@article{AIHPC_2008__25_5_865_0, author = {Denzler, Jochen and McCann, Robert J.}, title = {Nonlinear diffusion from a delocalized source : affine self-similarity, time reversal, & nonradial focusing geometries}, journal = {Annales de l'I.H.P. Analyse non lin\'eaire}, pages = {865--888}, publisher = {Elsevier}, volume = {25}, number = {5}, year = {2008}, doi = {10.1016/j.anihpc.2007.05.002}, mrnumber = {2457815}, zbl = {1146.76053}, language = {en}, url = {http://www.numdam.org/articles/10.1016/j.anihpc.2007.05.002/} }
TY - JOUR AU - Denzler, Jochen AU - McCann, Robert J. TI - Nonlinear diffusion from a delocalized source : affine self-similarity, time reversal, & nonradial focusing geometries JO - Annales de l'I.H.P. Analyse non linéaire PY - 2008 SP - 865 EP - 888 VL - 25 IS - 5 PB - Elsevier UR - http://www.numdam.org/articles/10.1016/j.anihpc.2007.05.002/ DO - 10.1016/j.anihpc.2007.05.002 LA - en ID - AIHPC_2008__25_5_865_0 ER -
%0 Journal Article %A Denzler, Jochen %A McCann, Robert J. %T Nonlinear diffusion from a delocalized source : affine self-similarity, time reversal, & nonradial focusing geometries %J Annales de l'I.H.P. Analyse non linéaire %D 2008 %P 865-888 %V 25 %N 5 %I Elsevier %U http://www.numdam.org/articles/10.1016/j.anihpc.2007.05.002/ %R 10.1016/j.anihpc.2007.05.002 %G en %F AIHPC_2008__25_5_865_0
Denzler, Jochen; McCann, Robert J. Nonlinear diffusion from a delocalized source : affine self-similarity, time reversal, & nonradial focusing geometries. Annales de l'I.H.P. Analyse non linéaire, Tome 25 (2008) no. 5, pp. 865-888. doi : 10.1016/j.anihpc.2007.05.002. http://www.numdam.org/articles/10.1016/j.anihpc.2007.05.002/
[1] Gradient Flows in Metric Spaces and in the Space of Probability Measures, Lecture Notes in Mathematics ETH Zürich, Birkhäuser Verlag, Basel, 2005. | MR | Zbl
, , ,[2] Large time asymptotics for the porous medium equation, in: Nonlinear Diffusion Equations and their Equilibrium States I, Math. Sci. Res. Inst. Publ., vol. 12, Springer, New York, 1988, pp. 21-34. | MR | Zbl
,[3] Optimal asymptotics for solutions to the initial value problem for the porous medium equation, in: , (Eds.), Nonlinear Problems in Applied Mathematics: in honor of Professor Ivar Stakgold on his 70th birthday, Society for Industrial and Applied Mathematics, Philadelphia, 1996, pp. 10-19. | MR | Zbl
, ,[4] Non-axial self-similar hole filling for the porous medium equation, J. Amer. Math. Soc. 14 (2001) 737-782. | MR | Zbl
, ,[5] Focusing of an elongated hole in porous medium flow, Physica D 151 (2001) 228-252. | MR | Zbl
, , , ,[6] A self-similar solution to the focusing problem for the porous medium equation, Eur. J. Appl. Math. 4 (1993) 65-81. | MR | Zbl
, ,[7] Parametric dependence of exponents and eigenvalues in focusing porous media flows, Eur. J. Appl. Math. 14 (2003) 485-512. | MR | Zbl
, , ,[8] On some unsteady motions of a liquid or gas in a porous medium, Akad. Nauk SSSR Prikl. Mat. Mekh. 16 (1952) 67-78. | MR | Zbl
,[9] Scaling, Cambridge University Press, Cambridge, 2003. | MR | Zbl
,[10] Self-similar intermediate asymptotics for a degenerate parabolic filtration absorption equation, Proc. Natl. Acad. Sci. USA 97 (2000) 9844-9848. | MR | Zbl
, , , ,[11] Higher order nonlinear degenerate parabolic equations, J. Differential Equations 83 (1990) 179-206. | MR | Zbl
, ,[12] Linear stability of source-type similarity solutions of the thin film equation, Appl. Math. Lett. 15 (2002) 599-606. | MR | Zbl
, ,[13] Finite-time blow-up of solutions of some long-wave unstable thin film equations, Indiana Univ. Math. J. 49 (2000) 1323-1366. | MR | Zbl
, ,[14] The thin viscous flow equation in higher space dimensions, Adv. Differential Equations 3 (1998) 417-440. | MR | Zbl
, , , ,[15] Discontinuous ‘viscosity' solutions of a degenerate parabolic equation, Trans. Amer. Math. Soc. 320 (1990) 779-798. | MR | Zbl
