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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.

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