可压缩与不可压缩Oldroyd-B模型的一些问题研究

本项目研究一类典型的粘弹性流体——Oldroyd-B模型的一些数学问题。该模型经常用来描述牛顿溶剂(Newtonian solvent)中的高分子聚合物(polymer)或以很快的松弛模式融化的高分子聚合物。无论从物理还是数学上讲，当流体速度和对称张量之间的耦合参数不小时，问题都更有意义。此时系统是强耦合的，在数学上会遇到更多困难。本项目就是在耦合参数不小的情形下，借助Littlewood-Paley分解等理论，研究可压缩与不可压缩Oldroyd-B模型在不同的临界空间解的存在性唯一性问题。除此之外，我们还将研究ill prepared初值条件下，可压缩Oldroyd-B模型与相应不可压缩模型之间的关系，即零马赫数极限问题。

This research project focuses on the study of some mathematical theories of one type of typical viscoelastic fluid: the Oldroyd-B model. This model may be used to describe solutions of polymers in a Newtonian solvent, or polymer melts with a fast relaxation mode. In both physical and mathematical point of view, the problems are more interesting when the coupling between the velocity of the fluids and the symmetric tensor of constrains is not small. In this case, the system is strongly coupled, and hence more difficult to study. In this project, under the condition that the coupling constant is not small, we consider the solutions of compressible and incompressible Oldroyd-B model in various scaling invariant critical spaces. Besides, for ill prepared initial data, we also investigate the relations between the compressible model and the corresponding incompressible one, i.e. the zero Mach number limit problem.