Some existence and regularity results for a non-local transport-diffusion equation with fractional derivatives in time and space

Abstract

We study the existence of global weak solutions of a nonlinear transport-diffusion equation with a fractional derivative in the time variable and, under some extra hypotheses, we also study some regularity properties for this type of solutions. In the system considered here, the diffusion operator is given by a fractional Laplacian and the nonlinear drift is assumed to be divergence free and it is assumed to satisfy some general stability and boundedness properties in Lebesgue spaces. In order to obtain global solutions, we first introduce an hyperviscosity perturbation and we perform a fixed-point argument to obtain a solution of the perturbed equation. Then, by using suitable a priori information, given by an energy inequality, we can extend the solutions and we finally obtain global weak solutions of the original problem.