粘性守恒律方程组解的渐近行为

We are interested in the asymptotic behaviors of the solutions to the system of the viscous conservation laws, including the vanishing viscosity limit problem and the large-time asymptotic stability. The asymptotic behavior of the solutions to the viscous conservation laws is closely related to the Riemann problem to the corresponding hyperbolic conservation laws and there are three fundamental hyperbolic waves, i. e., shock and rarefaction waves and contact discontinuity, to the Riemann problems to the inviscid hyperbolic conservation laws. When the wave strength is sufficiently small, there are many famous results concerned with the asymptotic behaviors to the solutions to the viscous conservation laws, for example, Oleinik, Goodman and Matsumura's stability results. However, if the wave strength is large, especially for the large shock wave case, there are few results to the asymptotic behaviors of the solutions to the viscous conservation laws. Furthermore, most existing results are concerned with the 1-dimensional spatial case and there are few results with the case when 1D fundamental wave are perturbed in the multi-dimensional conservation laws due to the essentially mathematical difficulties. In the present program, we are mainly concerned with the vanishing viscosity limit and large-time asymptotic stability of large shock wave to the 1D viscous conservation laws. Moreover, we will investigate the propagation of the 1D fundamental wave patterns in the multi-dimensional viscous conservation laws.

本项目主要研究粘性守恒律方程(组)解的渐近行为，包括解在粘性消失下的渐近极限和解的大时间渐近稳定性。粘性守恒律方程组解的渐近行为与对应的无粘双曲守恒律方程组的Riemann问题紧密相关，无粘双曲守恒律方程组的Riemann问题具有三种基本波，即激波、稀疏波和接触间断波。当波的强度充分小时，与三种基本波相关的粘性守恒律方程组解的渐近行为有很多著名的结果，如Oleinik, Goodman, Matsumura等人的结果。但是当波的强度充分大时，特别是大激波的情形，相应的结果很少。此外，对于1维基本波在高维粘性守恒律方程组中的传播，由于有本质性的困难，结果更少。在本项目中，我们拟主要研究1维粘性守恒律方程(组)大强度激波的大时间渐近稳定性和粘性消失极限以及1维基本波(包括激波、稀疏波、接触间断波)在高维粘性守恒律方程组中的传播和稳定性等。

本项目主要研究粘性守恒律方程(组)解的渐近行为，包括解的粘性消失极限和解的大时间渐近稳定性等。在本项目的资助下，我们取得的主要结果有：1. 证明了高维可压缩Navier-Stokes方程组(一类重要的粘性守恒律方程组)平面稀疏波的大时间渐近稳定性和粘性消失极限，被波兰科学院院士P. Biler在德国数学文摘上公开评论为：“这个二维稳定性的结果是具有物理粘性方程组平面稀疏波稳定性的第一个高维结果”；2.证明了高维单个粘性守恒律方程大强度激波的L^2压缩性；3.建立了双极Vlasov-Poisson-Boltzmann方程组新的微观－宏观分解，并利用这个新分解，证明了双极Vlasov-Poisson-Boltzmann方程组基本波的渐近稳定性。