Exact boundary controllability of the 3D incompressible ideal MHD system
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arXiv
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| Format: | Preprint |
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2024
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| author | Kukavica, Igor Ożański, Wojciech S. |
| author_facet | Kukavica, Igor Ożański, Wojciech S. |
| contents | We consider the three-dimensional ideal MHD system on a domain $Ω' \subset \mathbb{R}^3$ with a part $Γ$ of the boundary~$\partial Ω$, where we prescribe both $u\cdot n$ and $b\cdot n$, while $u\cdot n = b\cdot n =0$ on $\partial Ω' \setminus Γ$. We prove the boundary controllability of the system, namely that we can prescribe the boundary data such that the unique solution of the system with initial state $(u_0,b_0)$ achieves another state $(u_1,b_1)$ in finite time, where $u_0,b_0,u_1,b_1$ are arbitrary divergence-free vector fields satisfying impermeability boundary condition which are extendable to vector fields with the same properties on any bounded domain obtained by extension of $Ω'$ via $Γ$. As a byproduct, we give the first local well-posedness proof of incompressible, ideal MHD system, which does not use Elsasser variables and is thus applicable to any bounded domain with sufficient Sobolev regularity. We also provide a new and simple proof of the $2$D controllability. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2410_02588 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Exact boundary controllability of the 3D incompressible ideal MHD system Kukavica, Igor Ożański, Wojciech S. Analysis of PDEs We consider the three-dimensional ideal MHD system on a domain $Ω' \subset \mathbb{R}^3$ with a part $Γ$ of the boundary~$\partial Ω$, where we prescribe both $u\cdot n$ and $b\cdot n$, while $u\cdot n = b\cdot n =0$ on $\partial Ω' \setminus Γ$. We prove the boundary controllability of the system, namely that we can prescribe the boundary data such that the unique solution of the system with initial state $(u_0,b_0)$ achieves another state $(u_1,b_1)$ in finite time, where $u_0,b_0,u_1,b_1$ are arbitrary divergence-free vector fields satisfying impermeability boundary condition which are extendable to vector fields with the same properties on any bounded domain obtained by extension of $Ω'$ via $Γ$. As a byproduct, we give the first local well-posedness proof of incompressible, ideal MHD system, which does not use Elsasser variables and is thus applicable to any bounded domain with sufficient Sobolev regularity. We also provide a new and simple proof of the $2$D controllability. |
| title | Exact boundary controllability of the 3D incompressible ideal MHD system |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/2410.02588 |