, , ,[16] Nonuniqueness of solutions of a degenerate parabolic equation, Ann. Mat. Pura Appl. 161 (4) (1992) 57-81. | MR | Zbl
, , ,[17] Positivity properties of viscosity solutions of a degenerate parabolic equation, Nonlinear Anal. 14 (1990) 571-592. | MR | Zbl
, ,[18] Explicit solutions of a two-dimensional fourth order nonlinear diffusion equation, Math. Comput. Modelling 37 (2003) 395-403. | MR | Zbl
, ,[19] Kinetic approach to long time behavior of linearized fast diffusion equations, J. Statist. Phys. 128 (2007) 883-925. | MR | Zbl
, ,[20] Long-time behavior for a nonlinear fourth-order parabolic equation, Trans. Amer. Math. Soc. 357 (2005) 1161-1175. | MR | Zbl
, , ,[21] Long-time asymptotics for strong solutions of the thin film equation, Comm. Math. Phys. 225 (2002) 551-571. | MR | Zbl
, ,[22] Entropy dissipation methods for degenerate parabolic problems and generalized Sobolev inequalities, Monatsh. Math. 133 (2001) 1-82. | MR | Zbl
, , , , ,[23] Asymptotic -decay of solutions of the porous medium equation to self-similarity, Indiana Univ. Math. J. 49 (2000) 113-141. | MR | Zbl
, ,[24] Fine asymptotics for fast diffusion equations, Comm. Partial Differential Equations 28 (2003) 1023-1056. | MR | Zbl
, ,[25] Strict contractivity of the 2-Wasserstein distance for the porous medium equation by mass-centering, Proc. Amer. Math. Soc. 135 (2007) 353-363. | MR | Zbl
, , ,[26] Contractions in the 2-Wasserstein length space and thermalization of granular media, Arch. Rational Mech. Anal. 179 (2006) 217-263. | MR | Zbl
, , ,[27] Theory of extended solutions for fast-diffusion equations in optimal classes of data. Radiation from singularities, Arch. Rational Mech. Anal. 164 (2002) 133-187. | MR | Zbl
, ,[28] On the stability of a class of self-similar solutions to the filtration-absorption equation, Eur. J. Appl. Math. 13 (2002) 179-194. | MR | Zbl
,[29] The Ricci Flow: An Introduction, Mathematical Surveys and Monographs, vol. 110, American Mathematical Society, Providence, RI, 2004. | MR | Zbl
, ,[30] A degenerate diffusion problem not in divergence form, J. Differential Equations 69 (1987) 1-14. | MR | Zbl
, ,[31] Best constants for Gagliardo-Nirenberg inequalities and applications to nonlinear diffusions, J. Math. Pures Appl. 81 (2002) 847-875. | MR | Zbl
, ,[32] Phase transitions and symmetry breaking in singular diffusion, Proc. Natl. Acad. Sci. USA 100 (2003) 6922-6925. | MR | Zbl
, ,[33] Fast diffusion to self-similarity: complete spectrum, long time asymptotics, and numerology, Arch. Rational Mech. Anal. 175 (2005) 301-342. | MR | Zbl
, ,[34] Fluctuations of a stationary nonequilibrium interface, Phys. Rev. Lett. 67 (1991) 165-168. | MR | Zbl
, , , ,[35] Noncircular converging flows in viscous gravity currents, Phys. Rev. E 58 (1998) 6182-6187.
, , , , , , , ,[36] The asymptotic behaviour of a gas in an n-dimensional porous medium, Trans. Amer. Math. Soc. 262 (1980) 551-563. | MR | Zbl
, ,[37] Invariant subspaces and new explicit solutions to evolution equations with quadratic nonlinearities, Proc. Roy. Soc. Edinburgh Sect. A 125 (1993) 225-246. | MR | Zbl
,[38] L. Giacomelli, H. Knüpfer, Flat data solutions to the thin film equation do not rupture, Preprint SFB 611, n. 283, 2006.
[39] Variational formulation for the lubrication approximation of the Hele-Shaw flow, Calc. Var. Partial Differential Equations 13 (2001) 377-404. | MR | Zbl
, ,[40] Rigorous lubrication approximation, Interfaces Free Bound 5 (2003) 483-529. | MR | Zbl
, ,[41] U. Gianazza, G. Savaré, G. Toscani, The Wasserstein gradient flow of the Fisher information and the quantum drift-diffusion equation, Preprint at http://www.imati.cnr.it/~savare/pubblicazioni/preprints.html. | MR
[42] J. Graveleau, Quelques solutions auto-semblables pour l'équation dela chaleur non-linéaire, Rapport Interne C.E.A., 1972.
