AI 中文总结
该研究针对一维完全可压缩Navier-Stokes方程半空间出流问题，通过全局相平面分析将小振幅边界层解拓展到大振幅情形，完整刻画了大振幅解的存在性条件，发现其可非单调且存在区域可为开集。
AI 中文摘要
我们研究了半直线$\n\mathbb{R}_+$上一维完全可压缩Navier-Stokes方程出流问题的大振幅边界层解的存在性与不存在性。通过精细的全局相平面分析，我们将借助中心流形方法得到的小振幅边界层解实质性地拓展到大振幅情形。基于$M_+-1$的符号（其中$M_+$表示右端状态的马赫数）以及与普朗特数相关的参数$\zeta$，我们对这一半空间出流问题的大振幅边界层解的存在性与不存在性给出了完整刻画。与对应的入流问题形成鲜明对比的是，出流问题的边界层解可以是非单调的，且它们在相平面中的存在区域可以是一个开集。
英文摘要
We investigate the existence and non-existence of large-amplitude boundary layer solutions to the outflow problem for the one-dimensional full compressible Navier-Stokes equations on the half-line $\mathbb{R}_+$. Through a delicate global phase-plane analysis, we substantially extend the small-amplitude boundary layer solutions obtained via the center-manifold approach to the large-amplitude regime. Based on the sign of $M_+-1$ (with $M_+$ denoting the Mach number at the right end state) and the Prandtl-number-related parameter $ζ$, we provide a complete characterization of the existence and non-existence of large-amplitude boundary layer solutions for this half-space outflow problem. In sharp contrast to the corresponding inflow problem, boundary layer solutions to the outflow problem are permitted to be non-monotone, and their existence region in the phase plane can be an open set.
Comments27 pages, 11 figures
