Analysis of a non-linear phenomenon mainly on the issue of shape-memory alloy

[2004]在形状记忆合金动力学中，应变和应力之间的关系不能用通常的函数来描述，而是用滞后算子来描述。在本研究项目中，我们使用停运算符来描述这种关系，停运算符是滞后算子的一种。在前人的工作中，为了建立我们的数学模型的适定性，我们需要对粘度进行一些大的估计。在这一年里，我们可以去掉对粘度的假设。[2005]作为对我们的形状记忆合金数学模型的回顾过程，我们注意到没有关于弹性材料形状变化的数学结果。因此，我们提出了一种弹性材料的数学模型，它由一个自由边界问题给出。在这里，我们及时证明了自由边界问题解的局部存在性。在POF中使用了标准的近似方法。[2006]今年的结果是得到了我们的自由边界问题的唯一性结果。在整个研究项目中，我们不能证明解的整体存在性。我们继续研究弹性材料的动力学。

[2004] In dynamics of shape memory alloys the relationship between the strain and the stress can not be described by a usual function and is described by the hysteresis operator. In this research project we describe the relationship by the stop operator, which is one type of hysteresis operators. Throughout the previous works we needed some largeness for the viscosity in order to establish the well-posedness of our mathematical model. In this year we can remove the assumption for the viscosity.[2005] As a reviewing process for our mathematical model for shape memory alloys, we have noticed that there were no mathematical results concerned with change of shape of elastic materials. Hence, we have proposed a mathematical for elastic materials, which is given by a free boundary problem. Here, we proved the local existence of a solution in time for the free boundary problem. In the poof the standard approximation method was used.[2006] The result of this year is to obtain the uniqueness result for our free boundary problem. In the whole research project we can not prove the global existence of a solution. We continue to study the dynamics for the elastic material.