Equivalence between a time-fractional and an integer-order gradient flow: The memory effect reflected in the energy

Time-fractional partial differential equations are nonlocal-in-time and show an innate memory effect. Previously, examples like the time-fractional Cahn-Hilliard and Fokker-Planck equations have been studied. In this work, we propose a general framework of time-fractional gradient flows and we provide a rigorous analysis of well-posedness using the Faedo-Galerkin approach. Furthermore, we investigate the monotonicity of the energy functional of time-fractional gradient flows. Interestingly, it is still an open problem whether the energy is dissipating in time. This property is essential for integer-order gradient flows and many numerical schemes exploit this steepest descent characterization. We propose an augmented energy functional, which includes the history of the solution. Based on this new energy, we prove the equivalence of a time-fractional gradient flow to an integer-order one. This correlation guarantees the dissipating character of the augmented energy. The state function of the integer-order gradient flow acts on an extended domain similar to the Caffarelli-Silvestre extension for the fractional Laplacian. Additionally, we present a numerical scheme for solving time-fractional gradient flows, which is based on kernel compressing methods and reduces the problem to a system of ordinary differential equations. We illustrate the behavior of the original and augmented energy in the case of the Ginzburg-Landau energy.