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| Main Authors: | , , |
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| Format: | Preprint |
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2025
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| Online Access: | https://arxiv.org/abs/2503.24055 |
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| _version_ | 1866910899778879488 |
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| author | Cobb, Dimitri del Pino, Daniel Sánchez-Simón Velázquez, Juan J. L. |
| author_facet | Cobb, Dimitri del Pino, Daniel Sánchez-Simón Velázquez, Juan J. L. |
| contents | In this article we study a one dimensional model for Magnetic Relaxation. This model was introduced by Moffatt and describes a low resistivity viscous plasma, in which the pressure and the inercia are much smaller than the magnetic pressure. In the limit of resistivity $\varepsilon\rightarrow 0$, we prove the existence of two time scales for the evolution of the magnetic field: a fast one for times of order $\log(\varepsilon^{-1})$ in which the resistivity plays no role and the energy is dissipated only via viscosity; and a slow one for times of order $\varepsilon^{-1}$ characterized by the influence of the resistivity. We show that in this second time scale, as $\varepsilon\rightarrow 0$, the modulus of magnetic field approaches a function that depends only on time. We also prove that, in this regime, the magnetic field $b_\varepsilon(t,x)$ can be approximated as $\varepsilon \rightarrow 0$ by the solution of a PDE whose solutions exhibit blow up for some choices of initial data. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2503_24055 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Effective Dynamics and Blow Up in a Model of Magnetic Relaxation Cobb, Dimitri del Pino, Daniel Sánchez-Simón Velázquez, Juan J. L. Analysis of PDEs Mathematical Physics Plasma Physics 35Q35 (main), 76W05, 35B25, 35B40, 35B44 (secondary) In this article we study a one dimensional model for Magnetic Relaxation. This model was introduced by Moffatt and describes a low resistivity viscous plasma, in which the pressure and the inercia are much smaller than the magnetic pressure. In the limit of resistivity $\varepsilon\rightarrow 0$, we prove the existence of two time scales for the evolution of the magnetic field: a fast one for times of order $\log(\varepsilon^{-1})$ in which the resistivity plays no role and the energy is dissipated only via viscosity; and a slow one for times of order $\varepsilon^{-1}$ characterized by the influence of the resistivity. We show that in this second time scale, as $\varepsilon\rightarrow 0$, the modulus of magnetic field approaches a function that depends only on time. We also prove that, in this regime, the magnetic field $b_\varepsilon(t,x)$ can be approximated as $\varepsilon \rightarrow 0$ by the solution of a PDE whose solutions exhibit blow up for some choices of initial data. |
| title | Effective Dynamics and Blow Up in a Model of Magnetic Relaxation |
| topic | Analysis of PDEs Mathematical Physics Plasma Physics 35Q35 (main), 76W05, 35B25, 35B40, 35B44 (secondary) |
| url | https://arxiv.org/abs/2503.24055 |