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On the global well-posedness of the axisymmetric viscous Boussinesq system in critical Lebesgue spaces

  • * Corresponding author appreciates Taoufik Hmidi from Rennes 1 University for the fruitful discussions on the subject with the occasion of RAMA 11 organized by Sidi bel Abbess University, Algeria, November 2019. A particular greeting dedicates to the referee for the enormous efforts, careful reading and his appreciation of the work

    * Corresponding author appreciates Taoufik Hmidi from Rennes 1 University for the fruitful discussions on the subject with the occasion of RAMA 11 organized by Sidi bel Abbess University, Algeria, November 2019. A particular greeting dedicates to the referee for the enormous efforts, careful reading and his appreciation of the work

This work has been done while the second author is a PhD student at the University of Côte d’Azur-Nice-France, under the supervision of F. Planchon and P. Dreyfuss, in particular the second author would like to thank his supervisors, and the LJAD direction

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  • The contribution of this paper will be focused on the global existence and uniqueness topic in three-dimensional case of the axisymmetric viscous Boussinesq system in critical Lebesgue spaces. We aim at deriving analogous results for the classical two-dimensional and three-dimensional axisymmetric Navier-Stokes equations recently obtained in [19,20]. Roughly speaking, we show essentially that if the initial data $ (v_0, \rho_0) $ is axisymmetric and $ (\omega_0, \rho_0) $ belongs to the critical space $ L^1(\Omega)\times L^1( \mathbb{R}^3) $, with $ \omega_0 $ is the initial vorticity associated to $ v_0 $ and $ \Omega = \{(r, z)\in \mathbb{R}^2:r>0\} $, then the viscous Boussinesq system has a unique global solution.

    Mathematics Subject Classification: Primary: 76D03, 76D05; Secondary: 35B33, 35Q35.


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