Nonlinear Stability of Taylor-Couette Flows with Heat Buoyancy

This paper investigates the nonlinear stability of Taylor-Couette (TC) flows incorporating the thermal buoyancy within an annular domain characterized by small viscosity $\nu$ and thermal diffusivity $\mu$. It is well established that the buoyancy induced convection significantly impacts practical industrial applications of Taylor-Couette flow \cite{Chen2006}. In contrast to \cite{An.2024}, we specifically examines the influence of the temperature gradients and the gravity on the stability of Taylor-Couette flows in this article. The thermal buoyancy term introduces a destabilizing radial derivative $\partial_r$ into the rotating TC system. To mitigate this destabilizing effect, we employ estimates involving the negative derivatives. Consequently, the additional viscous damping becomes necessary to counterbalance the buoyancy induced instability. Our stability criterion requires that the initial perturbations from the Taylor-Couette flow are bounded by a suitable power of the viscosity. Under this condition, we prove that solutions to the 2D Boussinesq system on $[1, R] \times \mathbb{S}^1$ remain close to the Taylor-Couette flow at the same order.