Nonlinear diffusion from a delocalized source: affine self-similarity, time reversal, & nonradial focusing geometries

Annales de l'Institut Henri Poincare (C) Analyse Non Lineaire

Volumn 25, Issue 5, 2008, Pages 865-888

  Denzler, Jochen ^a   McCann, Robert J ^b  

^a UNIVERSITY OF TENNESSEE   (United States)

^b UNIVERSITY OF TORONTO   (Canada)

Author keywords

Affinely self similar dynamics; Asymptotic dynamics; Ellipsoidal level sets; Exact nonradial solutions; Fourth order diffusion; Hole filling; Invariant subspaces; Nonlinear diffusion; Porous medium equation

Indexed keywords

CONTROL NONLINEARITIES; POROUS MATERIALS; SEMICONDUCTOR DOPING;

AFFINE PROJECTIONS; AFFINELY SELF-SIMILAR DYNAMICS; ASYMPTOTIC DYNAMICS; ASYMPTOTICS; ELLIPSOIDAL LEVEL SETS; EXACT NONRADIAL SOLUTIONS; EXPLICIT SOLUTIONS; FOURTH ORDER DIFFUSION; HOLE-FILLING; INVARIANT SUBSPACES; LIMITING DISTRIBUTIONS; NON-LINEARITIES; NONLINEAR DIFFUSION; POROUS MEDIUM EQUATION; POROUS MEDIUMS; PRESSURE EQUATIONS; QUADRATIC FUNCTIONS; SELF SIMILARITIES; SHORT TIME; TIME REVERSAL; TIME REVERSAL SYMMETRIES;

CONTROL THEORY;

EID: 52149085929     PISSN: 02941449     EISSN: None     Source Type: Journal    
DOI: 10.1016/j.anihpc.2007.05.002     Document Type: Article

References (69)

