Large-time behavior of the 2D thermally non-diffusive Boussinesq equations with Navier-slip boundary conditions
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| Format: | Preprint |
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2023
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| _version_ | 1866916612443996160 |
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| author | Bleitner, Fabian Carlson, Elizabeth Nobili, Camilla |
| author_facet | Bleitner, Fabian Carlson, Elizabeth Nobili, Camilla |
| contents | This paper investigates the large-time behavior of a buoyancy-driven fluid without thermal diffusion under Navier-slip boundary conditions in a bounded domain with Lipschitz-continuous second derivatives. After establishing improved regularity for classical solutions, we analyze their large-time asymptotics. Specifically, we show that the solutions converge to a state where, as $t \rightarrow \infty$, $\|u\|_{W^{1,p}} \rightarrow 0$, and hydrostatic balance is achieved in the weak topology of $L^2$. Furthermore, we identify the necessary conditions under which stable stratification and hydrostatic balance can be achieved in the strong topology as time approaches infinity. We then analyze a particular steady state, the hydrostatic equilibrium, characterized by $ u = 0 $, $ θ= βx_2 + γ$, and $ p = \fracβ{2}x_2^2 + γx_2 + δ$. In a periodic strip, we establish the linear stability of this state for $β> 0$, indicating that the temperature is vertically stably stratified. This work builds upon the results in [Doering et al.], which focus on free-slip boundary conditions, as well as recent studies [Aydın, Kukavica, Ziane; Aydın, Jayanti] that address no-slip boundary conditions. Notably, the novelty of this study lies in the ability to directly bound the pressure term, made possible by the Navier-slip boundary conditions. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2309_05400 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Large-time behavior of the 2D thermally non-diffusive Boussinesq equations with Navier-slip boundary conditions Bleitner, Fabian Carlson, Elizabeth Nobili, Camilla Analysis of PDEs Mathematical Physics Fluid Dynamics 35B40, 35B65, 76D03, 76E20 This paper investigates the large-time behavior of a buoyancy-driven fluid without thermal diffusion under Navier-slip boundary conditions in a bounded domain with Lipschitz-continuous second derivatives. After establishing improved regularity for classical solutions, we analyze their large-time asymptotics. Specifically, we show that the solutions converge to a state where, as $t \rightarrow \infty$, $\|u\|_{W^{1,p}} \rightarrow 0$, and hydrostatic balance is achieved in the weak topology of $L^2$. Furthermore, we identify the necessary conditions under which stable stratification and hydrostatic balance can be achieved in the strong topology as time approaches infinity. We then analyze a particular steady state, the hydrostatic equilibrium, characterized by $ u = 0 $, $ θ= βx_2 + γ$, and $ p = \fracβ{2}x_2^2 + γx_2 + δ$. In a periodic strip, we establish the linear stability of this state for $β> 0$, indicating that the temperature is vertically stably stratified. This work builds upon the results in [Doering et al.], which focus on free-slip boundary conditions, as well as recent studies [Aydın, Kukavica, Ziane; Aydın, Jayanti] that address no-slip boundary conditions. Notably, the novelty of this study lies in the ability to directly bound the pressure term, made possible by the Navier-slip boundary conditions. |
| title | Large-time behavior of the 2D thermally non-diffusive Boussinesq equations with Navier-slip boundary conditions |
| topic | Analysis of PDEs Mathematical Physics Fluid Dynamics 35B40, 35B65, 76D03, 76E20 |
| url | https://arxiv.org/abs/2309.05400 |