# Nonlocal porous medium equation

(Difference between revisions)
 Revision as of 06:43, 2 June 2011 (view source)Nestor (Talk | contribs)← Older edit Revision as of 19:04, 2 June 2011 (view source)Nestor (Talk | contribs) Newer edit → Line 9: Line 9: $u_t +(-\Delta)^{s}(u^m) = 0$ $u_t +(-\Delta)^{s}(u^m) = 0$ - These equations agree when $s=1$ and $m=2$, otherwise they are not only different superficially, they also exhibit extremely different behaviors. They are both fractional order [[Quasilinear equations]]. The first of the two has the remarkable property (for nonlocal equations at least) that any initial data with compact support remains with compact support for all later times, the opposite is true of the second equation, for which [[instantaneous speed of propagation]] holds, thus, at least from the point of view of [[free boundary problems]], the first equation is the "truer" nonlocal porous medium equation. + These equations agree when $s=1$ and $m=2$, otherwise they are not only different superficially, they also exhibit extremely different behaviors. They are both fractional order [[Quasilinear equations]]. + The first of the two has the remarkable property (for nonlocal equations at least) that any initial data with compact support remains with compact support for all later times, the opposite is true of the second equation, for which [[instantaneous speed of propagation]]. This means that the first model presents us with a [[free boundary problem]], although there are some regularity results for the solution (add references), there is almost nothing known about the properties of its free boundary, this provides with a rich list of good open problems. + + For the second equation, both the Cauchy problem and long time behavior have been extensively studied in recent years (references). [[Category:Quasilinear equations]] [[Category:Evolution equations]] [[Category:Free boundary problems]] [[Category:Quasilinear equations]] [[Category:Evolution equations]] [[Category:Free boundary problems]]

## Revision as of 19:04, 2 June 2011

The nonlocal porous medium equation of order $\sigma$ is the name currently given to two very different equations, namely

$u_t = \nabla \cdot \left ( u \nabla \mathcal{K_\alpha} (u) \right )$

$\mbox{ where } \mathcal{K}_\alpha(u) := C_{n,\alpha}\; u * |x|^{-n+\alpha},\;\; \alpha+2=\sigma$

and

$u_t +(-\Delta)^{s}(u^m) = 0$

These equations agree when $s=1$ and $m=2$, otherwise they are not only different superficially, they also exhibit extremely different behaviors. They are both fractional order Quasilinear equations.

The first of the two has the remarkable property (for nonlocal equations at least) that any initial data with compact support remains with compact support for all later times, the opposite is true of the second equation, for which instantaneous speed of propagation. This means that the first model presents us with a free boundary problem, although there are some regularity results for the solution (add references), there is almost nothing known about the properties of its free boundary, this provides with a rich list of good open problems.

For the second equation, both the Cauchy problem and long time behavior have been extensively studied in recent years (references).