The Cauchy problem for 3D incompressible Hall-magnetohydrodynamics (Hall-MHD) system with fractional Laplacians \((-\Delta )^{\frac{1}{2}}\) is studied. The well-posedness of 3D incompressible Hall-MHD equations remains an open problem with fractional diffusion \((-\Delta )^{\frac{1}{2}}\). First, global well-posedness of small-energy solutions with general initial data in \(H^s\), \(s>\frac{5}{2}\), is proved. Second, a special class of large-energy initial data is constructed, with which the Cauchy problem is globally well-posed. The proofs rely upon a new global bound of energy estimates involving Littlewood–Paley decomposition and Sobolev inequalities, which enables one to overcome the \(\frac{1}{2}\)-order derivative loss of the magnetic field.

研究了具有分数拉普拉斯算子\((-\Delta )^{\frac{1}{2}}\) 的3D 不可压缩霍尔磁流体动力学 (Hall-MHD) 系统的柯西问题。3D 不可压缩 Hall-MHD 方程的适定性仍然是分数扩散\((-\Delta )^{\frac{1}{2}}\) 的一个悬而未决的问题。首先，证明了在\(H^s\) , \(s>\frac{5}{2}\) 中具有一般初始数据的小能量解的全局适定性。其次，构造了一类特殊的大能量初始数据，柯西问题是全局适定的。证明依赖于新的全球能量估计范围，包括 Littlewood-Paley 分解和 Sobolev 不等式，这使人们能够克服\(\frac{1}{2}\)- 磁场的阶导数损失。