Myths about nonlocal equations
The following myths are usually heard in the corridors of some math departments and conference coffee breaks.
- 1 There are no new difficulties in nonlocal equations and everything is proved analogously as in the classical case
- 2 Nonlocal equations is a field in which one replaces the Laplacian by the fractional Laplacian in whatever equation and writes a paper
- 3 Nonlocal equations are bizarre and unnatural objects
- 4 Most equations in nature are local
- 5 All statements and proofs in nonlocal equations involve gigantic formulas
There are no new difficulties in nonlocal equations and everything is proved analogously as in the classical case
Unfortunately, this is a common misconception. Nonlocal equations is a much richer class than the usual PDEs involving (local) differential operators of second order. Just look at the class of differential operators of order 2 with constant coefficients and the corresponding class of integro-differential operators of differentiability order 3/2 with constant coefficients (whatever this is).
Predictably, there are some intrinsic difficulties. A common difficulty comes from the fact that fractional order operators have different scaling properties and therefore interact differently with other terms. Moreover, in certain cases there are some surprising results which do not match what one would expect from local PDE intuition. We have a list of results that are fundamentally different to the local case.
Nonlocal equations is a field in which one replaces the Laplacian by the fractional Laplacian in whatever equation and writes a paper
One can certainly do this. In some cases the classical methods would work after a simple adaptation. In other cases there is a significant difference either in the methods or in the results. Naturally, the good papers are the ones that fit into the second category. This wiki should help people learn to differentiate one from the other.
Nonlocal equations are bizarre and unnatural objects
The Starting page of this wiki should clarify the importance of nonlocal equations.
Most equations in nature are local
In fact the opposite is true. In many cases local PDEs are a good simplification though.
All statements and proofs in nonlocal equations involve gigantic formulas
Nonlocal equations usually involve integral quantities that are larger to write than usual derivatives. This is a notation problem to a large extent. Many proofs in nonlocal equations deal with long integral quantities that come from the nonlocal character of the equation. These features are there, but are rarely at the essence of the arguments. Most statements and proofs are just as conceptual as in usual PDEs.