Analytical and numerical methods for time-fractional gradient flows
Dr. Marvin FritzJuly 18, 2022, 1:30 p.m. S2 416-2
We talk about time-fractional derivatives, which appear in applications where memory effects are present and hereditary properties of materials are studied, e.g., in viscoelasticity, viscoplasticity, diffusion and heat progression, signal processing, and cell growth. We discuss challenges and open problems in the analysis and numerics of time-fractional gradient flows. For example, there is no chain rule for fractional derivatives and compactness results such as the Aubin-Lions lemma are not directly applicable. Still, we provide remedies and new results to prove the well-posedness of the time-fractional Cahn-Hilliard equation. Moreover, there is a computational challenge since the nonlocal nature of time-fractional equations introduces a global history dependency. Still, there are numerical methods to circumvent this, which we will discuss.