Volume 1, Issue 3
How to Define Dissipation-Preserving Energy for Time-Fractional Phase-Field Equations

Chaoyu Quan, Tao Tang & Jiang Yang

CSIAM Trans. Appl. Math., 1 (2020), pp. 478-490.

Published online: 2020-09

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  • 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 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.

  • Keywords

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

  • AMS Subject Headings

65M06, 65M12, 74A50

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COPYRIGHT: © Global Science Press

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@Article{CSIAM-AM-1-478, author = {Chaoyu and Quan and and 9150 and and Chaoyu Quan and Tao and Tang and and 9151 and and Tao Tang and Jiang and Yang and and 9152 and and Jiang Yang}, 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 = {

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.

}, issn = {2708-0579}, doi = {https://doi.org/10.4208/csiam-am.2020-0024}, url = {http://global-sci.org/intro/article_detail/csiam-am/18304.html} }
TY - JOUR T1 - How to Define Dissipation-Preserving Energy for Time-Fractional Phase-Field Equations AU - Quan , Chaoyu AU - Tang , Tao AU - Yang , Jiang JO - CSIAM Transactions on Applied Mathematics VL - 3 SP - 478 EP - 490 PY - 2020 DA - 2020/09 SN - 1 DO - http://doi.org/10.4208/csiam-am.2020-0024 UR - https://global-sci.org/intro/article_detail/csiam-am/18304.html KW - Phase-field equation, energy dissipation, Caputo fractional derivative, Allen-Cahn equations, Cahn-Hilliard equations, positive definite kernel. AB -

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.

Chaoyu Quan, Tao Tang & Jiang Yang. (2020). How to Define Dissipation-Preserving Energy for Time-Fractional Phase-Field Equations. CSIAM Transactions on Applied Mathematics. 1 (3). 478-490. doi:10.4208/csiam-am.2020-0024
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