On a Generalized System with Applications to Ideal Magnetohydrodynamics
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866909968111763456 |
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| author | Sarria, Alejandro |
| author_facet | Sarria, Alejandro |
| contents | Finite-time blowup of solutions $(u(x,t),b(x,t))$ to a generalized system of equations with applications to ideal Magnetohydrodynamics (MHD) and one-dimensional fluid convection and stretching, among other areas, is investigated. The system is parameter-dependent, our spatial domain is the unit interval or the circle, and the initial data $(u_0(x),b_0(x))$ is assumed to be smooth. Among other results, we derive precise blowup criteria for specific values of the parameters by tracking the evolution of $u_x$ along Lagrangian trajectories that originate at a point $x_0$ at which $b_0(x)$ and $b_0'(x)$ vanish. We employ concavity arguments, energy estimates, and ODE comparison methods. We also show that for some values of the parameters, a non-vanishing $b_0'(x_0)$ suppresses finite-time blowup. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2507_14772 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | On a Generalized System with Applications to Ideal Magnetohydrodynamics Sarria, Alejandro Analysis of PDEs Mathematical Physics Finite-time blowup of solutions $(u(x,t),b(x,t))$ to a generalized system of equations with applications to ideal Magnetohydrodynamics (MHD) and one-dimensional fluid convection and stretching, among other areas, is investigated. The system is parameter-dependent, our spatial domain is the unit interval or the circle, and the initial data $(u_0(x),b_0(x))$ is assumed to be smooth. Among other results, we derive precise blowup criteria for specific values of the parameters by tracking the evolution of $u_x$ along Lagrangian trajectories that originate at a point $x_0$ at which $b_0(x)$ and $b_0'(x)$ vanish. We employ concavity arguments, energy estimates, and ODE comparison methods. We also show that for some values of the parameters, a non-vanishing $b_0'(x_0)$ suppresses finite-time blowup. |
| title | On a Generalized System with Applications to Ideal Magnetohydrodynamics |
| topic | Analysis of PDEs Mathematical Physics |
| url | https://arxiv.org/abs/2507.14772 |