On a maximum principle and its application to the 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 is a maximum principle whose proof is based on Askey theorem concerning characteristic functions and some tools from the theory of C₀-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^α(e⁴+D), with α [0,1/2]. This result improves on an earlier critical dissipation condition .(α = 0) needed for global well-posedness.