Hall-MHD System Research Articles

A Linearly Implicit Spectral Scheme for the Three-Dimensional Hall-MHD System

A Linearly Implicit Spectral Scheme for the Three-Dimensional Hall-MHD System

The inhomogeneous incompressible Hall-MHD system with only bounded density

The inhomogeneous incompressible Hall-MHD system with only bounded density

The two-and-a-half-dimensional incompressible Hall-MHD system with partial dissipation

In this paper, we investigate the Cauchy problem for the two-and-a-half-dimensional incompressible Hall-magnetohydrodynamic equations with partial dissipation. We prove the global regularity of a certain system of equations with horizontal or vertical dissipation and magnetic hyper-diffusion for large initial data.

On the regularity criteria for the 3D axisymmetric Hall-MHD system in Lorentz spaces

The current paper establishes the regularity criteria for the 3D Hall-MHD system in the axisymmetric setting. When the horizontal angular component of the velocity belongs to some Lorentz spaces, then strong solutions to this system can be smoothly extended beyond the possible blow-up time T. It should be pointed out that our result does not need any additional assumption on the magnetic field.

On the global well-posedness for the compressible Hall-MHD system

In this paper, we prove the global well-posedness of the Cauchy problem to the 3-D compressible Hall-magnetohydrodynamic (MHD) system supplemented with a class of large initial data in Besov spaces. By exploiting the intrinsic structure of the compressible Hall-MHD equations, we obtain the diffusion and the dispersive effects for the mixed hyperbolic-parabolic system. Based on this important observation, we prove that the compressible Hall-MHD system exists a global solution with the initial data close to a stable equilibrium.

Decay rates of the highest order spatial derivatives for solutions of compressible Hall MHD system with or without Coulomb force

In this thesis, we study the decay rates of the highest order (M-th order) spatial derivatives for solutions of both compressible Hall MHD system and that with Coulomb force by a pure energy method inspired by Guo and Wang (2012)[12] and Gao, Li and Yao (2023)[9]. When the norm of H3 for the initial perturbation is small enough, and the norms of both HM(M≥3) and H˙−α(α∈[0,32)) for that only need to be bounded, we first prove the decay rates of compressible Hall MHD system in L2 are (1+t)−(M+α)2. And then affected by the Coulomb force, for compressible Hall MHD system with Coulomb force, we obtain the decay rates are (1+t)−(M+α)2 for α∈[12,32) and (1+t)12[(14−12α)−(M+α)] for α∈[0,12), which are consistent with the decay estimates for compressible MHD system with Coulomb force. In particular, the results improve the original decay rates (1+t)−(M−1+α)2 established by Xu et al. (2016)[46] and Tan, Tong and Wang (2015)[31].

An efficient Hermite-Galerkin spectral scheme for three-dimensional incompressible Hall-magnetohydrodynamic system on infinite domain

The Hall-magnetohydrodynamic (Hall-MHD) system plays a significant role in the fluid description of an astrophysical plasma that features fast magnetic reconnection. Compared to the classical MHD system, the appearance of Hall term brings much more challenges in the field of numerical treatment. The aim of this paper is to construct an efficient spectral scheme for incompressible Hall-MHD system on three-dimensional infinite domain R3. Combining with a novel (HN+2,HN)-Hermite-Galerkin spectral scheme for spatial approximation, a new non-zero constant function approach for the nonlinear terms, a second-order projection method for the momentum equations, and BDF2 scheme for the decoupled system, here we establish, for the first time, a fully decoupled, completely linearized, and unconditionally energy stable scheme with directly spatial approximation in R3 and second-order temporal accuracy for incompressible Hall-MHD system. Numerical results are presented to illustrate the features of the proposed scheme, including convergence/stability tests and simulations of O- and X-points appearing in fast magnetic reconnection within the Hall-MHD regime.

