A research on a system of nonlinear partial differential equations describing phase transition phenomena

描述相变现象的非线性偏微分方程组的研究

基本信息

批准号：
12640166
负责人：
AIKI Toyohiko
金额：
$ 2.37万
依托单位：
Gifu University
依托单位国家：
日本
项目类别：
Grant-in-Aid for Scientific Research (C)
财政年份：
2000
资助国家：
日本
起止时间：
2000 至 2001
项目状态：
已结题

项目摘要

Several mathematicians and physicists already proposed some mathematical models consisting of partial differential equations in order to describe dynamics of shape memory alloy materials. In all of them they approximated the relationship between the strain and the stress by polynomials and functions and derived the models. However, in some experiments we know that the relationship is not a usual function and can be express by hysteresis operator, which depends on the historical data. Hence, in this research project we have proposed a new mathematical model including a hysteresis operator without polynomial approximation and studied the model by using the theory for evolution equations governed by time-dependent subdifferentials of convex functions on Hubert spaces. First, we considered the following problem. We already have known that the hysteresis operator is characterized by ordinary differential equations including the subdifferential operator of the indicator function. Then we added the ordinary differential equation to the system consisting of momentum balance law and internal energy balance law. A solution of this system may not satisfy the A smoothness condition so that we approximate the ordinary differential eauations by replacing the parabolic. Our first result is to prove the existence anu uniqueness theorem concerned witn sucn an approximated problem. Next, we applied the classical theory for parabolic equations to our system and showed the wellposedness of our problem without any approximations.

几位数学家和物理学家已经提出了一些由部分微分方程组成的数学模型，以描述形状记忆合金材料的动力学。在所有这些中，它们近似多项式和功能的应变与应力之间的关系，并得出模型。但是，在某些实验中，我们知道这种关系不是通常的功能，可以由滞后操作员表达，这取决于历史数据。因此，在该研究项目中，我们提出了一个新的数学模型，其中包括一个没有多项式近似的磁滞运算符，并通过使用该理论来研究该模型，用于由凸面上凸的时间依赖性亚分化函数控制的进化方程。首先，我们考虑了以下问题。我们已经知道，滞后操作员的特征是普通微分方程，包括指示器函数的亚分类操作员。然后，我们将普通的微分方程添加到由动量平衡法和内部能量平衡法组成的系统中。该系统的解决方案可能无法满足平滑度条件，因此我们通过更换抛物线代理来近似普通的差异。我们的第一个结果是证明存在anu唯一性定理与witn相关的问题。接下来，我们将经典理论应用于抛物线方程，并在我们的系统中显示出良好的问题，没有任何近似值。