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

Adalet Hanachi; Haroune Houamed; Mohamed Zerguine

doi:10.3934/dcds.2020287

Article Contents

2020, Volume 40, Issue 11: 6493-6526. Doi: 10.3934/dcds.2020287

This issue Previous Article A unified approach for energy scattering for focusing nonlinear Schrödinger equations Next Article Low regularity of solutions to the Rotation-Camassa-Holm type equation with the Coriolis effect

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

1.
LEDPA, Université de Batna 2, Faculté des Mathématiques et d'Informatique, Département de Mathématiques, 05000 Batna, Algérie
2.
Université Côte d'Azur, CNRS, LJAD, Nice, France

^* 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

Received: March 2020

Revised: April 2020

Published: July 2020

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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Abstract

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.

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

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