How to define dissipation-preserving energy for time-fractional phase-field equations?

：Dr. Quan Chaoyu
：2020-11-02 15:00
：数理大楼6楼天元会议室 (线下)

Speaker：Dr. Quan Chaoyu

      南方科技大学

Title: How to define dissipation-preserving energy for time-fractional phase-field equations?

Time：2^th, Nov., 2020, 15:00

Location：数理大楼6楼天元会议室 (线下) 

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.

Speaker Introduction：

权超禹，南方科技大学国际数学中心助理研究员。2013年本科毕业于中国科技大学，2017年博士毕业于巴黎六大，后在索邦大学（原巴黎六大）从事博士后研究一年。他的主要研究兴趣包括隐氏溶剂化模型中的数学方法、时间分数阶相场方程等。

 

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