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 ($2\le p<\mathrm{\infty }$) norm of the velocity can only grow at most algebraically as $t\to +\mathrm{\infty }$. 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 $t\to +\mathrm{\infty }$. Meanwhile, for any $m\ge 1$, 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 \cite{HR:2010AIHP} where the authors only provided rough temporal asymptotic upper bounds while proving the global well-posedness.