Degenerate Diffusions (ems Tracts In Mathematics)
by Panagiota Daskalopoulos /
2007 / English / PDF
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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.
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.
