Fully nonlinear integro-differential equations and Nonlocal porous medium equation: Difference between pages

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Fully nonlinear integro-differential equations are a nonlocal version of fully nonlinear elliptic equations of the form $F(D^2 u, Du, u, x)=0$. The main examples are the integro-differential [[Bellman equation]] from optimal control, and the [[Isaacs equation]] from stochastic games.
The nonlocal porous medium equation of order $\sigma$ is the name currently given to two very different equations, namely


A general definition of ellipticity can be given that does not require a specific form of the equation. However, the main two applications are the two above.
\[ u_t = \nabla \cdot \left ( u \nabla \mathcal{K_\alpha} (u) \right )\]


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


Given a family of linear integro-differential operators $\mathcal{L}$, we define the [[extremal operators]]$M^+_\mathcal{L}$ and $M^-_\mathcal{L}$:
and
\begin{align*}
M^+_\mathcal{L} u(x) &= \sup_{L \in \mathcal{L}} L u(x) \\
M^-_\mathcal{L} u(x) &= \inf_{L \in \mathcal{L}} L u(x)
\end{align*}


We define a nonlinear operator $I$ to be elliptic in a domain $\Omega$ with respect to the class $\mathcal{L}$ if it assigns a continuous function $Iu$ to every function $u \in L^\infty(\R^n) \cap C^2(\Omega)$, and moreover for any two such functions $u$ and $v$:
\[ u_t +(-\Delta)^{s}(u^m) = 0 \]
\[M^-_\mathcal{L} u(x)\leq Iu(x) - Iv(x) \leq M^+_\mathcal{L} u(x), \]
for any $x \in \Omega$.


A fully nonlinear elliptic equation with respect to $\mathcal{L}$ is an equation of the form $Iu=0$ in $\Omega$ with $I$ uniformly elliptic respect to $\mathcal{L}$.
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 <ref name="CV1"/>, 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 <ref name="PQRV" /ref>.
 
== 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>
 
<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>
}}
 
[[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