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

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 W 1 , 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 H 1 norm of the temperature fluctuation grows sub-exponentially as t → + ∞. Meanwhile, for any m ≥ 1 , we deduce that the H m -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. [ABSTRACT FROM AUTHOR]