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| Main Authors: | , |
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| Format: | Preprint |
| Published: |
2025
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2512.19281 |
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| _version_ | 1866909973488861184 |
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| author | Jiang, Song Wang, Quan |
| author_facet | Jiang, Song Wang, Quan |
| contents | This paper studies the two-dimensional inhomogeneous Navier--Stokes equations governing stratified flows in a bounded domain under a gravitational potential \(f\). Our main results are as follows. First, we provide a rigorous characterization of steady states, proving that under the Dirichlet condition \(\mathbf{u}|_{\partial Ω} = \mathbf{0}\), all admissible equilibria are hydrostatic and satisfy \(\nabla p_s = -ρ_s \nabla f\). Second, through a perturbative analysis around arbitrary hydrostatic profiles, we show that despite possible transient growth induced by the Rayleigh--Taylor mechanism, the system always relaxes to a hydrostatic equilibrium. Third, we identify a necessary and sufficient condition on the initial density perturbation for convergence to a linear hydrostatic density profile of the form \(ρ_s = -γf + β\), with \(γ> 0\) and \(β> 0\). Finally, we establish improved regularity estimates for strong solutions corresponding to initial data in the Sobolev space \(H^3(Ω)\). |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2512_19281 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | On the large time behavior of the 2D inhomogeneous incompressible viscous flows Jiang, Song Wang, Quan Analysis of PDEs This paper studies the two-dimensional inhomogeneous Navier--Stokes equations governing stratified flows in a bounded domain under a gravitational potential \(f\). Our main results are as follows. First, we provide a rigorous characterization of steady states, proving that under the Dirichlet condition \(\mathbf{u}|_{\partial Ω} = \mathbf{0}\), all admissible equilibria are hydrostatic and satisfy \(\nabla p_s = -ρ_s \nabla f\). Second, through a perturbative analysis around arbitrary hydrostatic profiles, we show that despite possible transient growth induced by the Rayleigh--Taylor mechanism, the system always relaxes to a hydrostatic equilibrium. Third, we identify a necessary and sufficient condition on the initial density perturbation for convergence to a linear hydrostatic density profile of the form \(ρ_s = -γf + β\), with \(γ> 0\) and \(β> 0\). Finally, we establish improved regularity estimates for strong solutions corresponding to initial data in the Sobolev space \(H^3(Ω)\). |
| title | On the large time behavior of the 2D inhomogeneous incompressible viscous flows |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/2512.19281 |