Liouville-type theorems for steady MHD and Hall-MHD equations in [formula omitted

In this paper, we study the Liouville-type theorems for three-dimensional stationary incompressible MHD and Hall-MHD systems in a slab with periodic boundary condition. We show that, under the assumptions that (uθ,bθ) or (ur,br) is axisymmetric, or (rur,rbr) is bounded, any smooth bounded solution to the MHD or Hall-MHD system with local Dirichlet integral growing as an arbitrary power function must be constant. This improves the result of Theorem 1.2 in Pan (2021) [33], where the Dirichlet integral of u is assumed to be finite. Inspired by Bang et al. (2022) [3], our proof relies on establishing Saint-Venant's estimates associated with our problem, and the result in the current paper extends that for stationary Navier-Stokes equations shown by [3] to MHD and Hall-MHD equations. To achieve this, more intricate estimates are needed to handle the terms involving b properly.

Well-Posedness for the Incompressible Hall-MHD System with Initial Magnetic Field Belonging to $$H^{\frac{3}{2}}({\mathbb {R}}^3)$$

In this paper, we first prove the local well-posedness of strong solutions to the incompressible Hall-MHD system for initial data $$(u_0,B_0)\in H^{\frac{1}{2}+\sigma }({\mathbb {R}}^3)\times H^{\frac{3}{2}}({\mathbb {R}}^3)$$ with $$\sigma \in (0,2)$$ . In particular, if the viscosity coefficient is equal to the resistivity coefficient, we can reduce $$\sigma $$ to 0 with the aid of the new formulation of the Hall-MHD system observed by Danchin and Tan (Commun Partial Differ Equ 46(1):31–65, 2021). Compared with the previous works, our local well-posedness results improve the regularity condition on the initial data. Moreover, we establish the global well-posedness for small initial data in $$H^{\frac{1}{2}+\sigma }({\mathbb {R}}^3)\times H^{\frac{3}{2}}({\mathbb {R}}^3)$$ with $$\sigma \in (0,2)$$ , and get the optimal time-decay rates of solutions.

A new blowup criterion for a generalized Hall-MHD system concerning the deformation tensor

The generalized Hall-MHD system in the 3D whole space is studied. A new blowup criterion just in term of the deformation tensor is given.

Global regularity for the 3D axisymmetric incompressible Hall-MHD system with partial dissipation and diffusion

In this paper, we study the Cauchy problem for the 3D incompressible axisymmetric Hall-MHD system with horizontal velocity dissipation and vertical magnetic diffusion.We obtain a unique global smooth solution of which in the cylindrical coordinate system the swirl velocity fields, the radial and the vertical components of the magnetic fields are trivial. This type of solution has been studied for the MHD system in [17][16] and [15] and for the Hall-MHD system with total dissipation and diffusion in [11]. Some new and fine estimates are obtained in this paper to overcome the difficulties raised from the Hall term and the loss of vertical velocity dissipation and horizontal magnetic diffusion. Finally we can show that the estimates $\int_0^T \ \nabla u(t)\ _{L^\infty} dt$ and $\int_0^T \ \nabla b(t)\ _{L^\infty} dt$ are finite in a priori way and hence obtain the global well-posedness to the system under considered.

On a Single-Component Regularity Criterion for the Non-resistive Axially Symmetric Hall-MHD System

On a Single-Component Regularity Criterion for the Non-resistive Axially Symmetric Hall-MHD System

Global Well-Posedness for the 3D Axisymmetric Hall-MHD System with Horizontal Dissipation

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.

Uniform regularity for a density-dependent incompressible Hall-MHD system

This paper proves uniform regularity for a density-dependent incompressible Hall-MHD system with positive density.

Global well-posedness and time-decay of solutions for the compressible Hall-magnetohydrodynamic system in the critical Besov framework

We consider the global existence of solution for the initial value problem for the compressible Hall-magnetohydrodynamic system in the whole space R3. The system consists of a hyperbolic-parabolic system of partial differential equations of the conservation laws type with non-symmetric diffusion. We show the existence of solution as a perturbation from a constant equilibrium state (ρ¯,0,B¯), where ρ¯>0 is a constant density, 0∈R3 is the zero velocity and B¯∈R3 is a constant magnetic field. The time-decay of the solution in the Besov spaces is also established. Our results show the pointwise estimate of the solution in the Fourier space for the linearized Hall-MHD system that related to the result obtained by Umeda–Kawashima–Shizuta [33] for a general class of linear symmetric hyperbolic-parabolic systems with symmetric diffusion. We utilize a systematic use of the product estimates in the Chemin–Lerner spaces and apply the energy method due to Matsumura–Nishida [27].

