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Abstract

In this paper, we investigate the nonlinear stability of the Couette flow for the two-dimensional compressible Navier--Stokes equations at high Reynolds numbers ($Re$) regime. It was proved that if the initial data $(\rho_{in},u_{in})$ satisfies $\|(\rho_{in},u_{in})-(1, y, 0)\|_{H^4(\mathbb{T}\times\mathbb{R})}\leq \epsilon Re^{-1}$ for some small $\epsilon$ independent of $Re$, then the corresponding solution exists globally and remains close to the Couette flow for all time. Formal asymptotics indicate that this stability threshold is sharp within the class of Sobolev perturbations. The proof relies on the Fourier-multiplier method and exploits three essential ingredients: (i) the introduction of ``good unknowns" that decouple the perturbation system; (ii) the construction of a carefully designed Fourier multiplier that simultaneously captures the enhanced dissipation and inviscid-damping effects while taming the lift-up mechanism; and (iii) the design of distinct energy functionals for the incompressible and compressible modes.