How to Define Dissipation-Preserving Energy for Time-Fractional Phase-Field Equations (CSIAM Transactions on Applied Mathematics, 2020)
There exists a well defined energy for classical phase-field equations underwhich the dissipation law is satisfied, i.e., the energy is non-increasing with respect totime. However, it is not clear how to extend the energy definition to time-fractionalphase-field equations so that the corresponding dissipation law is still satisfied. In thiswork, 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 bothnonlocal and nonlinear behavior, the dissipation analysis is challenging. To deal withthis, we propose a new theorem on judging the positive definiteness of a symmetricfunction, that is derived from a special Cholesky decomposition. Then, the nonlocalenergy 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.
Keywords: Phase-field equation; energy dissipation; Caputo fractional derivative; Allen-Cahn equations; Cahn-Hilliard equations; positive definite kernel.