超临界准地转方程的大时间性态

This project is devoted to studying the large time behavior for the H^{-1/2} weak solution of the supercritical quasi-geostrophic equation. We will prove the optimal upper and lower time decay rates for the H^{-1/2} weak solutions of the zero-forced supercritical quasi-geostrophic equation by developing the spectral decomposition methods of fractional Laplace operator. Applying Littlewood-Paley decompositon and iterative methods, we will examine the asymptotic stability for the H^{-1/2} weak solutions of the supercritical quasi-geostrophic equation in critical Besov spaces under large initial data and external forcing perturbation. By investigating the continuity of the time-weighted trilinear form and employing the fractional power of Laplace operator, we will further study the global solution in critical Besov space of the supercritical quasi-geostrophic equation converges to the H^{-1/2} weak solution with the optimal algebraic rate. The methods used are mainly from various combinations of partial differential equations techniques, function analysis and harmonic analysis.This project is of great significance to understand the large scale atmospheric and oceanic dynamics.

本项目致力于研究超临界准地转方程H^{-1/2}弱解的大时间性态。我们将利用分数阶Laplace算子的谱方法来刻画零外力下超临界准地转方程H^{-1/2}弱解最优上下界衰减估计；利用Littlewood-Paley分解方法和通过选择特殊的试验函数来研究在大的初值和外力扰动下超临界准地转方程H^{-1/2}弱解在临界Besov空间上的渐近稳定性；通过建立时间加权的三线型的连续性和利用Laplace算子分数幂来研究超临界准地转方程在临界Besov空间上的整体解以最优的代数速率收敛到H^{-1/2}弱解。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。本项目的研究对人们更深入地去认识和理解大尺度大气和海洋环流的渐近演化规律具有重要意义。

本项目主要研究了超临界准地转方程及其相关部分耗散流体动力学方程的大时间性态。具体来说，利用方程特殊的结构和广义Fourier分解方法，我们刻画了二维超临界准地转方程H^{-1/2}弱解的最优代数衰减；利用Littlewood-Paley方法和选取特殊的试验函数证明了三维Navier-Stokes方程解在临界Besov空间的渐近稳定性；证明了二维部分耗散MHD方程的整体光滑解和最优代数衰减性；刻画了若干复杂流体如Oldroyd-B流体模型，聚合物非牛顿流体和（磁）微极流体解的大时间性态。这些研究结果对人们进一步理解流体运动的演化规律具有重要意义。