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, \theta) \Vert _{L^{2}(\mathbb{R}^{n})} \rightarrow 0 $, as $ t \rightarrow \infty $. 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}-2\alpha \, (0 < \alpha < 1) $, the global solutions are derived. Furthermore, under the assumption that the initial data $ (u_{0} $, $ \theta_{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 $ \alpha $ = 0, we obtained the regularity and decay estimate of the damped Boussinesq equation in Besov space.
Citation: Xinli Wang, Haiyang Yu, Tianfeng Wu. Global well-posedness and optimal decay rates for the $ n $-D incompressible Boussinesq equations with fractional dissipation and thermal diffusion[J]. AIMS Mathematics, 2024, 9(12): 34863-34885. doi: 10.3934/math.20241660
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, \theta) \Vert _{L^{2}(\mathbb{R}^{n})} \rightarrow 0 $, as $ t \rightarrow \infty $. 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}-2\alpha \, (0 < \alpha < 1) $, the global solutions are derived. Furthermore, under the assumption that the initial data $ (u_{0} $, $ \theta_{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 $ \alpha $ = 0, we obtained the regularity and decay estimate of the damped Boussinesq equation in Besov space.
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Xinli Wang, Haiyang Yu, Tianfeng Wu. Global well-posedness and optimal decay rates for the $ n $-D incompressible Boussinesq equations with fractional dissipation and thermal diffusion[J]. AIMS Mathematics, 2024, 9(12): 34863-34885. doi: 10.3934/math.20241660
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