非线性热粘弹性中的正温度与线性化模型的推导
Positive temperature in nonlinear thermoviscoelasticity and the derivation of linearized models
AI总结：
本文研究非线性热粘弹性系统在正温度下的适定性，证明温度具有指数时间下界，并在变形接近恒等、温度接近正临界温度时建立向线性化热粘弹性系统的弱解收敛性，推广了已有线性化结果。
摘要：
根据能斯特定理，或等价地，热力学第三定律，绝对零度不可达到。从初始正温度出发，我们证明在有限应变情形下的准静态非线性热粘弹性的 Kelvin-Voigt 模型 [Mielke-Roubíček '20] 存在解，且温度满足指数时间下界。随后，我们关注变形接近恒等、温度接近某一临界正温度的情形，并证明该非线性系统的弱解在适当意义下收敛到线性化热粘弹性系统的解。我们的结果推广了 [Badal-Friedrich-Kružík '23] 中最近的线性化结果，因为它允许临界温度为正值。
英文摘要：
According to the Nernst theorem or, equivalently, the third law of thermodynamics, the absolute zero temperature is not attainable. Starting with an initial positive temperature, we show that there exist solutions to a Kelvin-Voigt model for quasi-static nonlinear thermoviscoelasticity at a finite-strain setting [Mielke-Roubíček '20], obeying an exponential-in-time lower bound on the temperature. Afterwards, we focus on the case of deformations near the identity and temperatures near a critical positive temperature, and we show that weak solutions of the nonlinear system converge in a suitable sense to solutions of a system in linearized thermoviscoelasticity. Our result extends the recent linearization result in [Badal-Friedrich-Kružík '23], as it allows the critical temperature to be positive.
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