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On a maximum principle and its application to the logarithmically critical Boussinesq system. (English) Zbl 1264.35173
Summary: 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 $$|D|/\log^\alpha(e^4 + D)$$, with $$\alpha \in[0, \frac12]$$. This result improves on an earlier critical dissipation condition $$(\alpha = 0)$$ needed for global well-posedness.

##### MSC:
 35Q35 PDEs in connection with fluid mechanics 76D03 Existence, uniqueness, and regularity theory for incompressible viscous fluids 35Q31 Euler equations
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