On the global solvability of the axisymmetric Boussinesq system with critical regularity

2020/02/28 by Houamed, Haroune, Zerguine, Mohamed
#35B33 #35Q35 #76D03 #76D05 #Analysis of PDEs (math.AP) #FOS: Mathematics

paper · doi:10.48550/arxiv.2002.12605

Abstract

The current paper is principally motivated by establishing the global well-posedness to the three-dimensional Boussinesq system with zero diffusivity in the setting of axisymmetric flows without swirling with v₀∈ H^\frac12(ℝ³)∩ B⁰_3,1(ℝ³) and density ρ₀∈ L²(ℝ³)∩ B⁰_3,1(ℝ³). This respectively enhances the two results recently accomplished in \citeDanchin-Paicu1, Hmidi-Rousset. Our formalism is inspired, in particular for the first part from \citeAbidi concerning the axisymmetric Navier-Stokes equations once v₀∈ H^\frac12(ℝ³) and external force f∈ L²_loc(ℝ₊;H^β(ℝ³)), with β>\frac14. This latter regularity on f which is the density in our context is helpless to achieve the global estimates for Boussinesq system. This technical defect forces us to deal once again with a similar proof to that of \citeAbidi but with f∈ L^β_loc(ℝ₊;L²(ℝ³)) for some β>4. Second, we explore the gained regularity on the density by considering it as an external force in order to apply the study already obtained to the Boussinesq system.

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