List of results that are fundamentally different to the local case: Difference between revisions

From nonlocal pde
Jump to navigation Jump to search
imported>Luis
(Created page with "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...")
(No difference)

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