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Global well-posedness for the Euler-Boussinesq system with axisymmetric data. (English) Zbl 1220.35127
This very interesting and deep paper deals with the Cauchy problem for the Euler-Boussinesq approximation in $$\mathbb R^3 \times\mathbb R^+$$, i.e., the incompressible Euler equations with buoyancy force, and the heat equation with convective term. Assuming the data sufficiently smooth and axisymmetric without swirl, the authors prove existence of globally defined strong solutions (axisymmetric without swirl). The result holds for arbitrarily large data.
The proof is based on deep commutator estimates in Lorentz and Lebesgue spaces together with the well-known a priori estimates for the incompressible Euler equations for axially symmetric solutions without swirl (due to Ukhovskii and Yudovich).

##### MSC:
 35Q31 Euler equations 76D03 Existence, uniqueness, and regularity theory for incompressible viscous fluids 42B35 Function spaces arising in harmonic analysis 42B37 Harmonic analysis and PDEs
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