Abstract
Abstract In the article, we are concerned with the three-dimensional anisotropic Boussinesq equations with the velocity dissipation in x 2 {x}_{2} and x 3 {x}_{3} directions and the thermal diffusion in only x 3 {x}_{3} direction. When the spatial domain is the whole space R 3 {{\mathbb{R}}}^{3} , the global well-posedness and stability problem for the partially dissipated Boussinesq system remain the extremely challenging open problems. Attention here focuses on the periodic domain Ω = R × T 2 \Omega ={\mathbb{R}}\times {{\mathbb{T}}}^{2} . We aim at establishing the stability for the problem of perturbations near hydrostatic equilibrium and the large-time behavior of the perturbed solution. We first obtain the global existence of some symmetric fluids in H 2 ( Ω ) {H}^{2}\left(\Omega ) for small initial data. Then the exponential decay rates for the oscillations u ˜ \widetilde{u} and θ \theta in H 1 ( Ω ) {H}^{1}\left(\Omega ) and the homogeneous Sobolev space H v 2 ˙ ( Ω ) \dot{{H}_{v}^{2}}\left(\Omega ) are also shown. The proof is based on a key observation that we can decompose the velocity u u into the average u ¯ \overline{u} on T 2 {{\mathbb{T}}}^{2} and the corresponding oscillation u ˜ \widetilde{u} . This enables us to establish the strong Poincaré-type inequalities on u ˜ \widetilde{u} , u 3 , θ {u}_{3},\theta and some anisotropic inequalities, which ensure the establishment of the closed priori estimates. In addition, we also prove the oscillations in one direction u ˜ ( 2 ) , u ˜ ( 3 ) {\widetilde{u}}^{\left(2)},{\widetilde{u}}^{\left(3)} in H 1 ( Ω ) {H}^{1}\left(\Omega ) decay to zero exponentially.
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