Besov and Triebel–Lizorkin space estimates for fractional diffusion

Space-time duality for fractional diffusion

Zolotarev proved a duality result that relates stable densities with different indices. In this paper, we show how Zolotarev duality leads to some interesting results on fractional diffusion. Fractional diffusion equations employ fractional derivatives in place of the usual integer order derivatives. They govern scaling limits of random walk models, with power law jumps leading to fractional de...

An Implicit Difference-ADI Method for the Two-dimensional Space-time Fractional Diffusion Equation

Fractional order diffusion equations are generalizations of classical diffusion equations which are used to model in physics, finance, engineering, etc. In this paper we present an implicit difference approximation by using the alternating directions implicit (ADI) approach to solve the two-dimensional space-time fractional diffusion equation (2DSTFDE) on a finite domain. Consistency, unconditi...

Sharp Energy Estimates for Nonlinear Fractional Diffusion Equations

We study the nonlinear fractional equation (−∆)u = f(u) in R, for all fractions 0 < s < 1 and all nonlinearities f . For every fractional power s ∈ (0, 1), we obtain sharp energy estimates for bounded global minimizers and for bounded monotone solutions. They are sharp since they are optimal for solutions depending only on one Euclidian variable. As a consequence, we deduce the one-dimensional ...

Strong Type Estimates for Homogeneous Besov Capacities

Continuous Dependence Estimates for Nonlinear Fractional Convection-diffusion Equations

We develop a general framework for finding error estimates for convection-diffusion equations with nonlocal, nonlinear, and possibly degenerate diffusion terms. The equations are nonlocal because they involve fractional diffusion operators that are generators of pure jump Lévy processes (e.g. the fractional Laplacian). As an application, we derive continuous dependence estimates on the nonlinea...