  We study the temperature front problem for the 3D viscous Boussinesq
equation. We prove that the $C^{k,\gamma}$ ($k\geq 1$, $0<\gamma< 1$) and
$W^{2,\infty}$ regularity of a temperature front is locally preserved along the
evolution as well as globally preserved under a smallness condition in a
critical space. In particular, beside giving another proof of the main result
in \cite{GGJ20}, we also extend it to a more general class of regular patch.
