Global existence for non-homogeneous incompressible inviscid fluids in presence of Ekman pumping
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arXiv
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| Format: | Preprint |
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2024
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| author | Bravin, Marco Fanelli, Francesco |
| author_facet | Bravin, Marco Fanelli, Francesco |
| contents | In this paper, we study the global solvability of the density-dependent incompressible Euler equations, supplemented with a damping term of the form $ \mathfrak{D}_α^γ(ρ, u) = αρ^γ u $, where $α>0$ and $ γ\in \{0,1\} $. To some extent, this system can be seen as a simplified model describing the mean dynamics in the ocean; from this perspective, the damping term can be interpreted as a term encoding the effects of the celebrated Ekman pumping in the system.
On the one hand, in the general case of space dimension $d\geq 2$, we establish global well-posedness in the Besov spaces framework, under a non-linear smallness condition involving the size of the initial velocity field $u_0$, of the initial non-homogeneity $ρ_0-1$ and of the damping coefficient $α$. On the other hand, in the specific situation of planar motions and damping term with $γ=1$, we exhibit a second smallness condition implying global existence, which in particular yields global well-posedness for arbitrarily large initial velocity fields, provided the initial density variations $ρ_0-1$ are small enough. The formulated smallness conditions rely only on the endpoint Besov norm $B^1_{\infty,1}$ of the initial datum, whereas, as a byproduct of our analysis, we derive exponential decay of the velocity field and of the pressure gradient in the high regularity norms $B^s_{p,r}$. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2402_12592 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Global existence for non-homogeneous incompressible inviscid fluids in presence of Ekman pumping Bravin, Marco Fanelli, Francesco Analysis of PDEs In this paper, we study the global solvability of the density-dependent incompressible Euler equations, supplemented with a damping term of the form $ \mathfrak{D}_α^γ(ρ, u) = αρ^γ u $, where $α>0$ and $ γ\in \{0,1\} $. To some extent, this system can be seen as a simplified model describing the mean dynamics in the ocean; from this perspective, the damping term can be interpreted as a term encoding the effects of the celebrated Ekman pumping in the system. On the one hand, in the general case of space dimension $d\geq 2$, we establish global well-posedness in the Besov spaces framework, under a non-linear smallness condition involving the size of the initial velocity field $u_0$, of the initial non-homogeneity $ρ_0-1$ and of the damping coefficient $α$. On the other hand, in the specific situation of planar motions and damping term with $γ=1$, we exhibit a second smallness condition implying global existence, which in particular yields global well-posedness for arbitrarily large initial velocity fields, provided the initial density variations $ρ_0-1$ are small enough. The formulated smallness conditions rely only on the endpoint Besov norm $B^1_{\infty,1}$ of the initial datum, whereas, as a byproduct of our analysis, we derive exponential decay of the velocity field and of the pressure gradient in the high regularity norms $B^s_{p,r}$. |
| title | Global existence for non-homogeneous incompressible inviscid fluids in presence of Ekman pumping |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/2402.12592 |