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.
翻译:在本说明中,我们在两个层面和三个层面都考虑到一个有限域中不可压缩的粘液,并研究确定自由度(预测、功能和节点)的问题。我们特别感兴趣的是非恒定的粘度,代表着一种具有粘度的流体,这种流体随着时间的变化而变化(如石油在降解时失去粘度),或一种流体,其空间的粘度不同(如两阶段或多阶段流体模型)。我们的目标是适用第二作者在以前关于纳维尔-斯托克斯方程式的薄弱解决方案的工作中所制定的确定预测框架,以便确定确定确定本案确定功能的界限,或相当于根据基本确定预测的近似性特征而变化的确定数据集的尺寸。时间变化的粘度反映了早期工作为恒定的粘度而确定的薄弱的解决方案。在单一浮度-斯托克斯方程式等方程式中处理的空间叠合的粘度案例,在单一浮度设想中处理,我们对这一初步的探索案例提出了挑战。