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 …

This paper deals with the fractional Sobolev spaces Ws, p. We analyze the relations among
some of their possible definitions and their role in the trace theory. We prove continuous and …

It has been observed in many numerical simulations, experiments and from various
theoretical treatments that heat transport in one-dimensional systems of interacting particles …

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 …

This research monograph is motivated by problems in mathematical physics that involve
fractional kinetic equations. These equations describe transport dynamics in complex …

We obtain the integration by parts formula for the regional fractional Laplacian which are
generators of symmetric α-stable processes on a subset of R^ n (0< α< 2). In this formula, a …

Consider a Markov chain {X n} n≥ 0 with an ergodic probability measure π. Let Ψ be a
function on the state space of the chain, with α-tails with respect to π, α∈(0, 2). We find …

We derive quantitatively the Harnack inequalities for kinetic integro-differential equations.
This implies Hölder continuity. Our method is based on trajectories and exploits a term …