#### Vol. 4, No. 2, 2011

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

### Taoufik Hmidi

Vol. 4 (2011), No. 2, 247–284
##### 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.

##### Keywords
Boussinesq system, logarithmic dissipation, global existence
Primary: 35Q35
Secondary: 76D03