nonlinear diffusion equation nt=(n−1nx)x,” J. Math. Phys. 23, 983 (1982).
- Blanchet, A., Bonforte, M., Dolbeault, J., Grillo, G., and Vázquez, J. L., “Asymptotics of the fast diffusion equation via entropy estimates,”
Arch. Ration. Mech. Anal. 191, 347 (2009) . - Bonforte, M. and Vázquez, J. L., “Positivity, local smoothing, and Harnack inequalities for very fast diffusion equations,” Adv. Math. (in press).
- Burelbach, J. P., Bankoff, S. G., and Davis, S. H., “Nonlinear stability of evaporating/condensating liquid films,”
J. Fluid Mech. 195, 463 (1988) . - Carrillo, J. A., Jüngel, A., Markowich, P. A., and Toscani, G., “G. and A, Unterreiter. Entropy dissipation methods for degenerate parabolic problems and generalized Sobolev inequalities,”
Monatsh. Math. 133, 1 (2001) . - Carrillo, J. A. and Toscani, G., “Asymptotic L1-decay of solutions of the porous medium equation to self-similarity,”
Indiana Univ. Math. J. 49, 113 (2000) . - Carrillo, J. A. and Toscani, G., “Long-time asymptotics for strong solutions of the thin film equation,”
Commun. Math. Phys. 225, 551 (2002) . - Carrillo, J. A. and Vázquez, J. L., “Fine asymptotics for fast diffusion equations,”
Commun. Partial Differ. Equ. 28, 1023 (2003) . - Daskalopoulos, P. and del Pino, M., “On the Cauchy problem for ut=
log u in higher dimensions,”
Math. Ann. 313, 189 (1999) . - Davis, S. H., DiBenedetto, E., and Diller, D. J., “Some a priori estimates for a singular evolution equation arising in thin-film dynamics,”
SIAM J. Math. Anal. 27, 638 (1996) . - Del Pino, M. and Dolbeault, J., “Best constants for Gagliardo-Nirenberg inequalities and applications to nonlinear diffusions,”
J. Math. Pures Appl. 81, 847 (2002) . - Denzler, J. and McCann, R. J., “Fast diffusion to self-similarity: complete spectrum, long-time asymptotics, and numerology,”
Arch. Ration. Mech. Anal. 175, 301 (2005) . - Desvillettes, L.,
Contemp. Math. 409, 33 (2006) . - DiBenedetto, E. and Diller, D. J., Partial Differential Equations and Applications, Lecture Notes in Pure and Applied Mathematics Vol. 177 (Dekker, New York, 1996), pp. 103–119.
- Esteban, J. R., Rodrí, A., and Vázquez, J. L., “A nonlinear heat equation with singular diffusivity,”
Commun. Partial Differ. Equ. 13, 985 (1988) . - Golse, F. and Salvarani, F., “The nonlinear diffusion limit for generalized Carleman models: the initial-boundary value problem,”
Nonlinearity 20, 927 (2007) . - Hamilton, R. S., Contemp. Math. 71, 237 (1988).
- Ladyženskaja, O. A., Solonnikov, V. A., and Uralceva, N. N., Translations of Mathematical Monographs (American Mathematical Society, Providence, RI, 1967), Vol. 23.
- Rodriguez, A., Vazquez, J. L., and Esteban, J. R., “The maximal solution of the logarithmic fast diffusion equation in two space dimensions,” Adv. Differ. Equ. 2, 867 (1997).
- Salvarani, F. and Toscani, G., “Large-time asymptotics for nonlinear diffusions: the initial-boundary value problem,” J. Math. Phys. 46, 023502 (2005).
- Vázquez, J. L., Smoothing and Decay Estimates for Nonlinear Diffusion Equations, Oxford Lecture Series in Mathematics and its Applications Vol. 33 (Oxford University Press, Oxford, 2006).
- Vázquez, J. L., “Finite-time blow-down in the evolution of point masses by planar logarithmic diffusion,”
Discrete Contin. Dyn. Syst. 19, 1 (2007) . - Vázquez, J. L., The Porous Medium Equation, Oxford Mathematical Monographs (Clarendon Press, Oxford, 2007).
- Vázquez, J. L., “Measure-valued solutions and the phenomenon of blow-down in logarithmic diffusion,”
J. Math. Anal. Appl. 352, 515 (2009) . - Williams, M. B. and Davis, S. H., “Non linear theory of film rupture,”
J. Colloid Interface Sci. 90, 220 (1982) .
