A refined long time asymptotic bound for 3D axially symmetric Boussinesq system with zero thermal diffusivity

Abstract

In this paper, we obtain a refined temporal asymptotic upper bound of the global axially symmetric solution to the Boussinesq system with no thermal diffusivity. We show the spacial W1,p-Sobolev (2≤p<∞) norm of the velocity can only grow at most algebraically as t→+∞. Under a signed potential condition imposed on the initial data, we further derive that the aforementioned norm is uniformly bounded at all times. Higher order estimates are also given: We find the H1 norm of the temperature fluctuation grows sub-exponentially as t→+∞. Meanwhile, for any m≥1, we deduce that the Hm-temporal growth of the solution is slower than a double exponential function. As a result, these improve the results in [11] where the authors only provided rough temporal asymptotic upper bounds while proving the global well-posedness.