SIAM Journal on Mathematical Analysis, Volume 53, Issue 4, Page 3985-4030, January 2021.
We prove the existence of large-data global-in-time weak solutions to an evolutionary PDE system describing flows of incompressible heat-conducting viscoelastic rate-type fluids with stress-diffusion, subject to a stick-slip boundary condition for the velocity and a homogeneous Neumann boundary condition for the extra stress tensor. In the introductory section we develop the thermodynamic foundations of the proposed model, and we document the role of thermodynamics in obtaining critical structural relations between the quantities of interest. These structural relations are then exploited in the mathematical analysis of the governing equations. In particular, the definition of weak solution is motivated by the thermodynamic basis of the model. The extra stress tensor describing the elastic response of the fluid is in our case purely spherical, which is a simplification from the physical point of view. The model nevertheless exhibits features that require novel mathematical ideas in order to deal with the technically complex structure of the associated internal energy and the more complicated forms of the corresponding entropy and energy fluxes. The paper provides the first rigorous proof of the existence of large-data global-in-time weak solutions to the governing equations for coupled thermo-mechanical processes in viscoelastic rate-type fluids.

中文翻译：

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具有应力扩散和纯球面弹性响应的不可压缩导热粘弹性速率型流体

SIAM 数学分析杂志，第 53 卷，第 4 期，第 3985-4030 页，2021 年 1 月。
我们证明了演化 PDE 系统的大数据全局实时弱解的存在，该系统描述具有应力扩散的不可压缩导热粘弹性速率型流体的流动，受速度和滑移边界条件的影响额外应力张量的齐次 Neumann 边界条件。在介绍部分，我们开发了所提出模型的热力学基础，并记录了热力学在获得感兴趣量之间的关键结构关系方面的作用。然后在控制方程的数学分析中利用这些结构关系。特别是，弱解的定义是由模型的热力学基础驱动的。描述流体弹性响应的额外应力张量在我们的例子中是纯球形的，这是从物理角度的简化。尽管如此，该模型表现出的特征需要新颖的数学思想来处理相关内能的技术复杂结构以及相应熵和能量通量的更复杂形式。该论文首次严格证明了粘弹性速率型流体中耦合热机械过程的控制方程存在大数据全局时间弱解。