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

How to Define Dissipation-Preserving Energy for Time-Fractional Phase-Field Equations
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Year:    2020

Author:    Chaoyu Quan, Tao Tang, Jiang Yang

CSIAM Transactions on Applied Mathematics, Vol. 1 (2020), Iss. 3 : pp. 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.

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Journal Article Details

Publisher Name:    Global Science Press

Language:    English

DOI:    https://doi.org/10.4208/csiam-am.2020-0024

CSIAM Transactions on Applied Mathematics, Vol. 1 (2020), Iss. 3 : pp. 478–490

Published online:    2020-01

AMS Subject Headings:    Global Science Press

Copyright:    COPYRIGHT: © Global Science Press

Pages:    13

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

Author Details

Chaoyu Quan

Tao Tang

Jiang Yang

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