Hölder continuous dissipative solutions of ideal MHD with nonzero helicity
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arXiv
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| Autori principali: | , , |
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| Natura: | Preprint |
| Pubblicazione: |
2025
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| _version_ | 1866911452653158400 |
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| author | Enciso, Alberto Peñafiel-Tomás, Javier Peralta-Salas, Daniel |
| author_facet | Enciso, Alberto Peñafiel-Tomás, Javier Peralta-Salas, Daniel |
| contents | We prove the existence of weak solutions to the 3D ideal MHD equations, of class $C^α$ with $α=1/200$, for which the total energy and the cross helicity (i.e., the so-called Elsässer energies) are not conserved. The solutions do not possess any symmetry properties and the magnetic helicity, which is necessarily conserved for Hölder continuous solutions, is nonzero. The construction, which works both on the torus $\mathbb{T}^3$ and on $\mathbb{R}^3$ with compact spatial support, is based on a novel convex integration scheme in which the magnetic helicity is preserved at each step. This is the first construction of continuous weak solutions at a regularity level where one conservation law (here, the magnetic helicity) is necessarily preserved while another (here, the total energy or cross helicity) is not, and where the preservation of the former is nontrivial in the sense that it does not follow from symmetry considerations. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2507_23749 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Hölder continuous dissipative solutions of ideal MHD with nonzero helicity Enciso, Alberto Peñafiel-Tomás, Javier Peralta-Salas, Daniel Analysis of PDEs We prove the existence of weak solutions to the 3D ideal MHD equations, of class $C^α$ with $α=1/200$, for which the total energy and the cross helicity (i.e., the so-called Elsässer energies) are not conserved. The solutions do not possess any symmetry properties and the magnetic helicity, which is necessarily conserved for Hölder continuous solutions, is nonzero. The construction, which works both on the torus $\mathbb{T}^3$ and on $\mathbb{R}^3$ with compact spatial support, is based on a novel convex integration scheme in which the magnetic helicity is preserved at each step. This is the first construction of continuous weak solutions at a regularity level where one conservation law (here, the magnetic helicity) is necessarily preserved while another (here, the total energy or cross helicity) is not, and where the preservation of the former is nontrivial in the sense that it does not follow from symmetry considerations. |
| title | Hölder continuous dissipative solutions of ideal MHD with nonzero helicity |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/2507.23749 |