Under study is some stationary model describing non-Newtonian fluid flows with the viscosity dependent on the strain rate and the heat transfer in a bounded 3D domain. The model is a strongly nonlinear system of coupled partial differential equations for the velocity field, temperature, and pressure. The system is supplemented with a no-slip condition and a linear Robin-type boundary condition for the temperature on the boundary of the flow domain. We propose an operator formulation of this boundary-value problem. Using the properties of d-monotone operators and the Leray-Schauder Fixed Point Theorem, we prove the existence of weak solutions under natural conditions on the data of the model. The solution set is shown to be bounded and closed.

中文翻译：

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非牛顿流体非等温流动模型的边值问题

正在研究中的是一个描述非牛顿流体流动的平稳模型，其粘度取决于应变速率和有限3D域中的热传递。该模型是耦合了速度场，温度和压力的偏微分方程的强非线性系统。对于流域边界上的温度，该系统补充有防滑条件和线性Robin型边界条件。我们提出了这个边值问题的算子公式。利用d-单调算子的性质和Leray-Schauder不动点定理，我们在模型数据上证明了自然条件下弱解的存在。解决方案集显示为有界且封闭。