纳米科学的发展为粘弹性流体的建模提出了新的挑战，物质的压缩性和热传导效应在这些材料粘弹性行为的描述中变得越来越重要。因此，推广经典的不可压缩粘弹性流体力学理论以反映这些性质成为当前的一个研究热点。另一方面，近年来非平衡态热力学的快速发展为这一问题的解决提供了重要的手段。然而，已有的非平衡态热力学理论似乎尚未成熟。如何完善已有的非平衡态热力学理论并将其应用于粘弹性流体的数学建模，是本文的主要研究目标。本文通过推广Yong、Zhu、Hong、Yang近年来发展的守恒－耗散理论，提出了几类非等温可压粘弹性流体模型：1、 推广了热传导的Guyer-Krumhansl理论；2、发展了含对流导数的守恒－耗散理论，由此提出了非等温可压上对流Maxwell模型；3、结合有限形变理论和守恒－耗散理论提出了有限形变守恒—耗散理论，由此推广了Lin的模型。通过这些工作，本文为非等温可压粘弹性流体的建模提供了新的手段。在数学分析方面，利用Yong的双曲平衡率方程组小解整体存在性理论和双曲方程松弛极限理论，证明了等温可压Maxwell模型和一维等温可压上对流Maxwell模型平衡态附近解的整体存在性，以及松弛参数趋于$0$时同经典Navier-Stokes方程的兼容性。最后验证了Lin等人的有限形变粘弹性模型不满足双曲－抛物方程组的Kawashima条件，并且通过对力学适应性条件的分析，给出了这一模型平衡态附近整体解存在性的一个新的证明。
New challenges occur in the modeling of viscoelastic fluids with the development of nanoscience. The compressibility and thermodynamical behaviors have become more and more important in the description of their viscoelasticity. Therefore, the promotion of classical incompressible viscoelastic theory to include these effects has become a research hotspot. On the other hand, rapid developments of non-equilibrium thermodynamics in recent years have provided important tools to solve this problem. However, the current theory of non-equilibrium thermodynamics is not yet perfect. How to improve the existing non-equilibrium thermodynamics theories and apply them to the mathematical modeling of viscoelastic fluid are the main goals of this thesis. We develop several non-isothermal compressible viscoelastic fluids models through a generalization of the conservation dissipation formalism of irreversible thermodynamics proposed by Yong, Zhu, Hong and Yang. First, a generalized Guyer-Krumhansl theory is developed. Second, a conservation dissipation theory including convective derivatives is developed and a non-isothermal compressible upper convected Maxwell model is proposed. Last, a conservation dissipation theory combing finite strain theory is developed and a generalized Lin‘s model is proposed. Thus, new tools are developed for the modeling of compressible viscoelastic fluids. In the aspect of mathematical analysis of viscoelastic models, the existences of smooth solutions near equilibrium states and the consistencies with Navier-Stokes systems of the isothermal compressible Maxwell model and the one dimensional isothermal compressible upper convected Maxwell model are proved using the global existence theory near equilibrium and singular limit theory of hyperbolic systems developed by Yong. Finally, we show that Lin‘s model fails to satisfy the Kawashima condition. However, it can be compensated by the mechanically compatibility conditions, enabling us to give a new proof of the global existence theorem of Lin‘s model.
