高维辐射流体动力学方程组解的适定性研究

In this project, we study the well-posedness of solutions for Radiation Hydrodynamics (RHD) system in multi-dimensional space. This system can be applied in some phenomena in astrophysics, such as nonlinear stellar pulsation, supernova explosions and stellar winds. The study of the mathematical theory about the system is one of important problems concerned by many researchers. Compared with the one-dimensional space and the 3-order approximates model (Hamer model), RHD system in multi-dimensional space is more difficult to deal with. Firstly, the standard method of energy estimate can not be used to obtain the existence of classical global solutions, even for the small initial data case. Secondly, we can't transform the system into a scale equation as the Hamer model.The characteristic of the equations is that the dissipative structure of the equations has some different characteristics between low-frequency and high-frequency domains, especially, this dissipative sturcture is very weak in the high-frequency domain, thus introducing the difficulty and challenging to studying this system. In this project, we will research on the following issues: (1). By using the Fourier transform to hyperbolic-elliptic coupled sysetm, we study the existence and optimal decay estimate of classical solutions with the energy method; (2). By combining the Green function with the method of energy estimate, we deal with the existence and large time behavior of solutions to RHD system in multi-dimensional space; (3). The large time behavior of solutions to RHD system when the perturbation of the initial data towards the planar waves is small.

本项目主要研究高维辐射流体动力学方程组解的适定性。该方程组在天体物理中有着广泛的应用，比如模拟非线性宇宙脉冲、超新星爆炸和宇宙风等等，其数学理论研究一直是国际数学界长期关注的问题之一。与一维的情形以及其三阶近似模型比较，高维方程组处理起来有较大困难。首先，利用标准的能量估计方法很难得到经典解的整体存在性，即使是小初值情形。其次，该方程组不能像其三阶近似模型一样转化成单个方程的形式，而且方程组本身空间耗散性在低频和高频部分呈现出完全不同的特性，特别是在高频部分其耗散结构很弱。这都给研究带来了新的困难和挑战。本项目拟围绕以下几个方面开展工作：(1). 利用Fourier变换结合能量估计的方法研究一般的双曲-椭圆耦合系统经典解的整体存在性和最佳衰减估计；(2).利用格林函数结合能量估计的方法处理高维辐射流体动力学方程组解的整体存在性；(3).当初值在平面波附近小扰动时方程组解的大时间行为。

本研究项目的成果主要包含两方面的内容: 1. 辐射流体动力学模型解的适定性问题; 2. 流体动力学方程组及相关耦合模型解的适定性问题. 项目对于Cauchy问题解的爆破, 经典解的整体存在性和大时间行为的研究取得了一系列成果.

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