  In this paper, we study the viscous Boussinesq equation in the whole space
$\mathbb{R}^n$, which describes the propagation of small amplitude and long
waves on the surface of water with viscous effects. Concerning the linearized
Cauchy problem, some qualitative properties of solutions including $L^m-L^q$
estimates with $1\leqslant m\leqslant q\leqslant \infty$, inviscid limits and
asymptotic profiles of solution with respect to the small viscosity are
investigated by means of the Fourier analysis and the WKB method. For another,
by applying some fractional order interpolations in the harmonic analysis, we
derive the $L^q$ well-posedness and estimates for small data solutions to the
nonlinear viscous Boussinesq equation under some conditions for the parameter
of nonlinearity.
