Porous media flow

Definition

The porous media flow with parameter $m$ is the flow equation corresponding to the Laplacian of the $m^{th}$ power. That is:

$\frac{\partial u}{\partial t}=\frac{1}{m}\mathrm{\Delta }(u^m)$

Particular cases

• When $m=1$, this is just the ordinary linear heat equation
• When $m>1$, it models ionized gases at very high temperatures
• When $0<m<1$, this appears in various models in plasma physics
• In the limit as $m=0$ we get the equation:

$\frac{\partial u}{\partial t}=\mathrm{\Delta }\log u$

which is an equation for the evolution of the scalar function relating two conformally equivalent metrics on a differential manifold.