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

TL;DR

This paper refines the understanding of the long-term behavior of solutions to the 3D axially symmetric Boussinesq system with zero thermal diffusivity, providing sharper bounds on solution growth over time.

Contribution

It offers improved asymptotic bounds for the solution's growth, including uniform bounds under certain conditions, advancing previous results on the system's global well-posedness.

Findings

Velocity's Sobolev norm grows at most algebraically

Temperature fluctuation's $H^1$ norm grows sub-exponentially

Solution's higher order temporal growth is slower than double exponential

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 $W^{1,p}$-Sobolev ($2\leq p<\infty$) norm of the velocity can only grow at most algebraically as $t\to+\infty$. 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 $t\to+\infty$. Meanwhile, for any $m\geq 1$, 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.