AI 中文总结
该研究针对带Navier边界条件的三维Boussinesq方程，通过Galerkin逼近构造弱解，利用加权Korn-Poincaré不等式等证明其能量指数衰减，解决了旋转固体上Navier-Stokes系统的遗留问题。
AI 中文摘要
我们研究带有光滑边界和Navier边界条件的有界区域上的三维不可压缩Boussinesq方程。通过Galerkin逼近构造全局弱解，并建立相关的Leray-Hopf能量不等式。对于非负边界摩擦，当摩擦系数在具有正表面测度的边界子集上为正时，总能量呈指数衰减；在无摩擦情形下，标量场与垂直于刚体运动核的速度分量呈指数衰减，当该核为平凡时，总能量呈指数衰减。当初始标量数据为零时，这也对旋转固体上的Navier-Stokes系统证明了指数衰减，其中对几乎处处满足α≥0且α≢0的α∈L^∞(∂Ω)，解决了文献[Kelliher2025]中遗留的对应情形。证明使用加权Korn-Poincaré不等式和带有指数衰减强迫项的双时间Gronwall型不等式。
英文摘要
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 $α\in L^\infty(\partialΩ)$ such that $α\ge0$ almost everywhere and $α\not\equiv0$, resolving the corresponding case left open in \cite{Kelliher2025}. The proof uses a weighted Korn--Poincaré inequality and a two-time Gronwall-type inequality with an exponentially decaying forcing term.
