Analysis of nonlinear phenomena describing hysteresis operators

【形状记忆合金问题】：在形状记忆合金材料的动力学中，应变和应力之间的关系不能用通常的函数来描述，而是用迟滞算子来表示。在这里，本研究项目的主要思想是使用一般停止算子来描述这种关系。此外，我们考虑了等价于广义停止算子的常微分方程，并提出了一个描述形状记忆合金动力学的系统。该系统由偏微分方程和常微分方程组成，本课题对其进行了研究。我们考虑一维问题，其中我们处理的常微分方程没有近似值。在这种情况下，常微分方程解的正则性在空间上是不够的，因此不可能证明经典解的存在性。然后证明了一个弱解的存在唯一性。此外，我们还考虑了三维形状记忆合金问题。在三维情况下，解的规律性不是很好。由此证明了仅近似问题的适定性。在这里，我们注意到，我们也有一个计划来分析我们的数学模型，数值。目前，我们还没有成功地得到一个完整问题的数值解，也没有一个求解动力学方程的算法，这在我们的过程中是最难计算的。[铁磁中的磁化]：利用广义Duhem模型提出了铁磁动力学的数学模型，并证明了该模型的适定性。

[Shape memory alloy problems] : In the dynamics of shape memory alloy materials the relationship between the strain and the stress can not be described by a usual function, and is represented by a hysteresis operator. Here, the main idea of this research project is to describe the relationship by using a general stop operator. Moreover, we consider the ordinary differential equation, which is equivalent to the generalized stop operator, and propose a system describing the dynamics of shape memory alloys. The system consists of partial differential equations and the ordinary differential equation and is studied in this project. We consider the one-dimensional problem, in which we dealt the ordinary differential equation without an approximation. In this case the regularity of a solution to the ordinary differential equation is not enough in space so that it is impossible to prove the existence of a classical solution. Then we showed the existence and the uniqueness of a weak solution. Also, we considered a shape memory alloy problem in three dimensions. In 3-d case the regularity of a solution is not good. Hence, we proved the well-posed ness for only approximated problem. Here, we note that we also have a plan to analyze our mathematical model, numerically. Now, we do not success to get a numerical solution of a complete problem and have an algorithm for solving a kinetic equation, which is most difficult to compute in our process.[magnetization in ferromagnetic] : By using a generalized Duhem model we propose a mathematical model for dynamics of ferromagnetic and prove the well-posedness.