Well-posedness of Hall-magnetohydrodynamics system forced by L $$\acute{\mathrm{e}}$$ e ´ vy noise

Abstract

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.