大扰动下不可压缩Navier-Stokes方程的稳定性态

项目名称： 大扰动下不可压缩Navier-Stokes方程的稳定性态

项目编号： No.11526032

项目类型： 专项基金项目

立项/批准年度： 2016

项目学科： 数理科学和化学

项目作者： 贾艳

作者单位： 安徽大学

项目金额： 3万元

中文摘要： 粘性不可压缩流体运动的渐近性态一直是流体动力学数学理论中最重要的前沿课题之一。本项目主要研究大扰动下三维不可压缩 Navier-Stokes 方程有限能量弱解的稳定性态。我们首先研究非衰减外力 Navier-Stokes 方程在大初值扰动下有限能量弱解在临界Besov空间中的最优收敛率，其次我们研究在大的初值和大的外力扰动下Navier-Stokes 方程有限能量弱解在临界Besov 空间中的渐近稳定性。本项目所用的方法主要依赖于灵活运用Stokes算子的谱分解方法，Fourier 局部化方法和偏微分方程的能量方法。经典不可压缩 Navier-Stokes方程作为描述粘性不可压缩流体运动的基本模型，本项目的研究对人们更深入地认识和理解粘性不可压缩流体的渐近演化规律具有重要意义。

中文关键词： ：Navier-Stokes方程；有限能量弱解；最优收敛率；渐近稳定性；

英文摘要： The asymptotic behaviors of viscous incompressible fluid motion is one of the most important issues in mathematical theory of the viscous incompressible flows. This project is devoted to the investigation of stability behaviors of finite energy weak solutions of the three-dimensional incompressible Navier-Stokes equations. For a weak solution of the Navier-Stokes equations in a critical Besov space, we will first investigate that the optimal asymptotic convergence rates of the perturbed Navier-Stokes equations with respect to a large initial data perturbation.Moreover,if both the initial data and external forcing perturbation is not small,then we will proved that the perturbed weak solution of Navier-Stokes equations converges asymptotically to the original weak solution in a critical Besov space as the time tends to the infinity.The findings are mainly based on the spectral decomposition methods of Stokes operator, Fourier location methods together with the energy methods.The investigation of this project is beneficial and important for understanding the evolution of the viscous incompressible fluid motion as the incompressible Navier-Stokes equations are basic model for describing the viscous incompressible fluid motion.

英文关键词： Navier-Stokes equations；finite energy weak solution；optimalconvergence rate；asymptotic stability；