Construction of a Family of Stable Finite-Time Blowup Solutions for the Viscous Boussinesq System

This paper concerns with the local dynamical behavior near explicit finite-time blowup solutions with the smooth initial data for the three-dimensional viscous Boussinesq system. More precisely, we show that there exists a family of explicit blowup solutions with the smooth initial data and infinite energy in whole space R-3. Meanwhile, we employ a suitable Nash-Moser iteration scheme by Yan (J Differ Equ 261:1973- 2005, 2016) to prove Lyapunov nonlinear stability of those explicit finite-time blowup solutions for three-dimensional viscous Boussinesq system in a smooth moving domain with the timelike boundary condition. This means that we find a family of stable finite-time blowup solutions for the three-dimensional viscous Boussinesq system in a smooth bounded domain.