Introduction to nonlocal equations
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 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".