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

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.

65M06, 65M12, 74A50

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@Article{CSIAM-AM-1-478, author = {Quan , ChaoyuTang , Tao and Yang , Jiang}, 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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