Nonlocal porous medium equation: Difference between revisions

From nonlocal pde
Jump to navigation Jump to search
imported>Nestor
mNo edit summary
imported>Nestor
mNo edit summary
Line 18: Line 18:


== References ==
== References ==
{{reflist|refs=
{{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>
<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>
Line 23: Line 24:
<ref name="PQRV">{{Citation | last1=Pablo | first1=Arturo de | last2=Quirós | first2=Fernando | last3=Rodríguez | first3=Ana | last4=Vazquez | first4=Juan Luis | title=A fractional porous medium equation | url=http://www.sciencedirect.com/science/article/pii/S0001870810003130 | doi=DOI: 10.1016/j.aim.2010.07.017 | year=2011 | journal=Advances in Mathematics | issn=0001-8708 | volume=226 | issue=2 | pages=1378–1409}}
<ref name="PQRV">{{Citation | last1=Pablo | first1=Arturo de | last2=Quirós | first2=Fernando | last3=Rodríguez | first3=Ana | last4=Vazquez | first4=Juan Luis | title=A fractional porous medium equation | url=http://www.sciencedirect.com/science/article/pii/S0001870810003130 | doi=DOI: 10.1016/j.aim.2010.07.017 | year=2011 | journal=Advances in Mathematics | issn=0001-8708 | volume=226 | issue=2 | pages=1378–1409}}
</ref>
</ref>
}}
}}


[[Category:Quasilinear equations]] [[Category:Evolution equations]] [[Category:Free boundary problems]]
[[Category:Quasilinear equations]] [[Category:Evolution equations]] [[Category:Free boundary problems]]

Revision as of 18:01, 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: Invalid <ref> tag; invalid names, e.g. too many

[2] }}

  1. Cite error: Invalid <ref> tag; no text was provided for refs named CV1
  2. Pablo, Arturo de; Quirós, Fernando; Rodríguez, Ana; Vazquez, Juan Luis (2011), "A fractional porous medium equation", Advances in Mathematics 226 (2): 1378–1409, doi:DOI: 10.1016/j.aim.2010.07.017, ISSN 0001-8708, http://www.sciencedirect.com/science/article/pii/S0001870810003130