[43] A. Jüngel, D. Matthes, The Derrida-Lebowitz-Speer-Spohn equation: existence, non-uniqueness, and decay rates of the solutions, SIAM J. Math. Anal. (2007), in press. | Zbl
[44] Global nonnegative solutions of a nonlinear fourth-order parabolic equation for quantum systems, SIAM J. Math. Anal. 32 (2000) 760-777. | MR | Zbl
, ,[45] Sharp decay rates for the fastest conservative diffusions, C. R. Acad. Sci. Paris, Ser. I 341 (2005) 157-162. | MR | Zbl
, ,[46] Potential theory and optimal convergence rates in fast nonlinear diffusion, J. Math Pures Appl. 86 (2006) 42-67. | MR | Zbl
, ,[47] Exact multidimensional solutions to some nonlinear diffusion equations, Quart. J. Mech. Appl. Math. 46 (1993) 419-436. | MR | Zbl
,[48] Heteroclinic orbits, mobility parameters and stability for thin film type equations, Electron. J. Differential Equations 95 (2002) 1-29. | MR | Zbl
, ,[49] Analysis, American Mathematical Society, Providence, RI, 1997. | MR | Zbl
, ,[50] Second-order asymptotics for the fast-diffusion equation, Int. Math. Res. Not. 24947 (2006) 1-22. | MR | Zbl
, ,[51] Thin films with high surface tension, SIAM Rev. 40 (1998) 441-462. | MR | Zbl
,[52] Long-scale evolution of thin liquid films, Rev. Modern Phys. 69 (1997) 931-980.
, , ,[53] Lubrication approximation with prescribed nonzero contact angle, Comm. Partial Differential Equations 23 (1998) 2077-2164. | MR | Zbl
,[54] The geometry of dissipative evolution equations: the porous medium equation, Comm. Partial Differential Equations 26 (2001) 101-174. | MR | Zbl
,[55] Diffusion from an instantaneous point source with concentration dependent coefficient, Quart. J. Mech. Appl. Math. 12 (1959) 407-409. | MR | Zbl
,[56] Exact multidimensional solutions of the nonlinear diffusion equation, J. Appl. Mech. Tech. Phys. 36 (1995) 169-176. | MR | Zbl
,[57] The construction of exact solutions of the multi-dimensional quasilinear heat-conduction equation, Comp. Math. Math. Phys. 33 (1993) 1087-1097. | MR | Zbl
, ,[58] Selfsimilar blowup of unstable thin-film equations, Indiana Univ. Math. J. 54 (2005) 1697-1738. | MR | Zbl
, ,[59] L. Tartar, Solutions particulières de et comportement asymptotique, Unpublished, 1986.
[60] A central limit theorem for solutions of the porous medium equation, J. Evol. Equ. 5 (2005) 185-203. | MR | Zbl
,[61] A degenerate parabolic equation modelling the spread of an epidemic, Ann. Mat. Pura Appl. 143 (4) (1986) 385-400. | MR | Zbl
,[62] Asymptotic behaviour and propagation of the one-dimensional flow of gas in a porous medium, Trans. Amer. Math. Soc. 277 (1983) 507-527. | MR | Zbl
,[63] An introduction to the mathematical theory of the porous medium equation, in: Shape Optimization and Free Boundaries, Montreal, PQ, 1990, NATO Adv. Sci. Inst. Ser. C Math. Phys. Sci., vol. 380, Kluwer Acad. Publ., 1992, pp. 347-389. | MR | Zbl
,[64] Asymptotic behaviour for the porous medium equation posed in the whole space, J. Evol. Equ. 3 (2003) 67-118. | MR | Zbl
,[65] The Porous Medium Equation. Mathematical Theory, Oxford University Press, Oxford, 2007. | MR | Zbl
,[66] Ordinary Differential Equations, Translated from the Sixth German (1996) Edition by Russell Thompson, Springer-Verlag, New York, 1998. | MR | Zbl
,[67] The asymptotic properties of self-modelling solutions of the nonstationary gas filtration equations, Sov. Phys. Doklady 3 (1989) 44-47.
, ,[68] Theory of heat transfer with temperature dependent thermal conductivity, in: Collection in Honour of the 70th Birthday of Academician A.F. Ioffe, Izdat. Akad. Nauk. SSSR, Moscow, 1950, pp. 61-71.
, ,[69] Investigation of polynomial solutions of the two-dimensional Leibenzon filtration equation with an integer adiabatic exponent, in: , (Eds.), Approximate Methods for Solving Boundary Value Problems of Continuum Mechanics, vol. 91, Akad. Nauk SSSR, Ural. Nauchn. Tsentr, Sverdlovsk, 1985, pp. 64-70, (in Russian). | MR
, ,Cité par Sources :