不可压磁粘弹性流体的数学研究

In this paper，we will mainly study the uniqueness and longtime behavior of the weak solution of incompressible magnetoviscoelastic fluids in two dimensions, and consider the well-posedness of the model in the critical Besov space for the Lp framework. Incompressible magnetoviscoelastic fluids are new mathematical model derived in recent years ( 2016 ), describing an important class of smart materials. smart materials are constructed materials that have the special properties that can be significantly changed in a controlled fashion by external stimuli, such as,stress, temperature,electric or magnetic fields. At present, the known conclusion is the existence of weak solution in 2-dimensional condition. this model has strong coupling and strong degeneration properties，there are some difficulties in research. we try to overcome these difficulties and give some meaningful mathematical conclusions. The mathematical study of this kind of differential system can undoubtedly reveal that essence of magnetize fluid, which is of great significance to engineering technology, medicine, national defense construction and people's daily life, and has wide scientific research and practical application value.

本文主要研究不可压磁粘弹性流体在二维时弱解的唯一性及长时间行为，并考虑在Lp框架下临界次Besov空间中的适定性问题。不可压磁粘弹性流体是近年来（2016）推导出来的数学新模型，描述一类重要的机敏材料。机敏材料是指可以感知环境刺激因素并以可控方式显著变化的材料，这些刺激包括压力、温度、电场、磁场等。目前已知的结论为二维时的弱解的存在性，此模型具有强耦合性、强退化性，具有一定的研究困难，我们试图克服这些困难，给出一些有意义的数学结论。对这类微分系统数学研究无疑可以揭示磁化流体的本质，对工程技术、医学、国防建设和人们的日常生活等方面具有重要的意义，具有广泛的科研和实际应用价值。

磁粘弹性流体作为可以应用到实际的重要流体，是一种机敏材料。由于其本身的磁敏感特性和弹性流动机理，近些年来越来越受到物理学和工业的重视，被应用到各类传感器领域，例如潜艇的水下探测，导弹制导领域的应用，具有很大的潜力。其数学理论是发展应用科学的前提条件，我们应用基本的偏微分理论，经过理论分析，获得了在二维和三维情况下的部分结论。 在二维情况下，对初值进行约束假设，建立弱解的整体适定性和长时间行为，在三维情况下，建立了适合空间的正则性准则和Besov空间下的整体适定性。我们分析很多与之相关的其他问题并做了一些尝试，总结和分析出磁粘弹性流体的一些特性和问题潜在解决方案。