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In this paper, we consider the initial boundary value problem of two-dimensional isentropic compressible Boussinesq equations with constant viscosity and thermal diffusivity in a square domain. Based on the time-independent lower-order and time-dependent higher-order a priori estimates, we prove that the classical solution exists globally in time provided the initial mass ∥ ρ 0 ∥ L 1 $\|\rho _{0}\|_{L^{1}}$ of the fluid is small. Here, we have no small requirements for the initial velocity and temperature.

Global existence of classical solutions to the two-dimensional compressible Boussinesq equations in a square domain