Energy conservation for the compressible ideal Hall-MHD equations

<abstract><p>In this paper, we study the regularity and energy conservation of the weak solutions for compressible ideal Hall-magnetohydrodynamic (Hall-MHD) system, where $ (t, x)\in(0, T)\times {\mathbb{T}}^d(d\geq\; 1) $. By exploring the special structure of the nonlinear terms in the model, we obtain the sufficient conditions for the regularity of the weak solutions for energy conservation. Our main strategy relies on commutator estimates.</p></abstract>

The global solvability of the Hall-magnetohydrodynamics system in critical Sobolev spaces

We are concerned with the 3D incompressible Hall-magnetohydrodynamic system (Hall-MHD). Our first aim is to provide the reader with an elementary proof of a global well-posedness result for small data with critical Sobolev regularity, in the spirit of Fujita–Kato’s theorem [On the Navier–Stokes initial value problem I, Arch. Ration. Mech. Anal. 16 (1964) 269–315] for the Navier–Stokes equations. Next, we investigate the long-time asymptotics of global solutions of the Hall-MHD system that are in the Fujita–Kato regularity class. A weak-strong uniqueness statement is also proven. Finally, we consider the so-called 2[Formula: see text]D flows for the Hall-MHD system (that is, 3D flows independent of the vertical variable), and establish the global existence of strong solutions, assuming only that the initial magnetic field is small. Our proofs strongly rely on the use of an extended formulation involving the so-called velocity of electron [Formula: see text] and as regards [Formula: see text]D flows, of the auxiliary vector-field [Formula: see text] that comes into play in the total magneto-helicity balance for the Hall-MHD system.

Global Small Solutions to the Inviscid Hall-MHD System

The local existence of smooth solutions to the inviscid Hall-MHD system has been obtained since Chae, Degond and Liu [Ann. Inst. H. Poincar\'e Anal. Non Lin\'eaire, {31} (2014), 555--565]. However, as we known, how to construct the global small solutions to the inviscid Hall-MHD system is still an open problem. In the present paper, we give a positive answer in $ \mathbb{T}^3$ when the initial magnetic field is close to a background magnetic field satisfying the Diophantine condition.

On Liouville type theorems for the stationary MHD and Hall-MHD systems

In this paper we prove a Liouville type theorem for the stationary magnetohydrodynamics (MHD) system in R 3 . Let ( v , B , p ) be a smooth solution to the stationary MHD equations in R 3 . We show that if there exist smooth matrix valued potential functions Φ , Ψ such that ∇ ⋅ Φ = v and ∇ ⋅ Ψ = B , whose L 6 mean oscillations have certain growth condition near infinity, namely ⨍ B ( r ) | Φ − Φ B ( r ) | 6 d x + ⨍ B ( r ) | Ψ − Ψ B ( r ) | 6 d x ≤ C r ∀ 1 < r < + ∞ , then v = B = 0 and p =constant. With additional assumption of r − 8 ∫ B ( r ) | B − B B ( r ) | 6 d x → 0 as r → + ∞ , similar result holds also for the Hall-MHD system.

Global smooth solutions to the 3D generalized Hall-magnetohydrodynamic equations with large data

In this paper, based on a perturbation argument and fully exploring the nonlinear structure of the considered system, we establish the global existence to the three-dimensional generalized incompressible Hall-MHD equations for a class of large initial data, whose norms can be arbitrarily large. Our obtained result improves considerably the recent result in Li et al. [A class large solution of the 3D Hall-magnetohydrodynamic equations. J Differ Equ. 2020;268(10):5811–5822] to the generalized Hall-MHD system with equipping with different viscosity and magnetic diffusion coefficients.