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

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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.