On Onsager-type conjecture for the Elsässer energies of the ideal MHD equations
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arXiv
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| Natura: | Preprint |
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2025
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| _version_ | 1866910906564214784 |
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| author | Miao, Changxing Nie, Yao Ye, Weikui |
| author_facet | Miao, Changxing Nie, Yao Ye, Weikui |
| contents | In this paper, we investigate the ideal magnetohydrodynamics (MHD) equations on tours $\TTT^d$. For $d=3$, we resolve the flexible part of Onsager-type conjecture for Elsässer energies of the ideal MHD equations. More precisely, for \(β< 1/3\), we construct weak solutions \((u, b) \in C^β([0,T] \times \mathbb{T}^3)\) with both the total energy dissipation and failure of cross helicity conservation. The key idea of the proof relies on a symmetry reduction that embeds the ideal MHD system into a 2$\frac{1}{2}$D Euler flow and the Newton-Nash iteration technique recently developed in \cite{GR}. For $d=2$, we show the non-uniqueness of Hölder-continuous weak solutions with non-trivial magnetic fields. Specifically, for \(β< 1/5\), there exist infinitely many solutions \((u, b) \in C^β([0,T] \times \mathbb{T}^2)\) with the same initial data while satisfying the total energy dissipation with non-vanishing velocity and magnetic fields. The new ingredient is developing a spatial-separation-driven iterative scheme that incorporates the magnetic field as a controlled perturbation within the convex integration framework for the velocity field, thereby providing sufficient oscillatory freedom for Nash-type perturbations in the 2D setting. As a byproduct, we prove that any Hölder-continuous Euler solution can be approximated by a sequence of $C^β$-weak solutions for the ideal MHD equations in the $L^p$-topology for $1\le p<\infty$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2504_06071 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | On Onsager-type conjecture for the Elsässer energies of the ideal MHD equations Miao, Changxing Nie, Yao Ye, Weikui Analysis of PDEs In this paper, we investigate the ideal magnetohydrodynamics (MHD) equations on tours $\TTT^d$. For $d=3$, we resolve the flexible part of Onsager-type conjecture for Elsässer energies of the ideal MHD equations. More precisely, for \(β< 1/3\), we construct weak solutions \((u, b) \in C^β([0,T] \times \mathbb{T}^3)\) with both the total energy dissipation and failure of cross helicity conservation. The key idea of the proof relies on a symmetry reduction that embeds the ideal MHD system into a 2$\frac{1}{2}$D Euler flow and the Newton-Nash iteration technique recently developed in \cite{GR}. For $d=2$, we show the non-uniqueness of Hölder-continuous weak solutions with non-trivial magnetic fields. Specifically, for \(β< 1/5\), there exist infinitely many solutions \((u, b) \in C^β([0,T] \times \mathbb{T}^2)\) with the same initial data while satisfying the total energy dissipation with non-vanishing velocity and magnetic fields. The new ingredient is developing a spatial-separation-driven iterative scheme that incorporates the magnetic field as a controlled perturbation within the convex integration framework for the velocity field, thereby providing sufficient oscillatory freedom for Nash-type perturbations in the 2D setting. As a byproduct, we prove that any Hölder-continuous Euler solution can be approximated by a sequence of $C^β$-weak solutions for the ideal MHD equations in the $L^p$-topology for $1\le p<\infty$. |
| title | On Onsager-type conjecture for the Elsässer energies of the ideal MHD equations |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/2504.06071 |