Global existence and uniqueness of weak solutions for the MHD equations with large $L^3$-initial values
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arXiv
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| Main Authors: | , , |
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| Format: | Preprint |
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2026
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| _version_ | 1866917254781730816 |
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| author | Lai, Baishun Tang, Ge Xu, Ziying |
| author_facet | Lai, Baishun Tang, Ge Xu, Ziying |
| contents | This paper is concerned with the weak solution theory for the MHD system with large $L^3$-initial data. Due to the fact that the natural boundary condition on the magnetic field $H$ is the slip boundary condition, the Leray-Schauder fixed-point theorem, which have used to investigate the weak solution theory of the Navier-Stokes system, becomes invalid. To address such difficulty, we will invoke the Leray's approximation technique and the perturbation theory to seek a global weak solution to the Cauchy problem for MHD equations with large $L^3$-initial data. Our strategy provides a simple alternative (self-contained) proof of weak $L^3$-solution theory of incompressible Navier-Stokes system. Moreover, this weak solution is unique under some restrictions. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2602_06979 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Global existence and uniqueness of weak solutions for the MHD equations with large $L^3$-initial values Lai, Baishun Tang, Ge Xu, Ziying Analysis of PDEs This paper is concerned with the weak solution theory for the MHD system with large $L^3$-initial data. Due to the fact that the natural boundary condition on the magnetic field $H$ is the slip boundary condition, the Leray-Schauder fixed-point theorem, which have used to investigate the weak solution theory of the Navier-Stokes system, becomes invalid. To address such difficulty, we will invoke the Leray's approximation technique and the perturbation theory to seek a global weak solution to the Cauchy problem for MHD equations with large $L^3$-initial data. Our strategy provides a simple alternative (self-contained) proof of weak $L^3$-solution theory of incompressible Navier-Stokes system. Moreover, this weak solution is unique under some restrictions. |
| title | Global existence and uniqueness of weak solutions for the MHD equations with large $L^3$-initial values |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/2602.06979 |