The book deals with the 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 $u_t = \Delta u^m$, $m \geq 0$, $u \geq 0$. Such models arise in plasma physics, diffusion 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 uses 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 ($m_c < m < 1$, $m_c=(n-2)_+/n$) while many problems remain in the range $m \leq m_c$. All of these aspects of the theory are discussed in the book.
Author(s): Panagiota Daskalopoulos and Carlos E. Kenig
Publisher: European Mathematical Society
Year: 2007
Language: English
Commentary: no
Pages: 210
Tags: Математика;Математическая физика;
Cover......Page 1
EMS Tracts in Mathematics 1......Page 2
Editorial Board......Page 3
Title......Page 4
ISBN 978-3-03719-033-36......Page 5
Dedication......Page 6
Preface......Page 8
Contents......Page 10
Introduction......Page 12
Maximum principle and approximation......Page 20
A priori L-bounds for slow diffusion......Page 26
Harnack inequality for slow diffusion......Page 34
Local L-bounds for fast diffusion......Page 39
Equicontinuity of solutions......Page 44
Existence of weak solutions......Page 63
Pointwise estimates and existence of initial trace......Page 69
Uniqueness of solutions......Page 71
Existence and blow up......Page 77
Proof of Pierre's uniqueness result......Page 79
Further results......Page 88
The Cauchy problem for super-critical fast diffusion......Page 95
The Cauchy problem for logarithmic fast diffusion......Page 104
Further results and open problems......Page 134
Preliminary results......Page 144
The friendly giant (slow diffusion)......Page 149
The trace (slow diffusion)......Page 154
Existence of solutions (slow diffusion)......Page 156
Asymptotic behavior (slow diffusion)......Page 159
A priori estimates (fast diffusion)......Page 161
The trace and uniqueness (fast diffusion)......Page 163
Existence of solutions (fast diffusion m_1 < m < 1)......Page 164
Further results and open problems......Page 165
Weak solutions of the porous medium equation in a cylinder......Page 168
Weak solutions of the porous medium equation......Page 177
Further results and open problems......Page 196
Bibliography......Page 198
Index......Page 208
Back Cover......Page 210
