Stability on 3D Boussinesq system with mixed partial dissipation

Abstract

In the article, we are concerned with the three-dimensional anisotropic Boussinesq equations with the velocity dissipation in x2{x}_{2} and x3{x}_{3} directions and the thermal diffusion in only x3{x}_{3} direction. When the spatial domain is the whole space R3{{\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×T2\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 H2(Ω){H}^{2}\left(\Omega ) for small initial data. Then the exponential decay rates for the oscillations u˜\widetilde{u} and θ\theta in H1(Ω){H}^{1}\left(\Omega ) and the homogeneous Sobolev space Hv2˙(Ω)\dot{{H}_{v}^{2}}\left(\Omega ) are also shown. The proof is based on a key observation that we can decompose the velocity uu into the average u¯\overline{u} on T2{{\mathbb{T}}}^{2} and the corresponding oscillation u˜\widetilde{u}. This enables us to establish the strong Poincaré-type inequalities on u˜\widetilde{u}, u3,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 H1(Ω){H}^{1}\left(\Omega ) decay to zero exponentially.

Keywords

• 3d boussinesq equations
• partial dissipation
• stability
• decay rates
• 35a05
• 35q35
• 76d03