We report on recent progress in the study of nonlinear diffusion equations involving
nonlocal, long-range diffusion effects. Our main concern is the so-called fractional porous …

We describe the mathematical theory of diffusion and heat transport with a view to including
some of the main directions of recent research. The linear heat equation is the basic …

We develop a theory of existence, uniqueness and regularity for the following porous
medium equation with fractional diffusion, with m> m⁎=(N− 1)/N, N⩾ 1 and f∈ L1 (RN). An …

We describe two models of flow in porous media including nonlocal (long-range) diffusion
effects. The first model is based on Darcy's law and the pressure is related to the density by …

We develop a theory of existence and uniqueness for the following porous medium equation
with fractional diffusion:\input amssym\left {∂ u ∂ t+\left (‐Δ\right)^ σ/2\left (\left| u\right|^ m‐1 …

We establish quantitative estimates for solutions u (t, x) to the fractional nonlinear diffusion
equation,∂ t u+(− Δ) s (um)= 0 in the whole range of exponents m> 0, 0< s< 1. The equation …

We obtain the large scale limit of the fluctuations around its hydrodynamic limit of the density
of particles of a weakly asymmetric exclusion process in dimension up to three. The proof is …

We introduce a fractional Yamabe flow involving nonlocal conformally invariant operators on
the conformal infinity of asymptotically hyperbolic manifolds, and show that on the conformal …

We consider an exclusion process with long jumps in the box $\Lambda\_N=\{1,\ldots, N-1\}
$, for $ N\ge 2$, in contact with infinitely extended reservoirs on its left and on its right. The …

We study the positivity and regularity of solutions to the fractional porous medium equations
in for m> 1 and s∈(0, 1), with Dirichlet boundary data u= 0 in and nonnegative initial …