In this paper, we generalize Chae’s Liouville-type rigidity theorems for the Navier–Stokes and Euler equations to the viscous Boussinesq system on \(\mathbb {R}^n\) . By testing the momentum equation against gradients of truncated quadratic polynomials and carefully estimating boundary contributions, we prove that if the pressure satisfies either a nonnegativity condition on its spatial integral or a Hardy space assumption ( \(p \in L^{1}(0, T; H_{q}(\mathbb {R}^{n}))\) for some \(q \in (0,1]\) ), and if the buoyancy field satisfies the weighted integrability condition \((1+|x|^2)\theta \in L^1(\mathbb {R}^n)\) with vanishing vertical first moment, then every weak solution must have identically vanishing velocity. Consequently, the temperature remains frozen at its initial profile and the pressure reduces to a vertical potential, yielding a complete Liouville-type theorem for the Boussinesq system.
