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 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|logα(e4 + D), with α [0, 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