Global existence and partial regularity for the Landau-Lifshitz-Gilbert equation with helicity. (English) Zbl 07922134
Summary: In this paper, the Landau-Lifshitz-Gilbert equation with helicity is considered. In \(\mathbb{R}^3\) or a bounded regular domain \(\Omega\) of \(\mathbb{R}^3\), we establish the global existence of a weak solution. In \(\mathbb{R}^n\), a global existence criterion and uniqueness of the smooth solution are given. In \(\mathbb{R}^1\), the local smooth solution is indeed global with large initial data. In \(\mathbb{R}^2\), we prove the existence of a global weak solution, which is smooth with the exception of at most finite singular points.
MSC:
| 35Qxx | Partial differential equations of mathematical physics and other areas of application |
| 35Kxx | Parabolic equations and parabolic systems |
| 58Exx | Variational problems in infinite-dimensional spaces |
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