TY - JOUR T1 - How to Define Dissipation-Preserving Energy for Time-Fractional Phase-Field Equations AU - Quan , Chaoyu AU - Tang , Tao AU - Yang , Jiang JO - CSIAM Transactions on Applied Mathematics VL - 3 SP - 478 EP - 490 PY - 2020 DA - 2020/09 SN - 1 DO - http://doi.org/10.4208/csiam-am.2020-0024 UR - https://global-sci.org/intro/article_detail/csiam-am/18304.html KW - Phase-field equation, energy dissipation, Caputo fractional derivative, Allen-Cahn equations, Cahn-Hilliard equations, positive definite kernel. AB -
There exists a well defined energy for classical phase-field equations under which the dissipation law is satisfied, i.e., the energy is non-increasing with respect to time. However, it is not clear how to extend the energy definition to time-fractional phase-field equations so that the corresponding dissipation law is still satisfied. In this work, we will try to settle this problem for phase-field equations with Caputo time-fractional derivative, by defining a nonlocal energy as an averaging of the classical energy with a time-dependent weight function. As the governing equation exhibits both nonlocal and nonlinear behavior, the dissipation analysis is challenging. To deal with this, we propose a new theorem on judging the positive definiteness of a symmetric function, that is derived from a special Cholesky decomposition. Then, the nonlocal energy is proved to be dissipative under a simple restriction of the weight function. Within the same framework, the time fractional derivative of classical energy for time-fractional phase-field models can be proved to be always nonpositive.