no. 3
Partial Differential Equations
Sharp decay rates for the fastest conservative diffusions
[Vitesse de convergence optimale pour les diffusions nonlinéaires conservatives les plus rapides]

Présenté par : Haïm Brezis

Kim, Yong Jung1 ; McCann, Robert J.2
1 Division of Applied Mathematics, KAIST, Gusong-dong 373-1, Yusong-gu, Taejon, 305-701 South Korea
2 Department of Mathematics, University of Toronto, 100 St. George Street, Toronto, ON, M5S 3G3, Canada
Comptes Rendus. Mathématique, Tome 341 (2005) no. 3, pp. 157-162.

Dans les milieux dissipatifs, les perturbations initiales disparaissent progressivement, et seuls sont preservés leurs traits les plus grossiers, comme leur taille et leur position. Estimer précisément la vitesse de cette « disparition » est parfois une question d'un interêt primordial. Ici, nous donnons cette vitesse pour les diffusions nonlinéaires les plus rapides qui préservent la masse, pour le modèle

ut=Δ(um),(n−2)+/n<m⩽n/(n+2),u,t⩾0,x∈Rn,
qui gouverne la diffusion d'une densité initiale, intégrable et à support compact, vers un profil autosimilaire. Pour cela, nous établissons une théorie de comparaison des potentiels, ce qui permet de montrer que la vitesse précise de décroissance est en 1/t pour la norme L1(Rn), et en fait uniforme pour l'erreur relative.

In many diffusive settings, initial disturbances will gradually disappear and all but their crudest features — such as size and location — will eventually be forgotten. Quantifying the rate at which this information is lost is sometimes a question of central interest. Here this rate is addressed for the fastest conservative nonlinearities in the singular diffusion equation

ut=Δ(um),(n−2)+/n<m⩽n/(n+2),u,t⩾0,x∈Rn,
which governs the decay of any integrable, compactly supported initial density towards a characteristically spreading self-similar profile. A potential theoretic comparison technique is outlined below which establishes the sharp 1/t conjectured power law rate of decay uniformly in relative error, and in weaker norms such as L1(Rn).

Reçu le :
Publié le :
DOI : 10.1016/j.crma.2005.06.025
Kim, Yong Jung 1 ; McCann, Robert J. 2

1 Division of Applied Mathematics, KAIST, Gusong-dong 373-1, Yusong-gu, Taejon, 305-701 South Korea
2 Department of Mathematics, University of Toronto, 100 St. George Street, Toronto, ON, M5S 3G3, Canada 
@article{CRMATH_2005__341_3_157_0,
     author = {Kim, Yong Jung and McCann, Robert J.},
     title = {Sharp decay rates for the fastest conservative diffusions},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {157--162},
     publisher = {Elsevier},
     volume = {341},
     number = {3},
     year = {2005},
     doi = {10.1016/j.crma.2005.06.025},
     language = {en},
     url = {http://www.numdam.org/articles/10.1016/j.crma.2005.06.025/}
}
TY  - JOUR
AU  - Kim, Yong Jung
AU  - McCann, Robert J.
TI  - Sharp decay rates for the fastest conservative diffusions
JO  - Comptes Rendus. Mathématique
PY  - 2005
SP  - 157
EP  - 162
VL  - 341
IS  - 3
PB  - Elsevier
UR  - http://www.numdam.org/articles/10.1016/j.crma.2005.06.025/
DO  - 10.1016/j.crma.2005.06.025
LA  - en
ID  - CRMATH_2005__341_3_157_0
ER  - 
%0 Journal Article
%A Kim, Yong Jung
%A McCann, Robert J.
%T Sharp decay rates for the fastest conservative diffusions
%J Comptes Rendus. Mathématique
%D 2005
%P 157-162
%V 341
%N 3
%I Elsevier
%U http://www.numdam.org/articles/10.1016/j.crma.2005.06.025/
%R 10.1016/j.crma.2005.06.025
%G en
%F CRMATH_2005__341_3_157_0
Kim, Yong Jung; McCann, Robert J. Sharp decay rates for the fastest conservative diffusions. Comptes Rendus. Mathématique, Tome 341 (2005) no. 3, pp. 157-162. doi : 10.1016/j.crma.2005.06.025. http://www.numdam.org/articles/10.1016/j.crma.2005.06.025/

