How to Define Dissipation-Preserving Energy for Time-Fractional Phase-Field Equations

Chaoyu Quan*, Tao Tang*, Jiang Yang*

*Corresponding author for this work

Research output: Contribution to journalJournal articlepeer-review

26 Citations (Scopus)


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.

Original languageEnglish
Pages (from-to)478-490
Number of pages13
JournalCSIAM Transactions on Applied Mathematics
Issue number3
Publication statusPublished - 1 Sept 2020

Scopus Subject Areas

  • Applied Mathematics

User-Defined Keywords

  • Allen-Cahn equations
  • Cahn-Hilliard equations
  • Caputo fractional derivative
  • energy dissipation
  • Phase-field equation
  • positive definite kernel


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