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* The denoising algorithms in [[nonlocal image processing]] are able to detect patterns in a better way than the PDE based models. A simple model for denoising is the [[nonlocal mean curvature flow]].
* The denoising algorithms in [[nonlocal image processing]] are able to detect patterns in a better way than the PDE based models. A simple model for denoising is the [[nonlocal mean curvature flow]].
* The [[Boltzmann equation]] models the evolution of dilute gases and it is intrinsically an integral equation. In fact, simplified [[kinetic models]] can be used to derive the [[fractional heat equation]] without resorting to stochastic processes.
* The [[Boltzmann equation]] models the evolution of dilute gases and it is intrinsically an integral equation. In fact, simplified [[kinetic models]] can be used to derive the [[fractional heat equation]] without resorting to stochastic processes.
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* In conformal geometry, the Paneitz operators $\mathcal{P}(s)$ encode information about the manifold, when the parameter is not an even integer they are fractional powers of the Laplacian, and therefore are nonlocal.  
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* In conformal geometry, the Paneitz operators encode information about the manifold, they include fractional powers of the Laplacian, which are nonlocal operators.
* In oceanography, the temperature on the surface may diffuse though the atmosphere giving rise to the [[surface quasi-geostrophic equation]].
* In oceanography, the temperature on the surface may diffuse though the atmosphere giving rise to the [[surface quasi-geostrophic equation]].
* Several stochastic models, in particular particle systems, can be used to derive nonlocal equations like the [[Nonlocal porous medium equation]], the [[Hamilton-Jacobi equation with fractional diffusion]], [[conservation laws with fractional diffusion]], etc...
* Several stochastic models, in particular particle systems, can be used to derive nonlocal equations like the [[Nonlocal porous medium equation]], the [[Hamilton-Jacobi equation with fractional diffusion]], [[conservation laws with fractional diffusion]], etc...

Revision as of 04:36, 7 February 2012

Welcome!
This is the Nonlocal Equations Wiki
(67 articles and counting)


In this wiki we collect several results about nonlocal elliptic and parabolic equations. If you want to know what a nonlocal equation refers to, a good starting point would be the Intro to nonlocal equations.

We also keep a list of open problems and of upcoming events.


The wiki has an assumed bias towards regularity results and consequently to equations for which some regularization occurs. But we also include some topics which are tangentially related, or even completely unrelated, to regularity.

Why nonlocal equations?

All partial differential equations are a limit case of nonlocal equations. A good understanding of nonlocal equations can ultimately provide a better understanding of their limit case: the PDEs. However, there are some cases in which a nonlocal equation gives a significantly better model than a PDE. Some of the most clear examples in which it is necessary to resort to nonlocal equations are

Suggested first reads



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