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In this wiki we collect several results about nonlocal elliptic and parabolic equations.
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<div style="font-size:162%; border:none; margin:0; padding:.1em; color:#000;">'''Welcome!'''</div>
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<div style="top:+0.2em; font-size:95%;">This is the Nonlocal Equations Wiki</div>
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<div id="articlecount" style="width:100%; text-align:center; font-size:85%;">([[Special:Statistics|{{NUMBEROFARTICLES}}]] articles and counting)</div>
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If you want to know what a nonlocal equation refers to, a good starting point would be the [[Intro to nonlocal equations]].
 
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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.
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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]]. If you want to find information on a specific topic, you may want to check the [[list of equations]] or use the search option on the left.
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We keep a list of [[open problems]] and also [[upcoming events]].
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We also keep a list of [[open problems]] and of [[upcoming events]].
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== Why nonlocal equations are important ==
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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.
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Some answers, including how to participate, can be found in the section about [[frequently asked questions]].
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<div style="font-size:150%; border:none; margin:0; padding:.1em; color:#000;">Why nonlocal equations?</div>
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
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
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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, nonlocal curvatures provide a very rich family of conformally invariant quantities.
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* In conformal geometry, the [[conformally invariant operators]] encode information about the manifold. They include fractional powers of the Laplacian.
* 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]].
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* Models for [[dislocation dynamics]] in crystals.
* 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...
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== Existence and uniqueness results ==
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For a variety of nonlinear elliptic and parabolic equations, the existence of [[viscosity solutions]] can be obtained using [[Perron's method]]. The uniqueness of solutions is a consequence of the [[comparison principle]].
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<div style="font-size:150%; border:none; margin:0; padding:.1em; color:#000;"> Suggested first reads </div>
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* [[Intro to nonlocal equations]]
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* [[Fractional Laplacian]]
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There are some equations for which this general framework does not work, for example the [[surface quasi-geostrophic equation]]. One could say that the underlying reason is that the equation is not ''purely'' parabolic, but it has one hyperbolic term.
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* [[Linear integro-differential operator]]
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== Regularity results ==
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* [[Fully nonlinear integro-differential equations]]
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The regularity tools used for nonlocal equations vary depending on the type of equation.
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* [[Myths about nonlocal equations]]
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=== Nonlinear equations ===
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* [[Surface quasi-geostrophic equation]]
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The starting point to study the regularity of solutions to a nonlinear elliptic or parabolic equation are the [[Holder estimates]] which hold under very weak assumptions and rough coefficients. They are related to the [[Harnack inequality]].
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For some [[fully nonlinear integro-differential equation]] with continuous coefficients, we can prove [[differentiability estimates|$C^{1,\alpha}$ estimates]].
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* [[Levy processes]]
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Under certain hypothesis, the nonlocal [[Bellman equation]] from optimal stochastic control has classical solutions due to the [[nonlocal Evans-Krylov theorem|nonlocal version of Evans-Krylov theorem]].
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* [[Obstacle problem]]
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=== Semilinear equations ===
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There are several interesting models that are [[semilinear equations]]. Those equations consists of either the [[fractional Laplacian]] or [[fractional heat equation]] plus a nonlinear term.
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There are challenging regularity questions especially when the Laplacian interacts with gradient terms in [[Drift-diffusion equations]]. A simple method that has been successful in proving the well posedness of some semilinear equations with drift terms in the critical case (when both terms have the same scaling properties) is the [[conserved modulus of continuity approach]], often called "nonlocal maximum principle method".
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Latest revision as of 19:09, 23 September 2013


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. If you want to find information on a specific topic, you may want to check the list of equations or use the search option on the left.

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.

Some answers, including how to participate, can be found in the section about frequently asked questions.

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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