Existence and uniqueness for a class of fractional drift-diffusion equations

Abstract

This work establishes the existence and uniqueness of solutions to the fractional diffusion equation $$\frac{\partial^\alpha u}{\partial t^{\alpha}} + K(-\Delta)^{\beta} u - \nabla \cdot (\nabla V u) = f$$ on a $d$-dimensional torus, subject to sufficient conditions on the input parameters. The focus is on fractional orders $\alpha$ and $\beta$ less than 1. The strategy uses a Galerkin method and focuses on the additional complexity that comes from the fractional-order derivatives. Additional Sobolev regularity of the solution is shown. The spectral approach to the existence proof suggests an algorithm to compute explicit solutions numerically, and the regularity results are used to support a rigorous convergence analysis of the proposed numerical scheme.