Existence of Weak Solutions to a Power-Law Model for Compressible Non-Newtonian Fluids on 1D Unbounded Domain

Abstract

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.