On a maximum principle and its application to logarithmically critical Boussinesq system

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 $\frac{\text{DD}}{{\log }^{\alpha }(e^4+\text{DD})}$, with $\alpha \in [0,\frac{1}{2}]$. This result improves the critical dissipation $(\alpha =0)$ needed for global well-posedness which was discussed in [15].