高雷诺数环流的解性质分析

调和分析中建立的许多分析方法诸如极大函数方法，算子插值，奇异积分算子及位势理论等是研究偏微分方程，复分析等领域的重要工具。Littlewood-paley分解，Bony的二次微局部分解技术以及仿积运算等在研究偏微分方程解的适定性等问题都体现了调和分析在偏微分方程研究中的关键作用。 本项目拟运用调和分析方法研究高雷诺数条件下，不同结构类型下Navier-Stokes方程组经典解在一类背景解（环流）附近小扰动时的整体存在性或不稳定性。我们将围绕如下问题进行:(1) 研究两个旋转圆柱面之间的流动，即泰勒-库埃特流，当外环流速度为零时，高雷诺数情形下环流的整体存在性。(2) 研究高雷诺数和非对称扰动的小初值下，环流在外区域问题上的全局存在性。（3）研究非等熵情形，外区域问题上的对称环流在高雷诺数和马赫数下的全局存在性。

Many methods established in harmonic analysis, such as maximum function method, operator interpolation, singular integral operator and potential theory, are important tools in the field of partial differential equation and complex analysis. In the study of the well-posedness of the partial differential equation solution, the Littlewood-paley decomposition, the Bony quadratic micro-local decomposition and the paracomponent operation all show the key role of the harmonic analysis in the partial differential equation research. In this paper, the harmonic analysis method is used to study the global existence or instability of classical solutions of Navier-Stokes equations with high Reynolds number near a kind of background solution (circulatory flows) . We will focus on the following problems: (1) the flow between two rotating cylinders, namely, the Taloy-Couttee flow, when the velocity of the outer circulation is zero, the global existence of the circulatory flows at high Reynolds number is studied. (2) the global existence of the circulatory flows in the outer domain is studied under the condition of High Reynolds number and small initial value of asymmetric perturbation. (3) we will consider the global existence of the symmetric and non-isentropic circulatory flows at high Reynolds number and Mach number.