On the global existence of strong solution to the 3D damped Boussinesq equations with zero thermal diffusion
type: Item
instance of: scholarly article
title: On the global existence of strong solution to the 3D damped Boussinesq equations with zero thermal diffusion
zbMATH Open document ID: 1397.35232
DOI: 10.4171/ZAA/1617
author: Y. Aharonov
published in: Zeitschrift für Analysis und ihre Anwendungen
publication date: 2018-08-10T00:00:00Z
review text: A particular type of 3D Navier-Stokes equations is studied, containing the damping term \(D= |u|^{\beta-1}u\) and ``exterior'' forces \(\theta (0,0,1)^T \), where \(u\) is the velocity and \(\theta\) is the temperature. The authors specify that `` zero thermal diffusion'' means \(D=0\). However, they study the case \(D \neq 0\). A state of art concerning the local existence and uniqueness of smooth solutions is given in the introduction. The new element of the paper is an existence and uniqueness theorem for the strong solution (see Definition 1.1 and Theorem 1.2) in the case \(\beta \geq 3\). The temperature equation is used to get some \textit{apriori} estimates for \(\theta, u\) in terms of the initial data. For this are used some interpolation inequalities and a result given in [\textit{Y. Zhou}, Appl. Math. Lett. 25, No. 11, 1822--1825 (2012; Zbl 1426.76095)]. The classical Riesz operator, the Duhamel's Principle, the Gronwall's inequality and some regularity results for the heat kernel given in [\textit{P. G. Lemarié-Rieusset}, Recent developments in the Navier-Stokes problem. Boca Raton, FL: Chapman \& Hall/CRC (2002; Zbl 1034.35093)] are used to estimate \( \Delta u\) and \(\nabla \theta\) in some specific spaces. The uniqueness result is obtained through Holder's inequality.
Mathematics Subject Classification ID: 35Q35
Mathematics Subject Classification ID: 35B65
Mathematics Subject Classification ID: 35D35
Mathematics Subject Classification ID: 35A01
Mathematics Subject Classification ID: 35A02
zbMATH DE Number: 6916003
zbMATH Keywords: 3D Boussinesq equations
zbMATH Keywords: damping
zbMATH Keywords: regularity
reviewed by: Gelu Paşa
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.4171/zaa/1617
OpenAlex ID: W2810723150
Wikidata QID: Q129548784
description: scientific article; zbMATH DE number 6916003