Global well-posedness and optimal decay rates for the $ n $-D incompressible Boussinesq equations with fractional dissipation and thermal diffusion - Xinli Wang; Haiyang Yu; Tianfeng Wu (AIMS Mathematics)
In this paper, $ n $-dimensional incompressible Boussinesq equations with fractional dissipation and thermal diffusion are investigated. Firstly, by applying frequency decomposition, we find that $ \Vert (u, θ) \Vert _{L^{2}(\mathbb{R}^{n})} → 0 $, as $ t → ∞ $. Secondly, by using energy methods, we can show that if the initial data is sufficiently small in $ H^{s}(\mathbb{R}^{n}) $ with s = 1+$ \frac{n}{2}-2α \, (0 < α < 1) $, the global solutions are derived. Furthermore, under the assumption that the initial data $ (u_{0} $, $ θ_{0}) $ belongs to $ L^{p}($where $ 1\le p < 2) $, using a more advanced frequency decomposition method, we establish optimal decay estimates for the solutions and their higher-order derivatives. Meanwhile, the uniqueness of the system can be obtained. In the case $ α $ = 0, we obtained the regularity and decay estimate of the damped Boussinesq equation in Besov space.