Abstract We investigate the existence and uniqueness issues of the 3D incompressible Hall-magnetohydrodynamic system supplemented with initial velocity u0 and magnetic field B0 in critical regularity spaces. In the case where u0, B0 and the current belong to the homogeneous Besov space and are small enough, we establish a global result and the conservation of higher regularity. If the viscosity is equal to the magnetic resistivity, then we obtain the global well-posedness provided u0, B0 and J0 are small enough in the larger Besov space If then we also establish the local existence for large data, and exhibit continuation criteria for solutions with critical regularity. Our results rely on an extended formulation of the Hall-MHD system that has some similarities with the incompressible Navier–Stokes equations.

中文翻译：

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临界空间霍尔磁流体动力学系统的适定性

摘要 我们研究了临界规律空间中辅以初速度 u0 和磁场 B0 的 3D 不可压缩霍尔磁流体动力系统的存在性和唯一性问题。在 u0、B0 和电流属于齐次 Besov 空间并且足够小的情况下，我们建立了全局结果和更高正则性的守恒。如果粘度等于磁阻率，那么我们得到全局适定性，前提是 u0、B0 和 J0 在较大的 Besov 空间中足够小 如果那么我们也建立大数据的局部存在性，并展示解的连续准则具有临界规律性。我们的结果依赖于 Hall-MHD 系统的扩展公式，该公式与不可压缩的 Navier-Stokes 方程有一些相似之处。