# Nonlocal porous medium equation

(Difference between revisions)
 Revision as of 23:01, 3 June 2011 (view source)Nestor (Talk | contribs)m← Older edit Revision as of 23:01, 3 June 2011 (view source)Nestor (Talk | contribs) Newer edit → Line 18: Line 18: == References == == References == - {{reflist|refs= {{reflist|refs= {{Citation | last1=Caffarelli | first1=Luis | last2=Vazquez | first2=Juan | title=Nonlinear Porous Medium Flow with Fractional Potential Pressure | url=http://dx.doi.org/10.1007/s00205-011-0420-4 | publisher=Springer Berlin / Heidelberg | year=2011 | journal=Archive for Rational Mechanics and Analysis | issn=0003-9527 | pages=1–29}} {{Citation | last1=Caffarelli | first1=Luis | last2=Vazquez | first2=Juan | title=Nonlinear Porous Medium Flow with Fractional Potential Pressure | url=http://dx.doi.org/10.1007/s00205-011-0420-4 | publisher=Springer Berlin / Heidelberg | year=2011 | journal=Archive for Rational Mechanics and Analysis | issn=0003-9527 | pages=1–29}}

## Revision as of 23:01, 3 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 holds.

This means that the first model presents us with a free boundary problem. For this model global existence and Hölder continuity of weak solutions have been recently obtained [1], there is almost nothing known about the properties of its free boundary, making it a rich source of open questions.

For the second equation, both the Cauchy problem and long time behavior have been extensively studied in recent years Cite error: Closing </ref> missing for <ref> tag

[2] }}

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