论文标题
关于具有关键规律性的轴对称BousSinesQ系统的全球可溶性
On the global solvability of the axisymmetric Boussinesq system with critical regularity
论文作者
论文摘要
当前的论文主要是通过建立全球良好的对三维BousSinesQ系统的良好性，但在轴对称流的情况下，零扩散率零扩散，而无需用$ v_0 \ in h^{\ frac12}旋转旋转， \ dot {b}^{0} _ {3,1}（\ Mathbb {r}^3）$和密度$ρ_0\ in L^2（\ Mathbb {r}^3）\ cap \ dot \ dot {b}这分别增强了最近在\ cite {danchin-paicu1，hmidi-rousset}中实现的两个结果。我们的形式主义受到启发，特别是第一部分，\ cite {abidi}涉及轴对称的navier-stokes方程，曾经$ v_0 \ in H^{\ frac12}（\ mathbb {r}^3）和外部力量$ f \ in In l^2_ {loc} \ big（\ mathbb {r} _ {+}; h^β（\ mathbb {r}^3）\ big）$，带有$β> \ frac14 $。在我们上下文中的密度是$ f $上的后一种规律性是为了实现BousSinesQ系统的全球估计。这种技术缺陷迫使我们再次处理与\ cite {abidi}的证据相似的证据，但是在l^β_{loc} \ big（\ m athbb {r} _ {+} _ {+}; l^2（\ mathbb {r}^3）中，$ f \ in l^β_{loc} \ big（\ mathbb {r} _ {+} _其次，我们通过将其视为外力来探讨其密度获得的规律性，以便将已经获得的研究应用于Boussinesq系统。
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.
