Global small-time approximate null and Lagrangian controllability of the viscous non-resistive MHD system in a $3D$ domain with Navier type boundary conditions
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| Format: | Preprint |
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2025
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| _version_ | 1866908360206450688 |
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| author | Liao, Jiajiang Sueur, Franck Zhang, Ping |
| author_facet | Liao, Jiajiang Sueur, Franck Zhang, Ping |
| contents | We consider the incompressible viscous MHD system without magnetic diffusion in a $3D$ bounded domain with Navier type boundary condition. We establish the global small-time approximate null controllability and the Lagrangian controllability of the system, in the class of smooth solutions, by following the approach initiated in \cite{CMS} to establish the global small-time null controllability of the incompressible Navier-Stokes equations in the class of weak solutions and extended in \cite{LSZ1} to establish the global small-time null and Lagrangian controllability of the incompressible Navier-Stokes equations in the class of strong solutions. This approach makes use of controls with an extra fast scale in time and some corresponding multi-scale asymptotic expansions of the controlled solution. This expansion is constructed by an iterative process which requires some regularity. The extra-difficulty here is that the MHD system at stake is mixed hyperbolic-parabolic, without any regularizing effect on the magnetic field. Despite our strategy makes use of a quite precise asymptotic expansion, we succeed to cover the case where the initial velocity belongs the Sobolev space $H^{24}$ and the initial magnetic field belongs to the Sobolev space $H^8$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2505_07366 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Global small-time approximate null and Lagrangian controllability of the viscous non-resistive MHD system in a $3D$ domain with Navier type boundary conditions Liao, Jiajiang Sueur, Franck Zhang, Ping Analysis of PDEs We consider the incompressible viscous MHD system without magnetic diffusion in a $3D$ bounded domain with Navier type boundary condition. We establish the global small-time approximate null controllability and the Lagrangian controllability of the system, in the class of smooth solutions, by following the approach initiated in \cite{CMS} to establish the global small-time null controllability of the incompressible Navier-Stokes equations in the class of weak solutions and extended in \cite{LSZ1} to establish the global small-time null and Lagrangian controllability of the incompressible Navier-Stokes equations in the class of strong solutions. This approach makes use of controls with an extra fast scale in time and some corresponding multi-scale asymptotic expansions of the controlled solution. This expansion is constructed by an iterative process which requires some regularity. The extra-difficulty here is that the MHD system at stake is mixed hyperbolic-parabolic, without any regularizing effect on the magnetic field. Despite our strategy makes use of a quite precise asymptotic expansion, we succeed to cover the case where the initial velocity belongs the Sobolev space $H^{24}$ and the initial magnetic field belongs to the Sobolev space $H^8$. |
| title | Global small-time approximate null and Lagrangian controllability of the viscous non-resistive MHD system in a $3D$ domain with Navier type boundary conditions |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/2505.07366 |