Nonlocal porous medium equation: Difference between revisions
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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]]. | 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]] | 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 H\"older continuity of weak solutions have been recently obtained <ref name="CV1"/>, 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). | For the second equation, both the Cauchy problem and long time behavior have been extensively studied in recent years (references). | ||
== References == | |||
{{reflist|refs= | |||
<ref name="CV1" {{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}} </ref> | |||
}} | |||
[[Category:Quasilinear equations]] [[Category:Evolution equations]] [[Category:Free boundary problems]] | [[Category:Quasilinear equations]] [[Category:Evolution equations]] [[Category:Free boundary problems]] | ||
Revision as of 16:56, 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 H\"older continuity of weak solutions have been recently obtained [1], 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).
References
- ↑ Cite error: Invalid
<ref>tag; no text was provided for refs namedCV1