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Large‐amplitude magnetic pulsations on ion inertial length scales are often observed in space plasmas, but their theoretical explanation is still controversial. We discuss a possible mechanism, different from ideas based on the classical plasma instabilities, for the generation of these pulsations. It is demonstrated that a competition between dispersion and wave steepening processes can lead to the transformation of a large‐scale magnetosonic structure into trains of solitons. This kind of longitudinal filamentation is possible for both slow and fast magnetosonic perturbations. Results of numerical simulations are compared with Cluster spacecraft measurements and show that the steepening filamentation mechanism can explain the emergence of a certain class of solitary waves observed in space plasmas. Key Points We show how solitary structures may evolve from magnetosonic perturbations Proposed mechanism works for both slow and fast structures We provide an analysis of experimental data supporting our simulation results

In turbulence, for neutral or conducting fluids, a large ratio of scales is excited because of the possible occurrence of inverse cascades to large, global scales together with direct cascades to small, dissipative scales, as observed in the atmosphere and oceans, or in the solar environment. In this context, using direct numerical simulations with forcing, we analyze scale dynamics in the presence of magnetic fields with a generalized Ohm’s law including a Hall current. The ion inertial length ϵ H serves as the control parameter at fixed Reynolds number. Both the magnetic and generalized helicity—invariants in the ideal case—grow linearly with time, as expected from classical arguments. The cross-correlation between the velocity and magnetic field grows as well, more so in relative terms for a stronger Hall current. We find that the helical growth rates vary exponentially with ϵ H , provided the ion inertial scale resides within the inverse cascade range. These exponential variations are recovered phenomenologically using simple scaling arguments. They are directly linked to the wavenumber power-law dependence of generalized and magnetic helicity, ∼ k − 2 , in their inverse ranges. This illustrates and confirms the important role of the interplay between large and small scales in the dynamics of turbulent flows.

In this note, we investigate Liouville-type theorems for the steady three-dimensional MHD and Hall-MHD equations and show that the velocity field u and the magnetic field B are vanishing provided that B ∈ L 6 , ∞ ( R 3 ) and u ∈ B M O - 1 ( R 3 ) , which state that the velocity field plays an important role. Moreover, the similar result holds in the case of partial viscosity or diffusivity for the three-dimensional MHD equations.

This paper concerns the global energy conservation for distributional solutions to incompressible Hall-MHD equations without resistivity. Motivated by the works of Tan and Wu in [arXiv:2111.13547v2] and Wu in [J. Math. Fluid Mech. 24, 111 (2022)], we establish the energy balance for a distributional solution in whole spaces ℝ d (d ≥ 2) provided that ∇ b ∈ L 8 3 L 8 3 . Moreover, as a corollary, we also obtain the energy conservation criterion for a Leray-Hopf weak solution.

In this paper, we focus on the well-posedness problem of the three-dimensional incompressible viscous and resistive Hall-magnetohydrodynamics system (Hall-MHD) with variable density. We mainly prove the existence and uniqueness issues of the density-dependent incompressible Hall-magnetohydrodynamic system in critical spaces on .

The Hall-magnetohydrodynamics (Hall-MHD) equations, rigorously derived from kinetic models, are useful in describing many physical phenomena in geophysics and astrophysics. This paper studies the local well-posedness of classical solutions to the Hall-MHD equations with the magnetic diffusion given by a fractional Laplacian operator, ( - Δ ) α . Due to the presence of the Hall term in the Hall-MHD equations, standard energy estimates appear to indicate that we need α ≥ 1 in order to obtain the local well-posedness. This paper breaks the barrier and shows that the fractional Hall-MHD equations are locally well-posed for any α > 1 2 . The approach here fully exploits the smoothing effects of the dissipation and establishes the local bounds for the Sobolev norms through the Besov space techniques. The method presented here may be applicable to similar situations involving other partial differential equations.

This paper is concerned with the Hall effect on the asymptotic stability of the global solutions to an initial-boundary value problem of the planar compressible MHD system. In the case when the heat conductivity depends on the temperature in the form κ ( θ ) = θ β with β ∈ ( 0 , + ∞ ) , we show that the global large solutions decay exponentially in time to the equilibrium states without any restriction on the Hall coefficient ε . The exponential stability of the global large solutions still holds when the heat conductivity is a positive constant, provided the Hall coefficient is suitably small. As by-products, the vanishing limit of Hall coefficient is also justified in both cases.

In this paper, we consider the initial value problem of the incompressible Hall-MHD system and prove the global well-posedness in the scaling critical class B ˙ p , ∞ - 1 + 3 p ( R 3 ) × ( B ˙ p , ∞ - 1 + 3 p ( R 3 ) ∩ L ∞ ( R 3 ) ) for 3 < p < ∞ . Moreover, we also refine the smallness conditions and show that our global well-posedness holds for initial data whose B ˙ p , ∞ - 1 + 3 p ( R 3 ) -norm is large, provided that some weaker norm is sufficiently small.

In this paper, we deal with the 2 1 2 dimensional Hall MHD by taking the velocity field u and the magnetic field B of the form u ( t , x , y ) = ∇ ⊥ ϕ ( t , x , y ) , W ( t , x , y ) and B ( t , x , y ) = ∇ ⊥ ψ ( t , x , y ) , Z ( t , x , y ) . We begin with the Hall equations (without the effect of the fluid part). In this case, we provide several results such as the long time behavior of weak solutions, weak-strong uniqueness, the existence of local and global in time strong solutions, decay rates of ( ψ , Z ) , the asymptotic profiles of ( ψ , Z ) , and the perturbation around harmonic functions. In the presence of the fluid field, the results, by comparison, fall short of the previous ones in the absence of the fluid part and we show the existence of local and global in time strong solutions.

Studied in this paper is the Cauchy problem for the 3D incompressible Hall-MHD system with horizontal dissipation. It is shown that if the initial data is axisymmetric and the swirl component of the velocity and the magnetic vorticity are trivial, such a system is globally well-posed for the large initial data. The key is to take full advantage of the structure of the Hall-MHD system in axisymmetric case to overcome the main difficulty due to the absence of vertical dissipation.