Book Review
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MathSciNet review:
2566451
Full text of review:
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Book Information:
Authors:
Panagiota Daskalopoulos and
Carlos E. Kenig
Title:
Degenerate diffusions
Additional book information:
EMS Tracts in Mathematics 1,
European Mathematical Society,
Zurich,
2007,
x+198 pp.,
ISBN 978-3-03719-033-36
Donald G. Aronson and Philippe Bénilan, Régularité des solutions de l’équation des milieux poreux dans $\textbf {R}^{N}$, C. R. Acad. Sci. Paris Sér. A-B 288 (1979), no. 2, A103–A105 (French, with English summary). MR 524760
D. G. Aronson and L. A. Caffarelli, The initial trace of a solution of the porous medium equation, Trans. Amer. Math. Soc. 280 (1983), no. 1, 351–366. MR 712265, DOI 10.1090/S0002-9947-1983-0712265-1
G. I. Barenblatt, On some unsteady motions of a liquid and gas in a porous medium, Akad. Nauk SSSR. Prikl. Mat. Meh. 16 (1952), 67–78 (Russian). MR 0046217
Philippe Bénilan, Michael G. Crandall, and Michel Pierre, Solutions of the porous medium equation in $\textbf {R}^{N}$ under optimal conditions on initial values, Indiana Univ. Math. J. 33 (1984), no. 1, 51–87. MR 726106, DOI 10.1512/iumj.1984.33.33003
J. Boussinesq, Recherches théoriques sur l’écoulement des nappes d’eau infiltrés dans le sol et sur le débit de sources, C. R. Acad. Sci./J. Math. Pures Appl., 10(1903/04), 5-78.
Björn E. J. Dahlberg and Carlos E. Kenig, Nonnegative solutions of the porous medium equation, Comm. Partial Differential Equations 9 (1984), no. 5, 409–437. MR 741215, DOI 10.1080/03605308408820336
Emmanuele DiBenedetto, Degenerate parabolic equations, Universitext, Springer-Verlag, New York, 1993. MR 1230384, DOI 10.1007/978-1-4612-0895-2
Miguel A. Herrero and Michel Pierre, The Cauchy problem for $u_t=\Delta u^m$ when $0<m<1$, Trans. Amer. Math. Soc. 291 (1985), no. 1, 145–158. MR 797051, DOI 10.1090/S0002-9947-1985-0797051-0
L. S. Leibenzon, The motion of a gas in a porous medium, Complete Works, Vol. 2, Acad. Sci. URSS, Moscow 1930 (in Russian).
M. Muskat, The Flow of Homogeneous Fluids Through Porous Media, McGraw-Hill, New York, 1937.
Michel Pierre, Uniqueness of the solutions of $u_{t}-\Delta \varphi (u)=0$ with initial datum a measure, Nonlinear Anal. 6 (1982), no. 2, 175–187. MR 651699, DOI 10.1016/0362-546X(82)90086-4
Paul E. Sacks, Continuity of solutions of a singular parabolic equation, Nonlinear Anal. 7 (1983), no. 4, 387–409. MR 696738, DOI 10.1016/0362-546X(83)90092-5
Juan Luis Vázquez, Smoothing and decay estimates for nonlinear diffusion equations, Oxford Lecture Series in Mathematics and its Applications, vol. 33, Oxford University Press, Oxford, 2006. Equations of porous medium type. MR 2282669, DOI 10.1093/acprof:oso/9780199202973.001.0001
Juan Luis Vázquez, The porous medium equation, Oxford Mathematical Monographs, The Clarendon Press, Oxford University Press, Oxford, 2007. Mathematical theory. MR 2286292
D. V. Widder, Positive temperatures on an infinite rod, Trans. Amer. Math. Soc. 55 (1944), 85–95. MR 9795, DOI 10.1090/S0002-9947-1944-0009795-2
References
- D. G. Aronson and Ph. Bénilan, Régularité des solutions de l’équation milieux poreux dans $\mathbf {R}^{n}$, C. R. Acad. Sci. Paris, 288(1979), 103-105. MR 524760 (82i:35090)
- D. G. Aronson and L. A. Caffarelli, The initial trace of a solution of the porous medium equation, Trans. Amer. Math. Soc., 280(1983), 351-366. MR 712265 (85c:35042)
- G. I. Barenblatt, On some unsteady motions of a liquid or a gas in a porous medium, Prikl. Mat. Mekh., 16(1952), 67-78 (in Russian). MR 0046217 (13:700a)
- P. Bénilan, M. G. Crandall and M. Pierre, Solutions of the porous medium equation in $\mathbf {R}^{n}$ under optimal conditions on the initial values, Indiana Univ. Math. J., 33(1984), 51-87. MR 726106 (86b:35084)
- J. Boussinesq, Recherches théoriques sur l’écoulement des nappes d’eau infiltrés dans le sol et sur le débit de sources, C. R. Acad. Sci./J. Math. Pures Appl., 10(1903/04), 5-78.
- B. E. Dahlberg and C. E. Kenig, Non-negative solutions of the porous medium equation, Comm. Partial Differential Equations, 9(1984), 409-437. MR 741215 (85j:35099)
- E. DiBenedetto, Degenerate Parabolic Equations, Springer Verlag, New York, 1993. MR 1230384 (94h:35130)
- M. A. Herrero and M. Pierre, The Cauchy problem for $u_{t}=\Delta u^{m}$ when $0<m<1$, Trans. Amer. Math. Soc., 291(1985), 145-158. MR 797051 (86i:35065)
- L. S. Leibenzon, The motion of a gas in a porous medium, Complete Works, Vol. 2, Acad. Sci. URSS, Moscow 1930 (in Russian).
- M. Muskat, The Flow of Homogeneous Fluids Through Porous Media, McGraw-Hill, New York, 1937.
- M. Pierre, Uniqueness of the solution of $u_{t}-\Delta \varphi (u)=0$ with initial datum a measure, Nonlinear Anal., 6(1982), 175-187. MR 651699 (83h:35062)
- P. Sacks, Continuity of solutions of a singular parabolic equation, Nonlinear Anal., 7(1983), 387-409. MR 696738 (84d:35081)
- Juan Luis Vazquez, Smoothing and Decay Estimates for Nonlinear Diffusion Equations, Oxford Lecture Series in Mathematics and Its Applications, Oxford University Press, Oxford, 2006. MR 2282669 (2007k:35008)
- Juan Luis Vazquez, The Porous Medium Equation. Mathematical theory, Oxford Mathematical Monographs, The Clarendon Press, Oxford University Press, Oxford, 2007. MR 2286292 (2008e:35003)
- D. V. Widder, Positive temperature on a infinite rod, Trans. Amer Math. Soc., 55(1944), 85-95. MR 0009795 (5:203f)
Review Information:
Reviewer:
D. G. Aronson
Affiliation:
School of Mathematics and, Institute for Mathematics and Its Applications, University of Minnesota
Email:
don@ima.umn.edu
Journal:
Bull. Amer. Math. Soc.
47 (2010), 171-176
DOI:
https://doi.org/10.1090/S0273-0979-09-01272-5
Published electronically:
July 21, 2009
Review copyright:
© Copyright 2009
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.
