  In this work, we revisit the study by M. E. Schonbek [11] concerning the
problem of existence of global entropic weak solutions for the classical
Boussinesq system, as well as the study of the regularity of these solutions by
C. J. Amick [1]. We propose to regularize by a "fractal" operator (i.e. a
differential operator defined by a Fourier multiplier of type $\epsilon |\xi
|^\lambda, \, (\epsilon,\lambda) \in\,\mathbb{R}_+\times ] 0,2]$). We first
show that the regularized system is globally unconditionally well-posed in
Sobolev spaces of type $H^s(\mathbb{R}),\,s > \frac {1}{2},$, uniformly in the
regularizing parameters $(\epsilon,\lambda) \in\,\mathbb{R}_+\times ]0,2]$. As
a consequence we obtain the global well-posedness of the classical Boussinesq
system at this level of regularity as well as the convergence in the strong
topology of the solution of the regularized system towards the solution of the
classical Boussinesq equation as the parameter e goes to 0. In a second time,
we prove the existence of low regularity entropic solutions of the Boussinesq
equations emanating from $u_0 \in H^1$ and $\zeta_0$ in an Orlicz class as weak
limits of regular solutions.