[1] Aronson, D.G.; Bénilan, P. Regularite des solutions de l'equation des milieux poreux dans RN, C. R. Acad. Sci. Paris Ser. A-B, Volume 288 (1979), p. A103-A105

[2] Barenblatt, G.I. On some unsteady motions of a liquid or gas in a porous medium, Akad. Nauk SSSR Prikl. Mat. Mekh., Volume 16 (1952), pp. 67-78

[3] Bénilan, P.; Crandall, M.G. The continuous dependence on φ of solutions of ut−Δφ(u)=0, Indiana Univ. Math. J., Volume 30 (1981), pp. 161-177

[4] Carrillo, J.A.; Jüngel, A.; Markowich, P.A.; Toscani, G.; Unterreiter, A. Entropy dissipation methods for degenerate parabolic problems and generalized Sobolev inequalities, Monatsh. Math., Volume 133 (2001), pp. 1-82

[5] Carrillo, J.A.; Vázquez, J.L. Fine asymptotics for fast diffusion equations, Comm. Partial Differential Equations, Volume 28 (2003), pp. 1023-1056

[6] Denzler, J.; McCann, R.J. Fast diffusion to self-similarity: complete spectrum, long time asymptotics, and numerology, Arch. Rational Mech. Anal., Volume 175 (2005), pp. 301-342

[7] Dolbeault, J.; del Pino, M. Best constants for Gagliardo–Nirenberg inequalities and applications to nonlinear diffusions, J. Math. Pures Appl., Volume 81 (2002), pp. 847-875

[8] Friedman, A.; Kamin, S. The asymptotic behaviour of a gas in an n-dimensional porous medium, Trans. Amer. Math. Soc., Volume 262 (1980), pp. 551-563

[9] Galaktionov, V.A.; Peletier, L.A.; Vázquez, J.L. Asymptotics of the fast-diffusion equation with critical exponent, SIAM J. Math. Anal., Volume 31 (2000), pp. 1157-1174

[10] Herrero, M.A.; Pierre, M. The Cauchy problem for ut=Δum when 0<m<1, Trans. Amer. Math. Soc., Volume 291 (1985), pp. 145-158

[11] Kim, Y.-J.; McCann, R.J. Potential theory and optimal convergences rates in fast nonlinear diffusion www.math.toronto.edu/~mccann (Preprint #23 at)

[12] Ladyženskaja, O.A.; Solonnikov, V.A.; Ural'ceva, N.N. Linear and Quasilinear Equations of Parabolic Type, Transl. Math. Monographs, vol. 23, American Mathematical Society, Providence, RI, 1967 (Translated from the Russian by S. Smith)

[13] Lederman, C.; Markowich, P.A. On fast-diffusion equations with infinite equilibrium entropy and finite equilibrium mass, Comm. Partial Differential Equations, Volume 28 (2001), pp. 301-332

[14] Lee, K.-A.; Vázquez, J.L. Geometrical properties of solutions of the porous medium equation for large times, Indiana Univ. Math. J., Volume 52 (2003), pp. 991-1016

[15] Otto, F. The geometry of dissipative evolution equations: the porous medium equation, Comm. Partial Differential Equations, Volume 26 (2001), pp. 101-174

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

[17] Pierre, M. Uniqueness of the solutions of ut−Δφ(u)=0 with measures as initial data, Nonlinear Anal., Volume 6 (1982), pp. 175-187

[18] Vázquez, J.L. An Introduction to the mathematical theory of the Porous Medium Equation, Shape Optimization and Free Boundaries (Montreal, PQ, 1990), NATO Adv. Sci. Inst. Ser. C Math. Phys. Sci., vol. 380, Kluwer Academic, Dordrecht, 1992, pp. 347-389

[19] Vázquez, J.L. Asymptotic behaviour for the porous medium equation posed in the whole space, J. Evol. Equ., Volume 3 (2003), pp. 67-118

[20] Zel'dovich, Ya.B.; Kompaneets, A.S. Towards a theory of heat conduction with thermal conductivity depending on temperature, Collection of Papers dedicated to 70th Anniversary of A.F. Ioffe, Izd. Akad. Nauk SSSR, Moscow, 1950, pp. 61-72

Cité par Sources :