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.

一些数学家和物理学家已经提出了一些由偏微分方程组成的数学模型来描述形状记忆合金材料的动力学。在所有这些模型中，他们通过多项式和函数来近似应变和应力之间的关系，并推导出模型。然而，在一些实验中我们知道这种关系不是通常的函数，可以用滞后算子来表达，这取决于历史数据。因此，在这个研究项目中，我们提出了一种新的数学模型，包括一个没有多项式逼近的磁滞算子，并利用休伯特空间上凸函数的时间相关次微分控制的演化方程理论来研究该模型。首先，我们考虑了以下问题。我们已经知道，滞环算子的特征是常微分方程，包括指示函数的次微分算子。然后我们将常微分方程添加到由动量平衡定律和内能平衡定律组成的系统中。该系统的解可能不满足A平滑条件，因此我们通过替换抛物线来逼近常微分方程。我们的第一个结果是证明与这样的近似问题有关的唯一性定理的存在性。接下来，我们将抛物线方程的经典理论应用到我们的系统中，并在没有任何近似的情况下证明了我们问题的适定性。