Comptes Rendus
Partial Differential Equations
Sharp decay rates for the fastest conservative diffusions
Comptes Rendus. Mathématique, Volume 341 (2005) no. 3, pp. 157-162.

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),(n2)+/n<mn/(n+2),u,t0,xRn,
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).

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),(n2)+/n<mn/(n+2),u,t0,xRn,
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.

Received:
Published online:
DOI: 10.1016/j.crma.2005.06.025

Yong Jung Kim 1; Robert J. McCann 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
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Yong Jung Kim; Robert J. McCann. Sharp decay rates for the fastest conservative diffusions. Comptes Rendus. Mathématique, Volume 341 (2005) no. 3, pp. 157-162. doi : 10.1016/j.crma.2005.06.025. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2005.06.025/

[1] D.G. Aronson; P. Bénilan 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] G.I. Barenblatt 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] P. Bénilan; M.G. Crandall The continuous dependence on φ of solutions of utΔφ(u)=0, Indiana Univ. Math. J., Volume 30 (1981), pp. 161-177

[4] 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., Volume 133 (2001), pp. 1-82

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

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

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

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

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

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

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

[12] O.A. Ladyženskaja; V.A. Solonnikov; N.N. Ural'ceva 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] C. Lederman; P.A. Markowich On fast-diffusion equations with infinite equilibrium entropy and finite equilibrium mass, Comm. Partial Differential Equations, Volume 28 (2001), pp. 301-332

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

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

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

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

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

[20] Ya.B. Zel'dovich; A.S. Kompaneets 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

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