Myths about nonlocal equations and List of results that are fundamentally different to the local case: Difference between pages

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The following myths are usually heard in the corridors of some math departments and conference coffee breaks.
In this page we collect some results in nonlocal equations when things behave very differently compared to the local counterpart. A result makes it to this list if it is somewhat suprising or counterintuitive.


=== Traveling fronts in Fisher-KPP equations with fractional diffusion ===
Let us consider the  reaction diffusion equation
\[ u_t + (-\Delta)^s u = f(u), \]
with a Fisher-KPP type of nonlinearity (for example $f(u) = u(1-u)$). In the local diffusion case, the stable state $u=1$ invades the unstable state $u=0$ at a constant speed. In the nonlocal case (any $s<1$), the invasion holds at an exponential rate.


=== There are no new difficulties in nonlocal equations and everything is proved analogously as in the classical case ===
The explanation of the difference can be understood intuitively from the fact that the fat tails in the fractional heat kernels make diffusion happen at a much faster rate <ref name="CR"/>.


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).  
=== Optimal regularity for the fractional obstacle problem ===
Given a function $\varphi$, the obstacle problem consists in the solution to an equation of the form
\[ \min((-\Delta)^s u , u-\varphi) = 0.\]


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]].
If $\varphi$ is smooth enough, the solution $u$ to the obstacle problem will be $C^{1,s}$ and no better. There is a big difference between the case $s=1$ and $s<1$ which makes the proof fundamentally different. In the classical case $s=1$, the optimal regularity matches the scaling of the equation. The classical proof of optimal regularity is to show an upper bound in the separation of $u$ from the obstacle and then just scale. In the fractional case $s<1$, this method only gives $C^{2s}$ regularity, which matches the scaling of the equation. It is somewhat surprising that a better regularity result holds and it requires a different method for the proof.


=== Nonlocal equations is a field in which one replaces the Laplacian by the fractional Laplacian in whatever equation and writes a paper ===
The intuitive explanation is that $(-\Delta)^s u$ satisfies an equation in terms of its Laplacian to the power $1-s$, and that equation provides the extra regularity <ref name="S"/>.


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.
== References ==
 
{{reflist|refs=
=== Nonlocal equations are bizarre and unnatural objects ===
<ref name="CR">{{Citation | last1=Cabré | first1=Xavier | last2=Roquejoffre | first2=Jean-Michel | title=Propagation de fronts dans les équations de Fisher-KPP avec diffusion fractionnaire | url=http://dx.doi.org/10.1016/j.crma.2009.10.012 | doi=10.1016/j.crma.2009.10.012 | year=2009 | journal=Comptes Rendus Mathématique. Académie des Sciences. Paris | issn=1631-073X | volume=347 | issue=23 | pages=1361–1366}}</ref>
 
<ref name="S">{{Citation | last1=Silvestre | first1=Luis | title=Regularity of the obstacle problem for a fractional power of the Laplace operator | url=http://dx.doi.org/10.1002/cpa.20153 | doi=10.1002/cpa.20153 | year=2007 | journal=[[Communications on Pure and Applied Mathematics]] | issn=0010-3640 | volume=60 | issue=1 | pages=67–112}}</ref>
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.

Revision as of 18:36, 15 July 2011

In this page we collect some results in nonlocal equations when things behave very differently compared to the local counterpart. A result makes it to this list if it is somewhat suprising or counterintuitive.

Traveling fronts in Fisher-KPP equations with fractional diffusion

Let us consider the reaction diffusion equation \[ u_t + (-\Delta)^s u = f(u), \] with a Fisher-KPP type of nonlinearity (for example $f(u) = u(1-u)$). In the local diffusion case, the stable state $u=1$ invades the unstable state $u=0$ at a constant speed. In the nonlocal case (any $s<1$), the invasion holds at an exponential rate.

The explanation of the difference can be understood intuitively from the fact that the fat tails in the fractional heat kernels make diffusion happen at a much faster rate [1].

Optimal regularity for the fractional obstacle problem

Given a function $\varphi$, the obstacle problem consists in the solution to an equation of the form \[ \min((-\Delta)^s u , u-\varphi) = 0.\]

If $\varphi$ is smooth enough, the solution $u$ to the obstacle problem will be $C^{1,s}$ and no better. There is a big difference between the case $s=1$ and $s<1$ which makes the proof fundamentally different. In the classical case $s=1$, the optimal regularity matches the scaling of the equation. The classical proof of optimal regularity is to show an upper bound in the separation of $u$ from the obstacle and then just scale. In the fractional case $s<1$, this method only gives $C^{2s}$ regularity, which matches the scaling of the equation. It is somewhat surprising that a better regularity result holds and it requires a different method for the proof.

The intuitive explanation is that $(-\Delta)^s u$ satisfies an equation in terms of its Laplacian to the power $1-s$, and that equation provides the extra regularity [2].

References

  1. ↑ Cabré, Xavier; Roquejoffre, Jean-Michel (2009), "Propagation de fronts dans les équations de Fisher-KPP avec diffusion fractionnaire", Comptes Rendus Mathématique. Académie des Sciences. Paris 347 (23): 1361–1366, doi:10.1016/j.crma.2009.10.012, ISSN 1631-073X, http://dx.doi.org/10.1016/j.crma.2009.10.012 
  2. ↑ Silvestre, Luis (2007), "Regularity of the obstacle problem for a fractional power of the Laplace operator", Communications on Pure and Applied Mathematics 60 (1): 67–112, doi:10.1002/cpa.20153, ISSN 0010-3640, http://dx.doi.org/10.1002/cpa.20153 
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