调和分析方法越来越多地被应用在非线性发展方程的研究中，受到了广泛的关注。申请人及其合作者曾利用Littlewood-Paley分解的相关理论等处理了趋化方程、液晶方程和广义磁流体方程等，获得了这几种方程解的基本性质。本项目拟应用Littlewood-Paley理论、多线性估计、权函数理论、交换子、乘子理论、半群理论以及函数空间的实变理论等调和分析方法进一步对（时间分数阶）Navier-Stokes流体方程和趋化流体方程解的性态进行研究，即建立相关函数空间上的解的局部和(几乎)整体适定性、不适定性、大时间衰减、Gevrey正则性和渐近极限。

Harmonic analysis methods have known a growing interest in the study of nonlinear evolution equations，which have been widely concerned.The applicant and his collaborators have used the theory of Littlewood-Paley decomposition to study several types of nonlinear evolution equations: chemotaxis system，liquid crystal flows，generalized Magneto-hydrodynamics equations etc，and we have obtained their basic properties.This project aims to apply Littlewood-Paley theory,multilinear estimates, commutator estimates, analytic semigroup theory, Muckenhoupt weight functions theory, multipliers theory and the real-variable theory of function spaces in harmonic analysis to the study the properties of the（time fractional）Navier-Stokes equations and chemotaxis-fluid equations. More precisely, we are focused on the local and (almost) global well-posedness, ill-posedness, large time behavior，Gevrey regularity, and asymptotic limit of solutions to the（time fractional）Navier-Stokes equations and Chemotaxis-fluid equations in some related function spaces.