In this wiki we collect several result 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.
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. One could even go further and boldly say that in nature all equations are nonlocal, and PDEs are a simplification. 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
- Optimal control problems with Levy processes give rise to the Bellman equation, or in general any equation derived from jump processes with be some fully nonlinear integro-differential equation.
- In financial mathematics it is particularly important to study models involving jump processes. This can be considered a particular case of the item above (stochastic control), but it is a very relevant one. The pricing model for American options involves the Obstacle problem.
- Nonlocal electrostatics is a very promising tool for drug design which could potentially have a strong impact in medicine in the future.
- 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.
- In conformal geometry, nonlocal curvatures provide a very rich family of conformally invariant quantities.
- 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...
Existence and uniqueness results
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.
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.
The regularity tools used for nonlocal equations vary depending on the type of equation.
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.
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. There are challenging regularity questions especially when the Laplacian interacts with gradient terms in Drift-diffusion equations.