# Nonlocal porous medium equation

(Difference between revisions)
 Revision as of 02:16, 24 January 2012 (view source)Nestor (Talk | contribs)← Older edit Latest revision as of 02:48, 16 April 2015 (view source)Luis (Talk | contribs) Line 11: Line 11: 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]]. 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, for which [[instantaneous speed of propagation]] holds. + 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 recently obtained , there is almost nothing known about the properties of its free boundary, making it a rich source of open questions. In the one-dimensional case, this equation is equivalent to a model for the [[dislocation dynamics  | dynamics of dislocation lines]], in this case uniqueness of solutions has been obtained , this is still unknown in higher dimensions. + 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 [[dislocation dynamics  | 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, both the Cauchy problem and long time behavior have been extensively studied in recent years . + For the second equation, the Cauchy problem, the regularity of solutions and long time behavior, have been extensively studied in recent years . == References == == References ==

## Latest revision as of 02:48, 16 April 2015

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 [1]. 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 [2]. 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 [3].

## References

1. Caffarelli, Luis; Vazquez, Juan (2011), "Nonlinear Porous Medium Flow with Fractional Potential Pressure", Archive for Rational Mechanics and Analysis (Springer Berlin / Heidelberg): 1–29, ISSN 0003-9527
2. Biler, Piotr; Monneau, Régis; Karch, Grzegorz (2009), "Nonlinear Diffusion of Dislocation Density and Self-Similar Solutions", Communications in Mathematical Physics 294 (1): 145–168, doi:10.1007/s00220-009-0855-8, ISSN 0010-3616
3. Pablo, Arturo de; Quirós, Fernando; Rodríguez, Ana; Vazquez, Juan Luis (2011), "A fractional porous medium equation", Advances in Mathematics 226 (2): 1378–1409, doi:DOI: 10.1016/j.aim.2010.07.017, ISSN 0001-8708