How to Define Dissipation-Preserving Energy for Time-Fractional Phase-Field Equations
How to Define Dissipation-Preserving Energy for Time-Fractional Phase-Field Equations
复制标题
如何定义时间分数相场方程的耗散守恒能量
DOI：
10.4208/csiam-am.2020-0024
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发表时间：
2020
期刊：
影响因子：
--
通讯作者：
Jiang Yang
中科院分区：
文献类型：
--
作者：
Chaoyu Quan;Tao Tang;Jiang Yang
对于经典的相场方程存在一个定义明确的能量，在该能量下，耗散律得到满足，即，能量相对于时间不增加。然而，目前尚不清楚如何将能量定义扩展到时间分数相场方程，以使相应的耗散律仍然满足。在这项工作中，我们将试图解决这个问题的相场方程与Caputo时间分数阶导数，通过定义一个非局部能量的平均的经典能量与时间相关的权重函数。由于控制方程具有非局部和非线性的特性，耗散分析具有挑战性。为了解决这个问题，我们提出了一个新的定理，判断一个对称函数的正定性，这是来自一个特殊的Cholesky分解。然后，在权函数的简单约束下，证明了非局部能量是耗散的。在同样的框架下，时间分数阶相场模型的经典能量的时间分数阶导数总是非正的。
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.