Nonlinear stability threshold for 3D compressible Couette flow

Rui Li; Fei Wang; Lingda Xu; Zeren Zhang

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arXiv · 2605.07538

Nonlinear stability threshold for 3D compressible Couette flow

Rui Li·Fei Wang·Lingda Xu·Zeren Zhang

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Abstract

We establish the nonlinear stability threshold $O(\nu^{3/2})$ for the three-dimensional Couette flow governed by the compressible Navier--Stokes equations. While stability thresholds are well understood in two dimensions for both compressible and incompressible flows, and in three dimensions for incompressible flows, the three-dimensional compressible case remains open due to additional structural features, strong mode interactions, and wave coupling. The proof is based on a refined frequency-space approach. For zero modes, we improve upon two-dimensional methods by clearly separating and precisely estimating the main contributions from diffusion waves, acoustic waves, and the lift-up mechanism, leading to a systematic way to handle their nonlinear coupling. For the non-zero modes, we introduce new multiplier estimates and a decomposition based on the structure of the compressible system, which allows us to track the interaction between dissipation and acoustic effects.

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Rui Li, Fei Wang, Lingda Xu, Zeren Zhang. 2026-05-08. Nonlinear stability threshold for 3D compressible Couette flow. https://arxiv.org/abs/2605.07538

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