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| HAL: hal-00605829, version 1 |
| arXiv: 0910.4991 |
| DOI: 10.2140/apde.2011.4.247 |
| Detailed view |  Export this paper |
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| Analysis & PDE 4, 2 (2011) 247-284 |
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| On a maximum principle and its application to logarithmically critical Boussinesq system |
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| Taoufik Hmidi 1 |
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| (2011) |
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| 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]. |
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| 1: | Institut de Recherche Mathématique de Rennes (IRMAR) |
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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 |
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| Equations aux dérivées partielles |
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| Subject | : | Mathematics/Analysis of PDEs |
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| Boussinesq system – logarithmic dissipation – global existence |
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| Fulltext link: |
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| http://fr.arXiv.org/abs/0910.4991 |
| hal-00605829, version 1 | |
| http://hal.archives-ouvertes.fr/hal-00605829 | |
| oai:hal.archives-ouvertes.fr:hal-00605829 | |
| From: Marie-Annick Guillemer | |
| Submitted on: Monday, 4 July 2011 14:49:32 | |
| Updated on: Wednesday, 13 June 2012 16:56:47 | |
