@Article{CSIAM-AM-1-3,
author = {Quan, Chaoyu and Tao, Tang and Yang, Jiang},
title = {How to Define Dissipation-Preserving Energy for Time-Fractional Phase-Field Equations},
journal = {CSIAM Transactions on Applied Mathematics},
year = {2020},
volume = {1},
number = {3},
pages = {478--490},
abstract = {<p style="text-align: justify;">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.</p>},
issn = {2708-0579},
doi = {https://doi.org/10.4208/csiam-am.2020-0024},
url = {https://global-sci.com/article/82383/how-to-define-dissipation-preserving-energy-for-time-fractional-phase-field-equations}
}