带拟微分算子粘性项的守恒律方程解的大时间行为研究

In this project, by combining methods and tools such as Green function, energy estimates and micro-local analysis, we will study the large-time behavior of solutions to conservation laws with pseudo-differential operator viscosity. We will focus on conservation laws with fractal viscosity and conservation laws with diffusion-type term and study the decay rates and pointwise estimates of solutions near constant states with large perturbation and near fundamental waves with small perturbation..Nonlinear evolution equations with fractal dissipation are among the hot topics of recent studies in the theory of fluid dynamics, and they have strong background in applications and are mathematically very rich. In this project, by applying tools from harmonic analysis and micro-local analysis, we will develop methods to study the large-time behavior of solutions to such equations. This will have a positive impact on the development of related theories and it is of great academic value.

本项目综合利用格林函数、能量估计、微局部分析等方法与工具，研究带拟微分算子粘性项的守恒律方程解的大时间行为。重点研究带分数阶粘性项的守恒律方程和带扩散型项的守恒律方程在常状态附近大扰动解和基本波附近小扰动解的衰减估计和逐点估计。.带分数阶耗散结构的非线性发展方程是近年来流体力学理论研究中的热点问题之一，具有很强的应用背景和丰富的数学内涵。本项目将深入应用调和分析与微局部分析工具来探索研究这类方程解的大时间行为的方法。这对相关理论的进一步发展有积极的促进作用，具有重要的学术价值。

项目主要研究了复微分方程与具有对称性的有界域上的复分析。在复微分方程方面，首先研究了Briot-Bouquet型方程，得到其全纯解存在的充分条件；同时证明高维的复向量场的投影化与庞加莱紧化是等价的。在复分析方面，首先证明了quasi-balanced域上的一个Alexander型定理，即其上的逆紧全纯映射必为自同构；另外深入研究了quasi-Reinhardt域，给出了双曲性、极小性等一些基本性质的刻画，并得到Cartan线性化定理的一个推广和本质证明。这些研究在方法和结果上均有较强的创新性，比如在Briot-Bouquet型方程的研究中克服特征值为正整数的困难情形，在quasi-balanced域的研究中运用复动力系统方法，在quasi-Reinhardt域的研究中运用Bergman representative coordinates等。