On the global existence of strong solution to the 3D damped Boussinesq equations with zero thermal diffusion
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.
scientific article; zbMATH DE number 6916003