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

Abstract

In this paper we study a transport-diffusion model with some logarithmic dissipations. We look for two kinds of estimates. The first 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 is a smoothing effect based on some results from harmonic analysis and submarkovian 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 $|D|∕{log}^{\alpha }\left(e^4+D\right)$, with $\alpha \in \left[0,\frac{1}{2}\right]$. This result improves on an earlier critical dissipation condition $\left(\alpha =0\right)$ needed for global well-posedness.