  In this paper we study a transport-diffusion model with some logarithmic
dissipations. We look for two kinds of estimates. The first one is a maximum
principle whose proof is based on Askey theorem concerning characteristic
functions and some tools from the theory of $C_0$-semigroups. The second one is
a smoothing effect based on some results from harmonic analysis and
sub-Markovian operators. As an application we prove the global well-posedness
for the two-dimensional Euler-Boussinesq system where the dissipation occurs
only on the temperature equation and has the form
$\frac{\DD}{\log^\alpha(e^4+\DD)}$, with $\alpha\in[0,\frac12]$. This result
improves the critical dissipation $(\alpha=0)$ needed for global well-posedness
which was discussed in [15].
