Haroune Houamed — Page 2

arXiv2020

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

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 \cite{Gallay,Gallay-Sverak}. Roughly speaking, we show essentially that if the initial data $(v_0,ρ_0)$ is axisymmetric and $(ω_0,ρ_0)$ belongs to the critical space $L^1(Ω)\times L^1(\mathbb{R}^3)$, with $ω_0$ is the initial vorticity associated to $v_0$ and $Ω=\{(r,z)\in\mathbb{R}^2:r>0\}$, then the viscous Boussinesq system has a unique global solution.

math.AP↗

arXiv2020

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

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 $v_0\in H^{\frac12}(\mathbb{R}^3)\cap \dot{B}^{0}_{3,1}(\mathbb{R}^3)$ and density $ρ_0\in L^2(\mathbb{R}^3)\cap \dot{B}^{0}_{3,1}(\mathbb{R}^3)$. This respectively enhances the two results recently accomplished in \cite{Danchin-Paicu1, Hmidi-Rousset}. Our formalism is inspired, in particular for the first part from \cite{Abidi} concerning the axisymmetric Navier-Stokes equations once $v_0\in H^{\frac12}(\mathbb{R}^3)$ and external force $f\in L^2_{loc}\big(\mathbb{R}_{+};H^β(\mathbb{R}^3)\big)$, 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 \cite{Abidi} but with $f\in L^β_{loc}\big(\mathbb{R}_{+};L^2(\mathbb{R}^3))$ 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.

math.AP↗