论文标题
如何定义耗散能量的时间分数相位方程
How to Define Dissipation-Preserving Energy for Time-Fractional Phase-Field Equations
论文作者
论文摘要
有一个明确的经典相位方程的能量，在该方程中满足了耗散定律，即，能量在时间上是不侵入的。但是，尚不清楚如何将能量定义扩展到时相相位场方程，以便仍然满足相应的耗散定律。在这项工作中，我们将尝试通过将非局部能量定义为具有时间依赖性权重函数的经典能量的平均经典能量，以解决具有CAPUTO时间折段衍生物的相位场方程。当管理方程式表现出非本地性和非线性行为时，耗散分析具有挑战性。为了解决这个问题，我们提出了一种新定理，以判断对称功能的积极确定性，该定理是从特殊的cholesky分解中得出的。然后，在简单地限制体重函数的情况下，非本地能量被证明是耗散的。在同一框架内，可以证明，时间分数相位模型的经典能量的时间分数衍生物始终是非阳性的。
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.
