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

Abstract

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 $\alpha\in L^\infty(\partial\Omega)$ such that $\alpha\ge0$ almost everywhere and $\alpha\not\equiv0$, 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.