一些流体力学方程的数学研究

This project is concerned with some hydrodynamic systems, by means of modern harmonic method, for instance, the incompressible viscoelastic hydrodynamic model, micropolar and magneto-micropolar hydrodynamic equations, which discribe the motion law of many complex fluids which are different from Newtonian fluids. These equations have a strong physical background and application prospects, whose well-posedness and the regularity of weak solutions is one of the hot issues of concerns by mathematicians and physicists. The local in time and global in time well-posedness in some critical Banch spaces for the Cauchy problems of the hydrodynamic systems will be investigated by harmonic ideas and techniques: Littlewood-Paley theory and Bony's paraproduct decomposition; The continous extension of a local smooth solution and regularity of a weak solution will be studied if the system is local well-posed; The large time behavior of a solution and its decay estimate and the relation with self-similar solution will be investigated if the system is globally well-posed. Applying the harmonic method to a non-Newtonian fluid system needs to overcome the strong coupling relationship of the equations and the difficulty that the equations do not satisfy the scaling invariance, whose mathematical theory has great challenge. This project will provide an important research method on the theory of partial differential equations.

本项目拟借助于现代调和分析方法研究一些流体力学方程组，如不可压粘弹性流体力学模型，微极和磁微极流体方程组。这些方程组刻画了一些不同于牛顿流体力学方程组的许多复杂流体的运动规律。这些方程组具有强烈的物理背景和应用前景，其适定性和弱解的正则性问题是当今数学家和物理学家非常关注的热点问题之一。利用调和分析的技术和思想，如Littlewood-Paley理论和Bony仿积分解方法，来研究这些流体力学方程组的Cauchy问题在某些临界Banach空间的局部适定性和整体适定性；在局部适定性的基础上研究局部光滑解的连续延拓和弱解的正则性问题；在整体解存在的基础上研究解的长时间行为和衰减估计以及与自相似解的关系。将调和分析的思想应用到非牛顿流体方程组需要克服方程组的强耦合关系以及不满足伸缩不变性等困难，其数学理论具有极大挑战性。本项目对偏微分方程的理论将提供一个重要研究方法。

本项目利用现代的调和分析方法研究了一些不可压缩流体力学方程组，包括粘弹性流体力学方程组，微极流体方程和磁微极流体力学方程组，磁流体方程组，带Hall项的磁流体方程组，Boussinesq方程组等。研究了三维不可压粘弹性流体力学模型Oldroyd-B方程组小初值光滑解的整体适定性和解的长时间行为和解的时间衰减估计，研究了二维和三维不可压粘弹性流体力学模型Oldroyd-B方程组局部光滑解的爆破准则，研究了微极和磁微极流体方程组光滑解的爆破准则和弱解的正则性准则，研究了三维磁流体力学方程组弱解的L^2渐进稳定性，研究了二维几乎具有电阻抗的磁流体方程组的整体正则性，其中速度场的粘性导数可以任意弱，研究了二维带部分方向分数阶导数的磁流体方程组的整体正则性，研究了研究了二维和三维Boussinesq方程组具有部分方向耗散导数时，整体光滑解的适定性，研究了具有分数阶耗散的霍尔磁流体方程组整体光滑解的适定性。四年发表学术论文31篇，其中SCI收录28篇，CSCD核心1篇，国际期刊2篇。项目组成员按计划进行了国内外的学术交流工作。举办一次国际会议“2018年数学流体力学方程国际研讨会”和开展系列讲座活动。课题组共培养13名研究生授予硕士学位，同时招收13名研究生。