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The stability threshold for Boussinesq equations around stratified Couette flow with unequal viscosity and thermal diffusivity on 𝕋×ℝ2
Abstract
We establish a stability threshold for the 3D Boussinesq equations around Couette flow (y,0,0) with linearly stratified temperature profile 1+y on 𝕋×ℝ2, with distinct viscosity κ and thermal diffusivity μ and arbitrary Brunt-Väisälä frequency β>0. Assume that μ and κ are comparable, and set ν=min{κ,μ}. For sufficiently small a0>0 and s≥4, we prove that if the initial perturbation satisfies |(uin,θin)|Hs+|(u0,in,θ0,in)|Ws,1≲ν2/3 |lnν|−2−2a0, then the corresponding solution exists globally in time. We also establish Strichartz-type estimates for zero mode. Moreover, our estimates further allow for a nonlinear amplification of order ν−1/6 in the high-regularity norm of the zero mode of the first component u10.
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BibTeX
@misc{chen2026stability,
title = {{The stability threshold for Boussinesq equations around stratified Couette flow with unequal viscosity and thermal diffusivity on $\mathbbT\times\mathbbR^2$}},
author = {Pengyu Chen and Zeren Zhang},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.29916},
eprint = {2609.29916},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.29916},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - The stability threshold for Boussinesq equations around stratified Couette flow with unequal viscosity and thermal diffusivity on 𝕋×ℝ^2
AU - Pengyu Chen
AU - Zeren Zhang
PY - 2026
DA - 2026/09/24
PB - arXiv
DO - 10.48550/arxiv.2609.29916
UR - https://arxiv.org/abs/2609.29916
AB - We establish a stability threshold for the 3D Boussinesq equations around Couette flow (y,0,0) with linearly stratified temperature profile 1+y on 𝕋×ℝ^2, with distinct viscosity κ and thermal diffusivity μ and arbitrary Brunt-Väisälä frequency β>0. Assume that μ and κ are comparable, and set ν=min{κ,μ}. For sufficiently small a_0>0 and s≥4, we prove that if the initial perturbation satisfies |(u_in,θ_in)|_H^s+|(u_0,in,θ_0,in)|_W^s,1≲ν^2/3 |lnν|^−2−2a_0, then the corresponding solution exists globally in time. We also establish Strichartz-type estimates for zero mode. Moreover, our estimates further allow for a nonlinear amplification of order ν^−1/6 in the high-regularity norm of the zero mode of the first component u^1_0.
ER -