Navier-Stokes方程和准地转方程中若干问题的研究

Understanding of the regularity and uniqueness of week solutions of the three-dimensional Navier-Stokes equations is fascinating and challenging. This problem introduced by Leray in 1934 has now become one of the most important unsolved mysteries in 21 century. Similar difficulty also appears in the examination of regularity of the supercritical quasi-geostrophic equation. Thus it is beneficial to examine the Navier-Stokes equations together with the supercritical quasi-geostrophic equation. This project is devoted to the study on the existence, smoothness, weak-strong uniqueness and asymptotic stability of the Navier-Stokes equations and the supercritical quasi-geostrophic equation. More precisely, we will prove the weak-strong uniqueness between Leray weak solutions and Koch-Tataru solution of the Navier-Stokes equations and establish their asymptotic stability under the large initial and external forcing perturbation. Moreover, we will show the existence and smoothness for the solution of the supercritical quasi-geostrophic equation in the largest critical Besov space. The weak-strong uniqueness between weak solutions and regular solutions in critical Besov spaces will be obtained. The methods used are mainly from various combinations of partial differential equations techniques, function analysis and harmonic analysis.

1934年Leray提出的三维Navier-Stokes 方程弱解的正则性和唯一性问题正越来越引人注目，现已成为本世纪数学领域中最重要的开问题之一。在研究这一问题中人们发现超临界准地转方程具有同三维Navier-Stokes方程一样本质的困难，但前者正则解的研究会给后者带来许多深刻的启示。作为理解这一问题的重要组成部分，本项目致力于研究Navier-Stokes方程和超临界准地转方程在最大临界空间上的存在性，光滑性,弱强唯一性和渐近稳定性。具体来说,对Navier-Stokes方程我们将证明Leray弱解和最大正则解Koch-Tataru解的弱强唯一性并在大的初始和外力扰动下建立其渐近稳定性。对超临界准地转方程，我们将在最大临界Besov空间上证明解的存在性和光滑性并讨论弱解与正则解之间的弱强唯一性。所用的方法主要源自于偏微分方程，泛函分析和调和分析等一些现代分析方法的有机结合和灵活运用。

流体运动的数学分析一直是非线性偏微分方程中非常重要和前沿的课题，本项目主要研究三维经典不可压缩Navier-Stokes方程和超临界准地准方程及其相关流体动力学方程的弱强唯一性和渐近稳定性。我们利用Fourier局部化方法和热核估计成功得到了三维经典不可压缩Navier-Stokes方程Leray-Hopf弱解和最大正则解Koch-tataur解的弱强唯一性，通过三线型在Besov空间的连续性刻画和选取特殊的试验函数证明了方程在最大临界齐次空间上弱解的渐近稳定性；利用Morrey空间技术和算子谱分解方法得到超临界准地转方程在临界Morrey空间上的正则准则和最优渐近收敛率，进一步刻画了其解在BMO空间上的渐近稳定性。通过构造新的变量和谱分析方法得到了二维部分耗散micropolar流体方程光滑解的整体适定性和最优代数衰减性，这一系列工作进一步推进了前人在这一领域的研究成果，丰富和发展了流体动力学的数学理论。