热粘弹性中一种耦合体-面过程的模型分析
Analysis of a model coupling volume and surface processes in thermoviscoelasticity
摘要：
我们专注于一个高度非线性演化的抽象PDE系统描述的体积过程与表面过程耦合的热粘弹性，具有准静态的动量平衡，表面上的内部变量的单向演化方程，以及在体域和表面上的温度方程。我们的系统的一个重要的例子发生在建模的单向演化的身体和刚性支撑之间的粘附力，受到热波动和接触摩擦。我们调查相关的初边值问题，特别是全球的时间解决方案的存在性问题，在一个抽象的水平。这使我们能够突出问题的分析特征，同时，利用各种方程之间的紧密耦合，以推导出适当的估计（近似）的问题。我们的存在性结果证明了通过一个精心定制的近似问题的限制，并通过延长所获得的本地时间的解决方案，通过一个精致的延长参数。
英文摘要：
We focus on a highly nonlinear evolutionary abstract PDE system describing volume processes coupled with surfaces processes in thermoviscoelasticity, featuring the quasi-static momentum balance, the equation for the unidirectional evolution of an internal variable on the surface, and the equations for the temperature in the bulk domain and the temperature on the surface. A significant example of our system occurs in the modeling for the unidirectional evolution of the adhesion between a body and a rigid support, subject to thermal fluctuations and in contact with friction. We investigate the related initial-boundary value problem, and in particular the issue of existence of global-in-time solutions, on an abstract level. This allows us to highlight the analytical features of the problem and, at the same time, to exploit the tight coupling between the various equations in order to deduce suitable estimates on (an approximation) of the problem. Our existence result is proved by passing to the limit in a carefully tailored approximate problem, and by extending the obtained local-in-time solution by means of a refined prolongation argument.
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