Nonlocal porous medium equation

The nonlocal porous medium equation of order $\sigma$ is the name currently given to two very different equations, namely

\[ u_t = \nabla \cdot \left ( u \nabla \mathcal{K_\alpha} (u) \right )\]

\[\mbox{ where } \mathcal{K}_\alpha(u) := C_{n,\alpha}\; u * |x|^{-n+\alpha},\;\; \alpha+2=\sigma \]

and

\[ u_t +(-\Delta)^{s}(u^m) = 0 \]

These equations agree when $s=1$ and $m=2$, otherwise they are not only different superficially, they also exhibit extremely different behaviors. They are both fractional order Quasilinear equations.

The first of the two has the remarkable property (for nonlocal equations at least) that any initial data with compact support remains with compact support for all later times. The opposite is true of the second equation: any nontrivial nonnegative solution becomes strictly positive instantaneously.

This means that the first model presents us with a free boundary problem. For this model global existence and Hölder continuity of weak solutions have been obtained . The properties of its free boundary are less well understood. In the one-dimensional case, this equation is equivalent to a model for the dynamics of dislocation lines. In this case, the uniqueness of solutions has been obtained . Uniqueness is still unknown in higher dimensions.

For the second equation, the Cauchy problem, the regularity of solutions and long time behavior, have been extensively studied in recent years .