Porous media equation

The porous media equation (also English as a porous medium equation referred to) is a non-linear degenerate parabolic partial differential equation . She owns the form

${\displaystyle {\frac {\partial u}{\partial t}}=\Delta u^{m}}$,

where is and denotes the spatial Laplace operator . ${\displaystyle m>1}$ ${\displaystyle \Delta :=\sum _{k=1}^{n}{\frac {\partial ^{2}}{\partial x_{k}^{2}}}}$

The porous media equation is used, for example, to describe the flow of an ideal gas in a homogeneous porous medium . In this case, the density of the gas is a solution of the porous media equation.

It is also sometimes referred to as the nonlinear conduction equation because it would be obtained by inserting . However, the heat conduction equation has unphysical properties, in particular an infinite speed of propagation. Due to the degeneracy ( ), the solutions have significantly different properties, namely, for example, a finite speed of propagation . ${\displaystyle m=1}$ ${\displaystyle m>1}$

literature

• Lawrence C. Evans : Partial Differential Equations. Reprinted with corrections. American Mathematical Society, Providence RI 1999, ISBN 0-8218-0772-2 ( Graduate Studies in Mathematics 19).
• Juan Luis Vázquez : The Porous Medium Equation. Mathematical theory. Clarendon Press, Oxford 2007, ISBN 978-0-19-856903-9 ( Oxford Mathematical Monographs ).