几类非线性发展方程大扰动解的整体适定性

In this project, we use Green’s function method, along with tools like energy method and harmonic analysis, to study the global well-posedness of solutions to some nonlinear evolution equations with large initial data. Specifically, we study the following three problems: 1) pointwise estimates of solutions to the large perturbation solutions to the conservation laws with nonlocal dissipation type terms; 2) pointwise estimates of large perturbation solutions to the two dimensional Navier-Stokes and Stokes approximation equations; 3) global well-posedness of large solutions to the “effective artificial viscosity” system canonically associated with the compressible Navier-Stokes equation. Using Green’s function method to study the pointwise estimate of solutions in the case of large perturbation is quite new. The purpose of this project is to provide a way to study the global well-posedness of the large perturbation solutions to some equations based on the understanding of the pointwise estimates of Green's function and nonlinear terms. The results of this project will contribute to a better understanding of the long-time behavior of solutions to the Stokes approximation equations, Navier-Stokes equation and even more general nonlinear evolution equations with dissipation, and to developing related theories, thus they are of important academic value.

本项目利用格林函数方法，结合传统的能量方法，同时融入微局部分析和调和分析等思想来研究非线性发展方程(组)大扰动解的整体适定性。具体来说，包含以下三个方面：1)带非局部耗散项的单个守恒律方程大扰动解的逐点估计，2)二维的可压缩流的Navier-Stokes方程及其简化方程常状态附近大扰动解的逐点估计，3)n维(n≥3)带人为粘性项的Navier-Stokes方程及其相关简化方程大扰动解的整体适定性。利用格林函数方法来研究大扰动问题的整体适定性是一个新的探索。本项目希望通过对方程(组)的线性部分的格林函数和非线性波互相作用的逐点形态的理解，为探讨研究一些带人为粘性项的方程组的大扰动解的整体存在性提供办法和途径。同时，也希望项目的研究成果对进一步了解Stokes近似方程、Navier-Stokes方程以及更一般的非线性发展方程的相关理论起到一定的推动作用。

本项目利用格林函数方法，结合传统的能量方法，同时融入微局部分析和调和分析等思想来研究非线性发展方程(组)大扰动解的整体适定性。具体来说，主要有以下三个方面:1)带非局部耗散项的单个守恒律方程大扰动解的逐点估计，2)二维的可压缩流的Navier-Stokes方程及其简化方程常状态附近大扰动解的逐点估计，3)3维Stokes近似方程常状态附近大扰动解的整体适定性。本项目希望通过对方程(组)的线性部分的格林函数和非线性波互相作用的逐点形态的理解，为探讨研究一些带人为粘性项的方程组的大扰动解的整体存在性提供办法和途径。目前已经得到了单个守恒律方程大扰动解的整体存在性和逐点估计，2维Stokes近似方程大扰动解的衰减估计和3维Stokes近似方程大扰动解的整体适定性。