关于确定非均匀不可压缩流体投影的注记
A Note On Determining Projections for Non-Homogeneous Incompressible Fluids
摘要：
在这篇文章中，我们考虑了二维和三维有限域中的粘性不可压缩流体，并研究了确定自由度（投影，泛函和节点）的问题。我们特别感兴趣的是非恒定粘度的情况，表示粘度随时间变化的流体（例如随着降解而失去粘度的油），或粘度在空间上变化的流体（如两相或多相流体模型的情况）。我们的目标是应用第二作者在以前的工作中开发的弱解的Navier-斯托克斯方程的确定投影框架，以建立在这种情况下，或等价地，一个确定集的尺寸上的确定泛函的数量的界限，基于一个基本的确定投影的近似性质。随时间变化的粘度的情况下，镜像的弱解在较早的工作中建立的恒定粘度的结果。在单流体Navier-Stokes模型中处理的空间变化粘度的情况下，分析起来相当具有挑战性，但我们探索了一些理解这种情况的初步想法。
英文摘要：
In this note, we consider a viscous incompressible fluid in a finite domain in both two and three dimensions, and examine the question of determining degrees of freedom (projections, functionals, and nodes). Our particular interest is the case of non-constant viscosity, representing either a fluid with viscosity that changes over time (such as an oil that loses viscosity as it degrades), or a fluid with viscosity varying spatially (as in the case of two-phase or multi-phase fluid models). Our goal is to apply the determining projection framework developed by the second author in previous work for weak solutions to the Navier-Stokes equations, in order to establish bounds on the number of determining functionals for this case, or equivalently, the dimension of a determining set, based on the approximation properties of an underlying determining projection. The results for the case of time-varying viscosity mirror those for weak solutions established in earlier work for constant viscosity. The case of space-varying viscosity, treated within a single-fluid Navier-Stokes model, is quite challenging to analyze, but we explore some preliminary ideas for understanding this case.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
