We establish the existence and uniqueness of a local smooth solution to the Cauchy problem for the Hall-magnetohydrodynamics system that is inviscid, resistive, and forced by multiplicative L\(\acute{\mathrm {e}}\)vy noise in the three dimensional space. Moreover, when the initial data is sufficiently small, we prove that the solution exists globally in time in probability; that is, the probability of the global existence of a unique smooth solution may be arbitrarily close to one given the initial data of which its expectation in a certain Sobolev norm is sufficiently small. The proofs go through for the two and a half dimensional case as well. To the best of the authors’ knowledge, an analogous result is absent in the deterministic case due to the lack of viscous diffusion, exhibiting the regularizing property of the noise. Our result may also be considered as a physically meaningful special case of the extension of work of Kim (J Differ Equ 250:1650–1684, 2011) and Mohan and Sritharan (Pure Appl Funct Anal 3:137–178, 2018) from the first-order quasilinear to the second-order quasilinear system because the Hall term elevates the Hall-magnetohydrodynamics system to the quasilinear class, in contrast to the Naiver–Stokes equations that has most often been studied and is semilinear.

中文翻译：

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L $$ \ acute {\ mathrm {e}} $$ e´vy噪声迫使霍尔磁流体动力学系统的适定性

我们建立了霍尔磁流体力学系统柯西问题的局部光滑解的存在性和唯一性，该霍尔奇磁流体力学系统是无形的，阻力性的，并且由乘法L \（\ acute {\ mathrm {e}} \）强迫三维空间中的vy噪声。此外，当初始数据足够小时，我们证明该解决方案有可能在时间上全局存在；也就是说，在给定初始数据的情况下，唯一光滑解的整体存在的概率可以任意接近某个特定Sobolev范数的期望。证明也适用于二维半个案例。据作者所知，由于缺乏粘性扩散，因此在确定性情况下没有类似的结果，表现出噪声的正则化特性。我们的结果也可能被认为是Kim（J Differ Equ 250：1650-1684，2011）和Mohan和Sritharan（Pure Appl Funct Anal 3：137-178，