The Mathematical Theories of Diffusion: Nonlinear and Fractional Diffusion

Abstract

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 mathematical model that has been thoroughly studied in the last two centuries. It was followed by the theory of parabolic equations of different types. In a parallel development, the theory of stochastic differential equations gives a foundation to the probabilistic study of diffusion.Nonlinear diffusion equations have played an important role not only in theory but also in physics and engineering, and we focus on a relevant aspect, the existence and propagation of free boundaries. Due to our research experience, we use the porous medium and fast diffusion equations as case examples.A large part of the paper is devoted to diffusion driven by fractional Laplacian operators and other nonlocal integro-differential operators representing nonlocal, long-range diffusion effects. Three main models are examined (one linear, two nonlinear), and we report on recent progress in which the author is involved.

Asymptotic behavior of solutions to fractional diffusion–convection equations

“…In the last 20 years there has been a great interest in models with nonlocal diffusion, specially fractional diffusion since the fractional Laplacian is the infinitesimal generator of a stable Levy process. There are many applications in physical sciences where models with anomalous diffusion are needed, see the survey for a description of possible applications, and the lecture notes for a presentation of recent models involving nonlocal diffusion.…”

The fractional Schrödinger equation with general nonnegative potentials. The weighted space approach

“…This plays an important role in many applications as for instance Quantum Mechanics, as already mentioned. Furthermore, since in bounded domains there are several different choices of (−∆) s present in the literature (see, e.g., [6,74]), it is relevant to study which choice represents the correct localization of a global problem (see Section 6).…”

Diffusion and kinetic transport with very weak confinement

“…Standard techniques use the Fourier transform, Green kernel estimates and integral representations: see for instance [14]. There are many other parabolic methods which provide decay rates and will not be reviewed here like, for instance, the Maximum Principle, Harnack inequalities and the parabolic regularity theory: see for instance [25].…”