  The global regularity problem for the Boussinesq system is a well known open
problem in mathematical fluid dynamics. As a follow up to our work \cite{EJSI},
we give examples of finite-energy and Lipschitz continuous velocity field and
density $(u_0,\rho_0)$ which are $C^\infty$-smooth away from the origin and
belong to a natural local well-posedness class for the Boussinesq equation
whose corresponding local solution becomes singular in finite time. That is,
while the sup norm of the gradient of the velocity field and the density remain
finite on the time interval $t\in [0,1)$, both quantities become infinite as
$t\rightarrow 1$. The key is to use scale-invariant solutions similar to those
introduced in \cite{EJSI}. The proof consists of three parts: local
well-posedness for the Boussinesq equation in critical spaces, the analysis of
certain special infinite-energy solutions belonging to those critical spaces,
and finally a cut-off argument to ensure finiteness of energy. All of this is
done on spatial domains $\{(x_1,x_2): x_1 \ge \gamma|x_2|\}$ for any $\gamma >
0$ so that we can get arbitrarily close to the half-space case. We remark that
the $2D$ Euler equation is globally well-posed in all of the situations we look
at, so that the singularity is not coming from the domain or the lack of
smoothness on the data but from the vorticity amplification due to the presence
of a density gradient. It is conceivable that our methods can be adapted to
produce finite-energy $C^\infty$ solutions on $\mathbb{R}^2_+$ which become
singular in finite time.
