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Degenerate Diffusions
EMS Tracts in Mathematics Vol. 1

Panagiota Daskalopoulos (Columbia University, New York, USA)
Carlos E. Kenig (University of Chicago, USA)

Degenerate Diffusions

Initial Value Problems and Local Regularity Theory

ISBN print 978-3-03719-033-3, ISBN online 978-3-03719-533-8
DOI 10.4171/033
May 2007, 207 pages, hardcover, 17.0 x 24.0 cm.
48.00 Euro

The book deals with existence, uniqueness, regularity and asymptotic behavior of solutions to the initial value problem (Cauchy problem) and the initial-Dirichlet problem for a class of degenerate diffusions modeled on the porous medium type equation ut = Δum, m ≥ 0, u ≥ 0. Such models arise in plasma physics, diffusions through porous media, thin liquid film dynamics as well as in geometric flows such as the Ricci flow on surfaces and the Yamabe flow. The approach presented to these problems is through the use of local regularity estimates and Harnack type inequalities, which yield compactness for families of solutions. The theory is quite complete in the slow diffusion case (m > 1) and in the supercritical fast diffusion case (mc < m < 1, mc = (n – 2)+/n) while many problems remain in the range m ≤ mc. All of these aspects of the theory are discussed in the book.
The book is addressed to both researchers and to graduate students with a good background in analysis and some previous exposure to partial differential equations.

Keywords: Degenerate diffusion, local regularity estimates, compactness for families of solutions, initial and boundary value problems for slow and fast diffusion

Further Information

Review in Zbl MATH 1205.35002

Review in MR 2338118 (2009b:35214)

Review in Bull. Amer. Math. Soc. 47 (2010), 171–176