On the existence and uniqueness of weak solutions to elliptic equations with a singular drift
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| Format: | Preprint |
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2024
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| _version_ | 1866929337153880064 |
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| author | Chernobai, Misha Shilkin, Tim |
| author_facet | Chernobai, Misha Shilkin, Tim |
| contents | In this paper we study the Dirichlet problem for a scalar elliptic equation in a bounded Lipschitz domain $Ω\subset \mathbb R^3$ with a singular drift of the form $b_0= b-α\frac {x'}{|x'|^2}$ where $x'=(x_1,x_2,0)$, $α\in \mathbb R$ is a parameter and $b$ is a divergence free vector field having essentially the same regularity as the potential part of the drift. Such drifts naturally arise in the theory of axially symmetric solutions to the Navier-Stokes equations. For $α<0$ the divergence of such drifts is positive which potentially can ruin the uniqueness of solutions. Nevertheless, for $α<0$ we prove existence and Hölder continuity of a unique weak solution which vanishes on the axis $Γ:=\{ ~x\in \mathbb R^3:~|x'|=0~\}$. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2405_04302 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | On the existence and uniqueness of weak solutions to elliptic equations with a singular drift Chernobai, Misha Shilkin, Tim Analysis of PDEs In this paper we study the Dirichlet problem for a scalar elliptic equation in a bounded Lipschitz domain $Ω\subset \mathbb R^3$ with a singular drift of the form $b_0= b-α\frac {x'}{|x'|^2}$ where $x'=(x_1,x_2,0)$, $α\in \mathbb R$ is a parameter and $b$ is a divergence free vector field having essentially the same regularity as the potential part of the drift. Such drifts naturally arise in the theory of axially symmetric solutions to the Navier-Stokes equations. For $α<0$ the divergence of such drifts is positive which potentially can ruin the uniqueness of solutions. Nevertheless, for $α<0$ we prove existence and Hölder continuity of a unique weak solution which vanishes on the axis $Γ:=\{ ~x\in \mathbb R^3:~|x'|=0~\}$. |
| title | On the existence and uniqueness of weak solutions to elliptic equations with a singular drift |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/2405.04302 |