Finite-time singularity formation at a magnetic neutral line in Hall magnetohydrodynamics

The formation of a current sheet in a weakly collisional plasma can be modelled as a finite-time singularity solution of magnetohydrodynamic equations. We use an exact self-similar solution to confirm and generalise a previous finding that, in sharp contrast to two-dimensional solutions in standard MHD, a finite-time collapse to a current sheet can occur in Hall MHD. We derive a criterion for the finite-time singularity in terms of initial conditions, and we use an intermediate asymptotic solution for the evolution of an axial magnetic field to obtain a general expression for the singularity formation time. We illustrate the analytical results by numerical solutions

On the blow-up criterion of strong solutions for the MHD equations with the Hall and ion-slip effects in {\mathbb{R}^{3}}

In this paper, we establish a blow-up criterion of strong solutions to the 3D incompressible magnetohydrodynamics equations including two nonlinear extra terms: the Hall term (quadratic with respect to the magnetic field) and the ion-slip term (cubic with respect to the magnetic field). This is an improvement of the recent results given by Fan et al. (Z Angew Math Phys, 2015)

Global existence and temporal decay for the 3D compressible Hall-magnetohydrodynamic system

In this paper, we are concerned with the 3D compressible Hall-magnetohydrodynamic system in the whole space. We prove the global existence and temporal decay rates of the solutions to the system when the initial data are close to a stable equilibrium state by using a pure energy method

Global weak solutions to the density-dependent Hall-magnetohydrodynamics system

We are concerned with the global existence of finite energy weak solutions to 3D density-dependent magnetohydrodynamics (MHD) system with Hall-effect set in a general smooth bounded domain. The perfectly conducting wall boundary condition is imposed on the magnetic field. Due to the degeneracy of Hall-effect term (a tri-linear term) in vacuum, we assumed initial density lies in the bounded function space and having a positive lower bound. Particularly, these bounds are preserved by the density transport equation which helps yield a satisfying compactness argument of the magnetic field

The Effects of Large-Scale Magnetic Fields on Disk Formation and Evolution

This is a draft chapter for a book, entitled Physical Processes in Circumstellar Disks around Young Stars, which is scheduled for publication by the University of Chicago Press as one of its Theoretical Astrophysics Series volumes. Sect. 1 presents the motivation for considering the effects of a large-scale, ordered magnetic field on the formation and evolution of protostellar disks. Sect. 2 outlines the physical principles that underlie the magnetohydrodynamics of disks that are threaded by such a field. Sect. 3 discusses the formation and early evolution of disks that result from the collapse of a rotating molecular cloud core that is coupled to the insterstellar magnetic field. Sect. 4 reviews the observational evidence for the disk--wind connection and describes the structure of magnetically accelerated disk outflows, focusing on centrifugally driven winds; it then goes on to discuss the equilibrium and stability properties of weakly ionized protostellar accretion disks in which the transport of angular momentum is dominated by a wind of this type. Sect. 5 considers the coupling between the central protostar and the surrounding disk through the protostellar magnetic field, covering, in turn, the phenomenology, basic concepts, and results of numerical simulations. The chapter is summarized in Sect. 6, which also contains a discussion of future research directions.Comment: 68 pages, 8 figures, to appear in Physical Processes in Circumstellar Disks around Young Stars, ed. P. J. V. Garcia (Chicago: University of Chicago Press), uses svmult.cl