Nonlinear stability threshold for 3D compressible Couette flow
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arXiv
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| Main Authors: | , , , |
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| Format: | Preprint |
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2026
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| _version_ | 1866909025926381568 |
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| author | Li, Rui Wang, Fei Xu, Lingda Zhang, Zeren |
| author_facet | Li, Rui Wang, Fei Xu, Lingda Zhang, Zeren |
| contents | We establish the nonlinear stability threshold $O(ν^{3/2})$ 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2605_07538 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Nonlinear stability threshold for 3D compressible Couette flow Li, Rui Wang, Fei Xu, Lingda Zhang, Zeren Analysis of PDEs We establish the nonlinear stability threshold $O(ν^{3/2})$ 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. |
| title | Nonlinear stability threshold for 3D compressible Couette flow |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/2605.07538 |