2020/07/29 by Chaoyu Quan, Tao Tang, Quan, Chaoyu +3 · 1 citation
Engineering · Materials Science · Mathematics · #65M06 #65M12 #74A50 #Analysis of PDEs (math.AP) #FOS: Mathematics #Fluid Dynamics and Thin Films #Fractional Differential Equations Solutions #Solidification and crystal growth phenomena
paper · pdf · doi:10.48550/arxiv.2007.14855
openalex publication_date 2020/07/29 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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.