Large time decay and growth for solutions of a viscous Boussinesq system

TL;DR

This paper investigates the long-term behavior of solutions to the 3D viscous Boussinesq system, revealing conditions under which solutions grow unbounded or dissipate, with detailed asymptotic profiles and estimates.

Contribution

It provides new insights into the decay and growth rates of solutions, including sharp estimates and explicit asymptotic profiles for both weak and strong solutions.

Findings

Generic solutions blow up as t→∞ with unbounded energy and norms.

Strong solutions have sharp upper and lower bounds with explicit asymptotic profiles.

Zero-mean initial temperature solutions dissipate energy and decay to zero over time.

Abstract

In this paper we analyze the decay and the growth for large time of weak and strong solutions to the three-dimensional viscous Boussinesq system. We show that generic solutions blow up as $t\to\infty$ in the sense that the energy and the $L^p$-norms of the velocity field grow to infinity for large time, for $1\le p<3$. In the case of strong solutions we provide sharp estimates both from above and from below and explicit asymptotic profiles. We also show that solutions arising from $(u_0,\theta_0)$ with zero-mean for the initial temperature $\theta_0$ have a special behavior as $|x|$ or $t$ tends to infinity: contrarily to the generic case, their energy dissipates to zero for large time.