A Decreasing Upper Bound of the Energy for Time-Fractional Phase-Field Equations

Abstract

In this article, we study the energy dissipation property of time-fractional Allen–Cahn equation. On the continuous level, we propose an upper bound of energy that decreases with respect to time and coincides with the original energy at $t=0$ and as $t$ tends to $ ∞.$ This upper bound can also be viewed as a nonlocal-in-time modified energy which is the summation of the original energy and an accumulation term due to the memory effect of time-fractional derivative. In particular, the decrease of the modified energy indicates that the original energy indeed decays w.r.t. time in a small neighborhood at $t=0.$ We illustrate the theory mainly with the time-fractional Allen–Cahn equation but it could also be applied to other time-fractional phase-field models such as the Cahn–Hilliard equation. On the discrete level, the decreasing upper bound of energy is useful for proving energy dissipation of numerical schemes. First-order L1 and second-order L2 schemes for the time-fractional Allen–Cahn equation have similar decreasing modified energies, so that stability can be established. Some numerical results are provided to illustrate the behavior of this modified energy and to verify our theoretical results.