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

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行为。在所有这些方法中，他们用多项式和函数来近似应变和应力之间的关系，并导出了模型。然而，在一些实验中，我们知道，这种关系不是一个通常的函数，可以表示为滞后算子，这取决于历史数据。因此，在这个研究项目中，我们提出了一个新的数学模型，包括一个滞后算子没有多项式逼近，并研究了该模型，通过使用理论的发展方程的时间依赖的次微分的凸函数的Hubert空间。首先，我们考虑以下问题。我们已经知道，滞后算子的特征在于常微分方程，包括指示函数的次微分算子。然后在由动量平衡定律和内能平衡定律组成的系统中加入常微分方程。该方程组的解可能不满足A光滑条件，因此我们用抛物型方程代替常微分方程来逼近。我们的第一个结果是证明了这类近似问题的存在性和唯一性定理。接下来，我们将抛物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。

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.