1. Ambrosio L.A., Gigli N., and Savaré G. Gradient Flows in Metric Spaces and in the Space of Probability Measures. Lecture Notes in Mathematics ETH Zürich (2005), Birkhäuser Verlag, Basel
2. Angenent S.B. Large time asymptotics for the porous medium equation. Nonlinear Diffusion Equations and their Equilibrium States I. Math. Sci. Res. Inst. Publ. vol. 12 (1988), Springer, New York 21-34
3. Angenent S.B., and Aronson D.G. Optimal asymptotics for solutions to the initial value problem for the porous medium equation. In: Angell T.S., et al. (Ed). Nonlinear Problems in Applied Mathematics: in honor of Professor Ivar Stakgold on his 70th birthday (1996), Society for Industrial and Applied Mathematics, Philadelphia 10-19
4. Angenent S.B., and Aronson D.G. Non-axial self-similar hole filling for the porous medium equation. J. Amer. Math. Soc. 14 (2001) 737-782
5. Angenent S.B., Aronson D.G., Betelú S.I., and Lowengrub J.S. Focusing of an elongated hole in porous medium flow. Physica D 151 (2001) 228-252
6. Aronson D.G., and Graveleau J. A self-similar solution to the focusing problem for the porous medium equation. Eur. J. Appl. Math. 4 (1993) 65-81
7. Aronson D.G., Bouwe van den Berg J., and Hulshof J. Parametric dependence of exponents and eigenvalues in focusing porous media flows. Eur. J. Appl. Math. 14 (2003) 485-512
8. Barenblatt G.I. On some unsteady motions of a liquid or gas in a porous medium. Akad. Nauk SSSR Prikl. Mat. Mekh. 16 (1952) 67-78
9. Barenblatt G.I. Scaling (2003), Cambridge University Press, Cambridge
10. Barenblatt G.I., Bertsch M., Chertock A.E., and Prostokishin V.M. Self-similar intermediate asymptotics for a degenerate parabolic filtration absorption equation. Proc. Natl. Acad. Sci. USA 97 (2000) 9844-9848
11. Bernis F., and Friedman A. Higher order nonlinear degenerate parabolic equations. J. Differential Equations 83 (1990) 179-206
12. Bernoff A.J., and Witelski T.P. Linear stability of source-type similarity solutions of the thin film equation. Appl. Math. Lett. 15 (2002) 599-606
13. Bertozzi A.L., and Pugh M.C. Finite-time blow-up of solutions of some long-wave unstable thin film equations. Indiana Univ. Math. J. 49 (2000) 1323-1366
14. Bertsch M., Dal Passo R., Garcke H., and Grün G. The thin viscous flow equation in higher space dimensions. Adv. Differential Equations 3 (1998) 417-440
15. Bertsch M., Dal Passo R., and Ughi M. Discontinuous 'viscosity' solutions of a degenerate parabolic equation. Trans. Amer. Math. Soc. 320 (1990) 779-798
16. Bertsch M., Dal Passo R., and Ughi M. Nonuniqueness of solutions of a degenerate parabolic equation. Ann. Mat. Pura Appl. 161 4 (1992) 57-81
17. Bertsch M., and Ughi M. Positivity properties of viscosity solutions of a degenerate parabolic equation. Nonlinear Anal. 14 (1990) 571-592
18. Betelú S., and King J.R. Explicit solutions of a two-dimensional fourth order nonlinear diffusion equation. Math. Comput. Modelling 37 (2003) 395-403
19. Cáceres M.J., and Toscani G. Kinetic approach to long time behavior of linearized fast diffusion equations. J. Statist. Phys. 128 (2007) 883-925
20. Cáceres M.J., Carrillo J.A., and Toscani G. Long-time behavior for a nonlinear fourth-order parabolic equation. Trans. Amer. Math. Soc. 357 (2005) 1161-1175
21. Carrillo J.A., and Toscani G. Long-time asymptotics for strong solutions of the thin film equation. Comm. Math. Phys. 225 (2002) 551-571
22. Carrillo J.A., Jüngel A., Markowich P.A., Toscani G., and Unterreiter A. Entropy dissipation methods for degenerate parabolic problems and generalized Sobolev inequalities. Monatsh. Math. 133 (2001) 1-82
23. 1-decay of solutions of the porous medium equation to self-similarity. Indiana Univ. Math. J. 49 (2000) 113-141
24. Carrillo J.A., and Vázquez J.L. Fine asymptotics for fast diffusion equations. Comm. Partial Differential Equations 28 (2003) 1023-1056
25. Carrillo J.A., Di Francesco M., and Toscani G. Strict contractivity of the 2-Wasserstein distance for the porous medium equation by mass-centering. Proc. Amer. Math. Soc. 135 (2007) 353-363
26. Carrillo J.A., McCann R.J., and Villani C. Contractions in the 2-Wasserstein length space and thermalization of granular media. Arch. Rational Mech. Anal. 179 (2006) 217-263
27. Chasseigne E., and Vázquez J.L. Theory of extended solutions for fast-diffusion equations in optimal classes of data. Radiation from singularities. Arch. Rational Mech. Anal. 164 (2002) 133-187
28. Chertock A. On the stability of a class of self-similar solutions to the filtration-absorption equation. Eur. J. Appl. Math. 13 (2002) 179-194
29. Chow B., and Knopf D. The Ricci Flow: An Introduction. Mathematical Surveys and Monographs vol. 110 (2004), American Mathematical Society, Providence, RI
30. Dal Passo R., and Luckhaus S. A degenerate diffusion problem not in divergence form. J. Differential Equations 69 (1987) 1-14
31. Del Pino M., and Dolbeault J. Best constants for Gagliardo-Nirenberg inequalities and applications to nonlinear diffusions. J. Math. Pures Appl. 81 (2002) 847-875
32. Denzler J., and McCann R.J. Phase transitions and symmetry breaking in singular diffusion. Proc. Natl. Acad. Sci. USA 100 (2003) 6922-6925
