On the global well-posedness of the two-dimensional Boussinesq system with a zero diffusivity
type: Item
instance of: scholarly article
title: On the global well-posedness of the two-dimensional Boussinesq system with a zero diffusivity
zbMATH Open document ID: 1154.35073
author: Taoufik Hmidi
author: Sahbi Keraani
published in: Advances in Differential Equations
publication date: 2008-09-12T00:00:00Z
review text: The authors consider the 2D Boussinesq system \[ \partial_t \vec v+\vec v \cdot \nabla \vec v -\nu \Delta \vec v+\nabla p=\theta \vec e, \] \[ \partial_t \theta+\vec v \cdot \nabla \theta -\kappa \Delta \theta=0, \] \[ \text{div}\;\vec v=0, \] with initial conditions \[ \left. \left(\vec v,\theta\right)\right| _{t=0}=\left(\vec v^0,\theta^0\right), \] where \(\vec e=(0,1)\), \(\vec v=(v_1,v_2)\) is the velocity, \(p\) is the pressure, the kinematic viscosity \(\nu\) is a positive parameter and the diffusivity coefficient \(\kappa\) is nonnegative. In this article, they study the specific situation where the temperature is only advected by the flow without diffusion (the molecular conductivity \(\kappa\) is zero). They first prove provided that \(\theta^0\in L^2\) and \(\vec v^0\) is a divergence-free \(H^s\)-vector-field with \(s\in [0,2)\), that system (BS) admits a global weak solution such that \[ \vec v\in C({\mathbb R}_+;H^s)\cap L^2_{\text{loc}}({\mathbb R}_+;H^{\min\{s+1,2\}}), \] and \[ \theta \in C_b({\mathbb R}_+; L^2). \] Moreover, if \(\theta^0\in B^0_{2,1}\cap B^0_{p,\infty}\) with \(p\in (2,\infty]\) and \(\vec v^0\) is a divergence-free \(H^s\)-vector-field with \(s\in (0,2]\), the system (BS) admits a unique global solution such that \[ \vec v\in C({\mathbb R}_+;H^s)\cap L^2_{\text{loc}}({\mathbb R}_+;H^{\min\{s+1,2\}})\cap L^1_{\text{loc}}({\mathbb R}_+;B^2_{2,1}), \] and \[ \theta \in C({\mathbb R}_+; B^0_{2,1}\cap B^0_{p,\infty}). \]
reviewed by: Bernard Ducomet
Mathematics Subject Classification ID: 35Q35
Mathematics Subject Classification ID: 35B05
Mathematics Subject Classification ID: 76B03
zbMATH DE Number: 5343694
zbMATH Keywords: Boussinesq system
zbMATH Keywords: zero diffusivity
zbMATH Keywords: global weak solution
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 5343694