Wellposedness and regularity of steady-state two-sided variable-coefficient conservative space-fractional diffusion equations

TL;DR

This paper investigates the mathematical properties of steady-state two-sided variable-coefficient space-fractional diffusion equations, establishing well-posedness and regularity results using novel formulations and analysis techniques.

Contribution

It introduces a Petrov-Galerkin formulation for variable-coefficient fractional diffusion equations and proves their well-posedness and high-order regularity.

Findings

The Galerkin weak formulation loses coercivity for variable coefficients.

The solution can be characterized via second-order diffusion and fractional integral equations.

The proposed Petrov-Galerkin formulation is weakly coercive and well-posed.

Abstract

We study the Dirichlet boundary-value problem of steady-state two-sided variable-coefficient conservative space-fractional diffusion equations. We show that the Galerkin weak formulation, which was proved to be coercive and continuous for a constant-coefficient analogue of the problem, loses its coercivity. We characterize the solution to the variable-coefficient problem in terms of the solutions of second-order diffusion equations along with a two-sided fractional integral equation. We then derive a Petrov-Galerkin formulation for this problem and prove that the weak formulation is weakly coercive and so the problem is well posed. We then prove high-order regularity estimates of the true solution in a properly chosen norm of Riemann-Liouville derivatives.