On the global stability and large time behavior of solutions of the Boussinesq equations
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arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2025
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| _version_ | 1866912812824002560 |
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| author | Jiang, Song Wang, Quan |
| author_facet | Jiang, Song Wang, Quan |
| contents | We study the two dimensional viscous Boussinesq equations, which model stratified flows in a circular domain under the influence of a general gravitational potential $f$. First, we show that the Boussinesq equations admit steady-state solutions only in the form of hydrostatic equilibria, $(\mathbf{u},ρ,p) = (0, ρ_s, p_s)$, where the pressure gradient satisfies $\nabla p_s = -ρ_s \nabla f$. Moreover, the relation between $ρ_s$ and $f$ is constrained by $(\partial_y ρ_s, -\partial_x ρ_s) \cdot (\partial_x f, \partial_y f) = 0$, which allows us to write $\nabla ρ_s = h(x,y) \nabla f$ for some scalar function $h(x,y)$. Second, we prove that any hydrostatic equilibrium $(0, ρ_s, p_s)$ is linearly unstable if $h(x_0, y_0) > 0$ at some point $(x, y) = (x_0, y_0)$. This instability coincides with the classical Rayleigh--Taylor instability. Third, by employing a series of regularity estimates, we reveal that although the presence of the Rayleigh--Taylor instability makes perturbations around the unstable equilibrium grow exponentially in time, the system ultimately converges to a state of hydrostatic equilibrium. The analysis is carried out for perturbations about an arbitrary hydrostatic equilibrium, covering both stable and unstable configurations. Finally, we derive a necessary and sufficient condition on the initial density perturbation under which the density converges to a profile of the form $-γf + β$ with constants $γ, β> 0$. This result underscores the system's inherent tendency to settle into a hydrostatic state, even in the presence of Rayleigh--Taylor instability. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2502_16226 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | On the global stability and large time behavior of solutions of the Boussinesq equations Jiang, Song Wang, Quan Analysis of PDEs We study the two dimensional viscous Boussinesq equations, which model stratified flows in a circular domain under the influence of a general gravitational potential $f$. First, we show that the Boussinesq equations admit steady-state solutions only in the form of hydrostatic equilibria, $(\mathbf{u},ρ,p) = (0, ρ_s, p_s)$, where the pressure gradient satisfies $\nabla p_s = -ρ_s \nabla f$. Moreover, the relation between $ρ_s$ and $f$ is constrained by $(\partial_y ρ_s, -\partial_x ρ_s) \cdot (\partial_x f, \partial_y f) = 0$, which allows us to write $\nabla ρ_s = h(x,y) \nabla f$ for some scalar function $h(x,y)$. Second, we prove that any hydrostatic equilibrium $(0, ρ_s, p_s)$ is linearly unstable if $h(x_0, y_0) > 0$ at some point $(x, y) = (x_0, y_0)$. This instability coincides with the classical Rayleigh--Taylor instability. Third, by employing a series of regularity estimates, we reveal that although the presence of the Rayleigh--Taylor instability makes perturbations around the unstable equilibrium grow exponentially in time, the system ultimately converges to a state of hydrostatic equilibrium. The analysis is carried out for perturbations about an arbitrary hydrostatic equilibrium, covering both stable and unstable configurations. Finally, we derive a necessary and sufficient condition on the initial density perturbation under which the density converges to a profile of the form $-γf + β$ with constants $γ, β> 0$. This result underscores the system's inherent tendency to settle into a hydrostatic state, even in the presence of Rayleigh--Taylor instability. |
| title | On the global stability and large time behavior of solutions of the Boussinesq equations |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/2502.16226 |