可压Navier-Stokes方程的初边值问题

In recent years,the asymptotics of global solutions for the initial-boundary value problem of the compressible Navier-Stokes equations have been focused on. For the initial-boundary value problems, to study its large time behavior, in addition to the three type of elementary waves such as viscous shock waves, rarefaction waves, contact discontinuities and their superpositons which are sufficient for the corresponding Cauchy problem, a new type nonlinear wave, the so-called boundary layer wave,should be taken into consideration which is due to the occurrence of the boundary. In this project, we are concerned with the precise description of the large time behaviors of solutions to the inflow and outflow problems for compressible Navior-Stokes equations. For the isentropic case, a complete classification of its asymptotics is given and some of them have been justified mathematically by the professor Matsumura and other workers recently. Based on our previous works, this project will be concentrated on these problems:(1)global stability of the boundary layer wave, rarefaction waves,viscous shock waves and their superpostion for inflow problems;(2)nonlinear stability of viscous shock waves and the superposition for outflow problems; (3)the large time behavior of solutions to the compressible Navier-Stokes equation in multidimensional space.

可压Navier-Stokes方程组整体解的大时间性态的研究是国内外数学工作者所关心的一个焦点问题.对Cauchy问题, 利用稀疏波, 粘性激波，粘性接触间断及其所构成的复合波就足以刻画其解的大时间渐进行为。对初边值问题，由于边界的出现，为准确描述其大时间渐进行为，除上述基本波外，还需引入一类新的非线性波即边界层解。本项目拟研究可压Navier-Stokes方程的内/外流问题整体解的大时间渐进性态。这一问题，在等熵的情形，已有解大时间性态的完整分类. 近年来, 合作者Matsumura教授和其他同行已对其中的部分情形从数学上给出了严格的证实，我们拟在前期工作的基础上围绕上述问题开展研究:(1)内流问题边界层,稀疏波,激波以及它们所构成的复合波的整体稳定性;(2)外流问题粘性激波及激波与其他基本波构成的复合波的非线性稳定性; (3)高维Navier-Stokes方程组基本波的稳定性.

可压Navier-Stokes 方程组整体解的大时间性态的研究是国内外数学工作者所关心的一个焦点问题. 本项目拟研究可压Navier-Stokes 方程的内/外流问题整体解的大时间渐进性态。我们在前期工作的基础上围绕上述问题开展研究:(1)内流问题边界层, 激波以及它们所构成的复合波的一类大初值整体稳定性; 在这一方面,我已经取得一些成熟结果. (2) 激波与其他基本波构成的复合波的非线性稳定性. 这一方面, 我们得到了非等熵无粘Navier-Stokes 方程组的第一类和第三类激波所构成的复合波的小初值整体解的稳定性, 进一步分析了Navier-Stokes 方程组得性质.