AI 中文总结
本文研究一维无界实数域上可压缩非牛顿流体幂律模型，通过区域截断与紧性论证证明p→∞时解的收敛性，严格论证极限方程弱解存在性，将已有结果从周期域推广到全实轴。
AI 中文摘要
本文研究了定义在实数域$\n\b\b{R}$上的一维可压缩流体动力学幂律模型的分析问题，该模型的切应力形式为$\nμ|\partial_{x}u|^{p-2}\partial_{x}u$，其中$μ$为黏度系数，$u$为速度。我们证明了在奇异极限$p\rightarrow\infty$下，解收敛于满足以下条件的函数对$(ρ,u)$：在$\n\mathbb{R}$上几乎处处成立$|\partial_{x}u|\leq 1$、$τ= π\partial_{x}u$、$π\geq 0$以及$π(1 - |\partial_{x}u|) = 0$。此外，我们严格证明了极限方程弱解的存在性。$p \to \infty$时的收敛性通过区域截断与紧性论证得到，其中的关键挑战是证明密度在任意紧子集上始终保持有界且远离零与无穷大。本文将Bresch、Burtea与Szlenk近期发表于《Nonlinearity》2026年第26卷第5期（论文编号055010）的研究结果从一维周期区域推广到了整个实数轴。
英文摘要
This paper is concerned with the analysis of a one-dimensional power-law model for compressible fluid dynamics on $\mathbb{R}$, in which the shear stress takes the form $μ|\partial_{x}u|^{p-2}\partial_{x}u$, where $μ$ is the viscosity coefficient and $u$ is the velocity. We prove that, in the singular limit $p\rightarrow\infty$, the solutions converge to functions $(ρ,u)$ satisfying $|\partial_{x}u|\leq 1$, $τ= π\partial_{x}u$, $π\geq 0$, and $π(1 - |\partial_{x}u|) = 0$ a.e. on $\mathbb{R}$. Moreover, we rigorously justify the existence of weak solutions to the limiting equation. The convergence as $p \to \infty$ is obtained via domain truncation and compactness arguments, of which the key challenge is to show that the density remains bounded away from zero and infinity on any compact subset. This extends the recent result of Bresch, Burtea, and Szlenk [Nonlinearity 26 (2026), no. 5, Paper No. 055010.] from one-dimensional periodic domain to the whole real line.
Comments33 pages
