Exponential decay for the 3D Boussinesq equations with Navier boundary conditions

We study the three-dimensional incompressible Boussinesq equations on a bounded domain with smooth boundary and Navier boundary conditions. We construct global weak solutions by a Galerkin approximation and establish the associated Leray–Hopf energy inequalities. For nonnegative boundary friction, the total energy decays exponentially when the friction coefficient is positive on a boundary subset of positive surface measure. In the frictionless case, the scalar field and the velocity component orthogonal to the rigid-motion kernel decay exponentially; when the kernel is trivial, this is exponential decay of the total energy. When the scalar initial datum vanishes, this also proves exponential decay for the Navier–Stokes system on solids of revolution for every friction coefficient $\alpha \in L^{\mathrm{\infty }}(\mathrm{\partial }\mathrm{\Omega })$ such that $\alpha \ge 0$ almost everywhere and $\alpha \equiv ̸0$, resolving the corresponding case left open in \cite{Kelliher2025}. The proof uses a weighted Korn–Poincar'e inequality and a two-time Gronwall-type inequality with an exponentially decaying forcing term.