Degenerate diffusions

✍ Scribed by Daskalopoulos P., Kenig C.E.

• Publisher: EMS
• Year: 2007
• Tongue: English
• Leaves: 208
• Series: EMS Tracts in Mathematics
• Category: Library

✦ Synopsis

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.

✦ Table of Contents

Preface......Page 7
Contents......Page 9
Introduction......Page 11
Maximum principle and approximation......Page 19
A priori L-bounds for slow diffusion......Page 25
Harnack inequality for slow diffusion......Page 33
Local L-bounds for fast diffusion......Page 38
Equicontinuity of solutions......Page 43
Existence of weak solutions......Page 62
Pointwise estimates and existence of initial trace......Page 68
Uniqueness of solutions......Page 70
Existence and blow up......Page 76
Proof of Pierre's uniqueness result......Page 78
Further results......Page 87
The Cauchy problem for super-critical fast diffusion......Page 94
The Cauchy problem for logarithmic fast diffusion......Page 103
Further results and open problems......Page 133
Preliminary results......Page 143
The friendly giant (slow diffusion)......Page 148
The trace (slow diffusion)......Page 153
Existence of solutions (slow diffusion)......Page 155
Asymptotic behavior (slow diffusion)......Page 158
A priori estimates (fast diffusion)......Page 160
The trace and uniqueness (fast diffusion)......Page 162
Existence of solutions (fast diffusion m_1 < m < 1)......Page 163
Further results and open problems......Page 164
Weak solutions of the porous medium equation in a cylinder......Page 167
Weak solutions of the porous medium equation......Page 176
Further results and open problems......Page 195
Bibliography......Page 197
Index......Page 207