Wasserstein geometry of porous medium equation
Annales de l'I.H.P. Analyse non linéaire, Tome 29 (2012) no. 2, pp. 217-232.

We study the porous medium equation with emphasis on q-Gaussian measures, which are generalizations of Gaussian measures by using power-law distribution. On the space of q-Gaussian measures, the porous medium equation is reduced to an ordinary differential equation for covariance matrix. We introduce a set of inequalities among functionals which gauge the difference between pairs of probability measures and are useful in the analysis of the porous medium equation. We show that any q-Gaussian measure provides a nontrivial pair attaining equality in these inequalities.

DOI : 10.1016/j.anihpc.2011.10.003
Classification : 60D05, 46E27
Mots-clés : q-Gaussian measure, Porous medium equation, Functional inequality 
@article{AIHPC_2012__29_2_217_0,
     author = {Takatsu, Asuka},
     title = {Wasserstein geometry of porous medium equation},
     journal = {Annales de l'I.H.P. Analyse non lin\'eaire},
     pages = {217--232},
     publisher = {Elsevier},
     volume = {29},
     number = {2},
     year = {2012},
     doi = {10.1016/j.anihpc.2011.10.003},
     mrnumber = {2901195},
     zbl = {1276.35106},
     language = {en},
     url = {http://www.numdam.org/articles/10.1016/j.anihpc.2011.10.003/}
}
TY  - JOUR
AU  - Takatsu, Asuka
TI  - Wasserstein geometry of porous medium equation
JO  - Annales de l'I.H.P. Analyse non linéaire
PY  - 2012
SP  - 217
EP  - 232
VL  - 29
IS  - 2
PB  - Elsevier
UR  - http://www.numdam.org/articles/10.1016/j.anihpc.2011.10.003/
DO  - 10.1016/j.anihpc.2011.10.003
LA  - en
ID  - AIHPC_2012__29_2_217_0
ER  - 
%0 Journal Article
%A Takatsu, Asuka
%T Wasserstein geometry of porous medium equation
%J Annales de l'I.H.P. Analyse non linéaire
%D 2012
%P 217-232
%V 29
%N 2
%I Elsevier
%U http://www.numdam.org/articles/10.1016/j.anihpc.2011.10.003/
%R 10.1016/j.anihpc.2011.10.003
%G en
%F AIHPC_2012__29_2_217_0
Takatsu, Asuka. Wasserstein geometry of porous medium equation. Annales de l'I.H.P. Analyse non linéaire, Tome 29 (2012) no. 2, pp. 217-232. doi : 10.1016/j.anihpc.2011.10.003. http://www.numdam.org/articles/10.1016/j.anihpc.2011.10.003/

[1] M. Agueh, N. Ghoussoub, X. Kang, Geometric inequalities via a general comparison principle for interacting gases, Geom. Funct. Anal. 14 no. 1 (2004), 215-244 | MR | Zbl

[2] S. Amari, Differential–Geometrical Methods in Statistics, Lecture Notes in Statist. vol. 28, Springer-Verlag, New York (1985) | MR | Zbl

[3] S. Amari, H. Nagaoka, Methods of Information Geometry, Transl. Math. Monogr. vol. 191, Amer. Math. Soc., Providence, RI (2000) | MR

[4] G.I. Barenblatt, On self-similar motions of a compressible fluid in a porous medium, Akad. Nauk SSSR Prikl. Mat. Meh. 16 (1952), 679-698 | MR | Zbl

[5] Y. Brenier, Polar factorization and monotone rearrangement of vector-valued functions, Comm. Pure Appl. Math. 44 no. 4 (1991), 375-417 | MR | Zbl

[6] J.A. Carrillo, A. Jüngel, P.A. Markowich, G. Toscani, A. Unterreiter, Entropy dissipation methods for degenerate parabolic problems and generalized Sobolev inequalities, Monatsh. Math. 133 no. 1 (2001), 1-82 | MR | Zbl

[7] J.A. Carrillo, R.J. Mccann, C. Villani, Kinetic equilibration rates for granular media and related equations: entropy dissipation and mass transportation estimates, Rev. Mat. Iberoam. 19 no. 3 (2003), 971-1018 | EuDML | MR | Zbl

[8] J.A. Carrillo, R.J. Mccann, C. Villani, Contractions in the 2-Wasserstein length space and thermalization of granular media, Arch. Ration. Mech. Anal. 179 no. 2 (2006), 217-263 | MR | Zbl

[9] D. Cordero-Erausquin, W. Gangbo, C. Houdré, Inequalities for generalized entropy and optimal transportation, Recent Advances in the Theory and Applications of Mass Transport, Contemp. Math. vol. 353, Amer. Math. Soc., Providence, RI (2004), 73-94 | MR | Zbl

[10] L.C. Evans, R.F. Gariepy, Measure Theory and Fine Properties of Functions, Stud. Adv. Math., CRC Press, Boca Raton, FL (1992) | MR | Zbl

[11] R.J. Mccann, A convexity principle for interacting gases, Adv. Math. 128 no. 1 (1997), 153-179 | MR | Zbl

[12] A. Ohara, T. Wada, Information geometry of q-Gaussian densities and behaviors of solutions to related diffusion equations, J. Phys. A 43 (2010), 035002 | MR | Zbl

[13] S. Ohta, A. Takatsu, Displacement convexity of generalized relative entropies, Adv. Math. 228 (2011), 1742-1787 | MR | Zbl

[14] F. Otto, The geometry of dissipative evolution equations: the porous medium equation, Comm. Partial Differential Equations 26 no. 1–2 (2001), 101-174 | MR | Zbl

[15] F. Otto, C. Villani, Generalization of an inequality by Talagrand and links with the logarithmic Sobolev inequality, J. Funct. Anal. 173 no. 2 (2000), 361-400 | MR | Zbl

[16] R.E. Pattle, Diffusion from an instantaneous point source with a concentration-dependent coefficient, Quart. J. Mech. Appl. Math. 12 (1959), 407-409 | MR | Zbl

[17] A. Takatsu, Wasserstein geometry of Gaussian measures, Osaka J. Math. 48 (4) (2011), in press. | MR

[18] A. Takatsu, Wasserstein geometry of non-linear Fokker–Planck type equations, Sūrikaisekikenkyūsho Kōkyūroku 1671 (2009), 20-36

[19] A. Takatsu, Behaviors of φ-exponential distributions in Wasserstein geometry and an evolution equation, arXiv:1109.6776 (2011) | MR

[20] C. Tsallis, Possible generalization of Boltzmann–Gibbs statistics, J. Stat. Phys. 52 no. 1–2 (1988), 479-487 | MR | Zbl

[21] C. Tsallis, Introduction to Nonextensive Statistical Mechanics—Approaching a Complex World, Springer-Verlag, Berlin (2009) | MR | Zbl

[22] C. Vignat, A. Hero, J. Costa, About closedness by convolution of the Tsallis maximizers, Phys. A 340 no. 1–3 (2004), 147-152 | MR

[23] C. Villani, Topics in Optimal Transportation, Grad. Stud. Math. vol. 58, Amer. Math. Soc., Providence, RI (2003) | MR | Zbl

[24] C. Villani, Optimal Transport, Old and New, Grundlehren Math. Wiss. vol. 338, Springer-Verlag, Berlin (2009) | MR | Zbl

Cité par Sources :