Osgood type blow-up criterion for the 3D Boussinesq equations with partial viscosity

Abstract

This paper is dedicated to studying the blow-up criterion of smooth solutions to the three-dimensional Boussinesq equations with partial viscosity. By means of the Littlewood-Paley decomposition, we give an improved logarithmic Sobolev inequality and through this, we obtain the corresponding blow-up criterion in a space larger than $\dot{B}^0_{\infty,\infty}$, which extends several previous works.