Low regularity solutions for the Cauchy problem of the ideal incompressible Magnetohydrodynamics equations
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866917247206817792 |
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| author | Zhang, Huali |
| author_facet | Zhang, Huali |
| contents | In Lagrangian coordinates, the local well-posedness of low regularity solutions is established for an ideal incompressible magnetohydrodynamic (MHD) system subject to a homogeneous background magnetic field. First, the MHD system is reformulated into a degenerate wave-elliptic system with a particular null structure. By introducing a suitably defined solution space, several refined product estimates are derived. Next, using the inherent null structure, a Klainerman-Machedon type bilinear estimate is obtained for the nonlinear terms. These nice structures and estimates yield the local well-posedness of the ideal incompressible MHD equations in Lagrangian coordinates for initial velocity fields $\bv_0 \in H^{s}(\mathbb{R}^n)$ with $s > \frac{n+1}{2}$ $(n=2,3,4)$. Moreover, the regularity requirement is lowered by half a derivative compared with the classical exponent $s > \frac{n}{2}+1$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2407_20531 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Low regularity solutions for the Cauchy problem of the ideal incompressible Magnetohydrodynamics equations Zhang, Huali Analysis of PDEs In Lagrangian coordinates, the local well-posedness of low regularity solutions is established for an ideal incompressible magnetohydrodynamic (MHD) system subject to a homogeneous background magnetic field. First, the MHD system is reformulated into a degenerate wave-elliptic system with a particular null structure. By introducing a suitably defined solution space, several refined product estimates are derived. Next, using the inherent null structure, a Klainerman-Machedon type bilinear estimate is obtained for the nonlinear terms. These nice structures and estimates yield the local well-posedness of the ideal incompressible MHD equations in Lagrangian coordinates for initial velocity fields $\bv_0 \in H^{s}(\mathbb{R}^n)$ with $s > \frac{n+1}{2}$ $(n=2,3,4)$. Moreover, the regularity requirement is lowered by half a derivative compared with the classical exponent $s > \frac{n}{2}+1$. |
| title | Low regularity solutions for the Cauchy problem of the ideal incompressible Magnetohydrodynamics equations |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/2407.20531 |