The stability threshold α=1/3 for 2D Boussinesq-Couette flow holds for unequal viscosity and thermal diffusivity, with H^{s+1/2} regularity for s>3/2.
Suppression of Fluid Echoes and Sobolev Stability Threshold for 2D Dissipative Fluid Equations Around Couette Flow
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abstract
We study the Sobolev stability thresholds of 2d dissipative fluid equations around Couette flow on the domain $\mathbb T\times \mathbb R$. We prove a bound for general nonlinear interactions, which, for several fluid equations, reduces the proof of nonlinear stability to a linear stability analysis. We apply this approach to the examples of Navier-Stokes, Boussinesq and magnetohydrodynamic equations around Couette flow. This improves the Sobolev stability threshold for the Boussinesq equations around Couette flow and large affine temperature to $1/3$ and for the MHD equations around Couette flow and constant magnetic field to $1/3^+$.
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Transition threshold of Couette flow for 2D Boussinesq equations
The stability threshold α=1/3 for 2D Boussinesq-Couette flow holds for unequal viscosity and thermal diffusivity, with H^{s+1/2} regularity for s>3/2.