33. Denzler J., and McCann R.J. Fast diffusion to self-similarity: complete spectrum, long time asymptotics, and numerology. Arch. Rational Mech. Anal. 175 (2005) 301-342
34. Derrida B., Lebowitz J.L., Speer E., and Spohn H. Fluctuations of a stationary nonequilibrium interface. Phys. Rev. Lett. 67 (1991) 165-168
35. Diez J.A., Thomas L.P., Betelú S., Gratton R., Marino B., Gratton J., Aronson D.G., and Angenent S.B. Noncircular converging flows in viscous gravity currents. Phys. Rev. E 58 (1998) 6182-6187
36. Friedman A., and Kamin S. The asymptotic behaviour of a gas in an n-dimensional porous medium. Trans. Amer. Math. Soc. 262 (1980) 551-563
37. Galaktionov V.A. Invariant subspaces and new explicit solutions to evolution equations with quadratic nonlinearities. Proc. Roy. Soc. Edinburgh Sect. A 125 (1993) 225-246
38. L. Giacomelli, H. Knüpfer, Flat data solutions to the thin film equation do not rupture, Preprint SFB 611, n. 283, 2006
39. Giacomelli L., and Otto F. Variational formulation for the lubrication approximation of the Hele-Shaw flow. Calc. Var. Partial Differential Equations 13 (2001) 377-404
40. Giacomelli L., and Otto F. Rigorous lubrication approximation. Interfaces Free Bound 5 (2003) 483-529
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, 2006
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
44. Jüngel A., and Pinnau R. Global nonnegative solutions of a nonlinear fourth-order parabolic equation for quantum systems. SIAM J. Math. Anal. 32 (2000) 760-777
45. Kim Y.J., and McCann R.J. Sharp decay rates for the fastest conservative diffusions. C. R. Acad. Sci. Paris, Ser. I 341 (2005) 157-162
46. Kim Y.J., and McCann R.J. Potential theory and optimal convergence rates in fast nonlinear diffusion. J. Math Pures Appl. 86 (2006) 42-67
47. King J.R. Exact multidimensional solutions to some nonlinear diffusion equations. Quart. J. Mech. Appl. Math. 46 (1993) 419-436
48. Laugesen R.S., and Pugh M.C. Heteroclinic orbits, mobility parameters and stability for thin film type equations. Electron. J. Differential Equations 95 (2002) 1-29
49. Lieb E.H., and Loss M. Analysis (1997), American Mathematical Society, Providence, RI
50. McCann R.J., and Slepčev D. Second-order asymptotics for the fast-diffusion equation. Int. Math. Res. Not. 24947 (2006) 1-22
51. Myers T.G. Thin films with high surface tension. SIAM Rev. 40 (1998) 441-462
52. Oron A., Davis S.H., and Bankoff S.G. Long-scale evolution of thin liquid films. Rev. Modern Phys. 69 (1997) 931-980
53. Otto F. Lubrication approximation with prescribed nonzero contact angle. Comm. Partial Differential Equations 23 (1998) 2077-2164
54. Otto F. The geometry of dissipative evolution equations: the porous medium equation. Comm. Partial Differential Equations 26 (2001) 101-174
55. Pattle R.E. Diffusion from an instantaneous point source with concentration dependent coefficient. Quart. J. Mech. Appl. Math. 12 (1959) 407-409
56. Pukhnachev V.V. Exact multidimensional solutions of the nonlinear diffusion equation. J. Appl. Mech. Tech. Phys. 36 (1995) 169-176
57. Rudykh G.A., and Semenov E.I. The construction of exact solutions of the multi-dimensional quasilinear heat-conduction equation. Comp. Math. Math. Phys. 33 (1993) 1087-1097
58. Slepčev D., and Pugh M.C. Selfsimilar blowup of unstable thin-film equations. Indiana Univ. Math. J. 54 (2005) 1697-1738
59. m et comportement asymptotique, Unpublished, 1986
60. Toscani G. A central limit theorem for solutions of the porous medium equation. J. Evol. Equ. 5 (2005) 185-203
61. Ughi M. A degenerate parabolic equation modelling the spread of an epidemic. Ann. Mat. Pura Appl. 143 4 (1986) 385-400
62. Vazquez J.L. Asymptotic behaviour and propagation of the one-dimensional flow of gas in a porous medium. Trans. Amer. Math. Soc. 277 (1983) 507-527
63. 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 (1992), Kluwer Acad. Publ. 347-389
64. Vázquez J.L. Asymptotic behaviour for the porous medium equation posed in the whole space. J. Evol. Equ. 3 (2003) 67-118
65. Vázquez J.L. The Porous Medium Equation. Mathematical Theory (2007), Oxford University Press, Oxford
66. Walter W. Ordinary Differential Equations. Translated from the Sixth German (1996) Edition by Russell Thompson (1998), Springer-Verlag, New York
67. Zel'dovich Ya.B., and Barenblatt G.I. The asymptotic properties of self-modelling solutions of the nonstationary gas filtration equations. Sov. Phys. Doklady 3 (1989) 44-47
68. Zel'dovich Ya.B., and Kompaneets A.S. Theory of heat transfer with temperature dependent thermal conductivity. Collection in Honour of the 70th Birthday of Academician A.F. Ioffe (1950), Izdat. Akad. Nauk. SSSR, Moscow 61-71
69. Titov S.S., and Ustinov V.A. Investigation of polynomial solutions of the two-dimensional Leibenzon filtration equation with an integer adiabatic exponent. In: Sidorov A.F., and Vershinin S.V. (Eds). Approximate Methods for Solving Boundary Value Problems of Continuum Mechanics, vol. 91 (1985), Akad. Nauk SSSR, Ural. Nauchn. Tsentr, Sverdlovsk 64-70 (in Russian)