940 articles – 1212 references  [version française]
 HAL: hal-00605829, version 1
 arXiv: 0910.4991
 Analysis & PDE 4, 2 (2011) 247-284
 On a maximum principle and its application to logarithmically critical Boussinesq system
 (2011)
 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].
 1: Institut de Recherche Mathématique de Rennes (IRMAR) CNRS : UMR6625 – Université de Rennes 1 – École normale supérieure de Cachan - ENS Cachan – Institut National des Sciences Appliquées (INSA) : - RENNES – Université de Rennes II - Haute Bretagne
 Research team: Equations aux dérivées partielles
 Subject : Mathematics/Analysis of PDEs
 Keyword(s): Boussinesq system – logarithmic dissipation – global existence