Navier--Stokes flow in the exterior of a moving obstacle with a Lipschitz boundary
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866909099066654720 |
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| author | Takahashi, Tomoki Watanabe, Keiichi |
| author_facet | Takahashi, Tomoki Watanabe, Keiichi |
| contents | Consider the three-dimensional Navier--Stokes flow past a moving rigid body $\mathscr{O} \subset \mathbb{R}^3$ with prescribed translational and angular velocities, where $\mathscr{O}$ stands for a bounded Lipschitz domain. We prove that the solution to the linearized problem is governed by a $C_0$-semigroup on solenoidal $L^q$-vector spaces with the $L^q$-$L^r$ estimates provided that $|1/q-1/2|<1/6+\varepsilon$ with some $\varepsilon>0$, where $r \ge q$ may be taken arbitrary large. As an application, we prove the existence and uniqueness of global mild solutions to the Navier--Stokes problem if the translational and angular velocities as well as the initial are sufficiently small. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2402_05618 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Navier--Stokes flow in the exterior of a moving obstacle with a Lipschitz boundary Takahashi, Tomoki Watanabe, Keiichi Analysis of PDEs Consider the three-dimensional Navier--Stokes flow past a moving rigid body $\mathscr{O} \subset \mathbb{R}^3$ with prescribed translational and angular velocities, where $\mathscr{O}$ stands for a bounded Lipschitz domain. We prove that the solution to the linearized problem is governed by a $C_0$-semigroup on solenoidal $L^q$-vector spaces with the $L^q$-$L^r$ estimates provided that $|1/q-1/2|<1/6+\varepsilon$ with some $\varepsilon>0$, where $r \ge q$ may be taken arbitrary large. As an application, we prove the existence and uniqueness of global mild solutions to the Navier--Stokes problem if the translational and angular velocities as well as the initial are sufficiently small. |
| title | Navier--Stokes flow in the exterior of a moving obstacle with a Lipschitz boundary |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/2402.05618 |