Speaker：Chaoyu Quan(Southern University of Science and Technology)

Time：2020-11-02, 15:00

Location：Conference Room 686 at the 6th floor of Shuli Building at Haiyun Campus

Abstract:

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 settle this problem for phase-field equations with Caputo time_ fractional derivative, by defining a nonlocal energy as an averaging of the classical en_ergy 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.