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

Hmidi, Taoufik

doi:10.2140/apde.2011.4.247

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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

DOI: 10.2140/apde.2011.4.247

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 $| D | ∕ {log}^{α} (e^{4} + D)$ , with $α \in [0, \frac{1}{2}]$ . This result improves on an earlier critical dissipation condition $(α = 0)$ needed for global well-posedness.

Keywords

Boussinesq system, logarithmic dissipation, global existence

Mathematical Subject Classification 2000

Primary: 35Q35

Secondary: 76D03

Milestones

Received: 13 November 2009

Accepted: 18 March 2010

Published: 18 November 2011

Authors

Taoufik Hmidi
Institut de recherche mathématique de Rennes Université de Rennes 1 Campus de Beaulieu 35042 Rennes Cedex France

Vol. 4, No. 2, 2011