Three-dimensional MHD Equations Research Articles

Liouville-type theorems for the stationary inhomogeneous incompressible MHD equations

In this paper, we investigate the Liouville-type problem for the three-dimensional stationary inhomogeneous incompressible MHD equations in the Lorentz spaces without any integrability condition for (∇u,∇B). More precisely, we prove that the velocity and magnetic field (u,B), being in some Lorentz spaces, must be zero provided that the density ρ is essentially bounded.

Space–time decay of solutions to three-dimensional MHD equations with Hall and ion-slip effects

In this paper, we study the space–time decay rate of solutions to three-dimensional incompressible MHD equations with Hall and ion-slip effects in the whole space R3. Based on a parabolic interpolation inequality, bootstrap argument, and some weighted estimates, we obtain the higher order mixed spatial and time derivative estimates for such a system.

Singular limit of the nonisentropic compressible ideal MHD equations in a domain with boundary

ABSTRACT In this paper, we study the singular limit of three-dimensional nonisentropic compressible ideal MHD equations with horizontal boundary conditions. It is rigorously justified that, when the Mach number and Alfvén number are same and tend to zero, the solution of original equations converges to that of two-dimensional incompressible flow.

A regularity criterion of the 3D MHD equations involving one velocity and one current density component in Lorentz space

In this paper, we study the regularity criterion of weak solutions to the three-dimensional (3D) MHD equations. It is proved that the solution $(u,b)$ becomes regular provided that one velocity and one current density component of the solution satisfy% \begin{equation} u_{3}\in L^{\frac{30\alpha }{7\alpha -45}}\left( 0,T;L^{\alpha ,\infty }\left( \mathbb{R}^{3}\right) \right) \text{ \ \ \ with \ \ }\frac{45}{7}% \leq \alpha \leq \infty , \label{eq01} \end{equation}% and \begin{equation} j_{3}\in L^{\frac{2\beta }{2\beta -3}}\left( 0,T;L^{\beta ,\infty }\left( \mathbb{R}^{3}\right) \right) \text{ \ \ \ with \ \ }\frac{3}{2}\leq \beta \leq \infty , \label{eq02} \end{equation}% which generalize some known results.

Energy and cross-helicity conservation for the three-dimensional ideal MHD equations in a bounded domain

In this paper, we prove the energy and cross-helicity conservation of weak solutions to the three-dimensional ideal MHD equations in a bounded domain under the interior Besov regularity conditions which are exactly same as the three-dimensional periodic domain case in [3], and the boundedness and the Besov-type continuity conditions for both the velocity and magnetic fields near the boundary, which seem crucial for the bounded domain case due to the boundary effect. Note that our Besov-type continuity conditions near the boundary are consistent with the interior Besov regularity conditions.

Global well-posedness and asymptotic behavior of solutions for the three-dimensional MHD equations with Hall and ion-slip effects

In this paper, we consider the small initial data global well-posedness of solutions for the magnetohydrodynamics with Hall and ion-slip effects in $$\mathbb {R}^3$$ . In addition, we also establish the temporal decay estimates for the weak solutions. With these estimates in hand, we study the algebraic time decay for higher-order Sobolev norms of small initial data solutions.

Large Time Existence of Special Strong Solutions to MHD Equations in Cylindrical Domains

We investigate the problem of the existence of regular solutions to the three-dimensional MHD equations in cylindrical domains with perfectly conducting boundaries and under the Navier boundary conditions for the velocity field. We show that if the initial and external data do not change too rapidly along the axis of the cylinder, then there exists a unique regular solution for any finite time.

Global solution of 3D axially symmetric nonhomogeneous incompressible MHD equations

We construct a global smooth solution of the nonhomogeneous (density not to be a constant) incompressible axially symmetric MHD equations in three space dimensions. Here we do not require some components of velocity field and magnetic field to be zero, and no other regularity assumptions. Firstly, we propose a new one-dimensional model which approximates the incompressible MHD equations. The nonlinear structure of the one-dimensional has some very interesting properties. Secondly, we find that this one-dimensional model is the one which can construct a family solutions of the three-dimensional axially symmetric MHD equations, what's more we prove the global regularity of the one-dimensional model for a family of initial data. Finally, we prove global regularity of the three-dimensional MHD equations by the special nonlinear structure of one-dimensional model.

Local 4/5-law and energy dissipation anomaly in turbulence of incompressible MHD Equations

In this paper, we establish the longitudinal and transverse local energy balance equation of distributional solutions of the incompressible three-dimensional MHD equations. In particular, we find that the functions \(D_L^\varepsilon (\mathbf {u},\mathbf {B})\) and \(D_T^\varepsilon (\mathbf {u},\mathbf {B})\) appeared in the energy balance, all converging to the defect distribution (in the sense of distributions) \(D(\mathbf {u},\mathbf {B})\) which has been defined in Gao et al. (Acta Math Sci 33:865–871, 2013). Furthermore, we give a simpler form of defect distribution term, which is similar to the relation in turbulence theory, called the “4 / 3-law.” As a corollary, we give the analogous “4 / 5-law” holds in the local sense.

Property of Solution to Axially Symmetric Incompressible MHD Equations in Three Dimensions

In this paper, we study the property of solutions to axially symmetric incompressible MHD equations in three dimensions. First, we present the three-dimensional axially symmetric incompressible MHD equations. We propose a new one-dimensional model that approximates the MHD equations along the symmetric axis. This one-dimensional model can construct a family of exact solutions to the three-dimensional MHD equations. Second, we give a family of particular solutions to the three-dimensional and a prior estimates of one-dimensional MHD equations. Finally, we construct a family of global smooth solutions to the three-dimensional MHD equations by applying the one-dimensional solutions.

A regularity criterion for the three-dimensional MHD equations in terms of one directional derivative of the pressure

In this work, we will establish a sufficient condition for the regularity criterion to the 3D MHD equations in terms of one directional derivative of the pressure. More precisely, we prove that if ∂π∂x3(x,t)∈L21−r(0,T;M⋅2,3r(R3)) for 0<r<1, then the solution (u,b) is regular. We obtain some improvement of the related results in Berselli and Galdi (2002), Cao and Wu (2010), Jia and Zhou (2012), Lin and Du (2013) and Yamazaki (2013).

Axially symmetric incompressible MHD in three dimensions

In this paper, we study the property of solutions to axially symmetric incompressible MHD equations in three dimensions. Firstly, we present the three-dimensional axially symmetric incompressible MHD equations. We propose a new one-dimensional model that approximates the MHD equations along the symmetric axis. To our surprise, this one-dimensional model can construct a family of exact solutions to the three-dimensional MHD equations. Secondly, we give a family of particular solutions to the three-dimensional and the prior estimates of one-dimensional MHD equations. Finally, we construct a family of global smooth solutions to the three-dimensional MHD equations by applying the one-dimensional solutions.

Unconditional convergence of the Euler semi-implicit scheme for the three-dimensional incompressible MHD equations

Unconditional convergence of the Euler semi-implicit scheme for the three-dimensional incompressible MHD equations

Regularity criteria in terms of the pressure for the three-dimensional MHD equations

In this paper, we consider the Cauchy problem for the 3D incompressible MHD equations. A regularity criterion by providing sufficient condition in terms of the pressure is established. More precisely, we show that if p∈Lβ([0,T],Lα(R3)) with 3α+2β⩽74 and 125⩽α⩽4, then any weak solutions (u,b) of the 3D MHD equations is regular on [0,T].

Magnetohydrodynamic simulation of a solar flare: 1. Current sheet in the corona

The results of modeling the preflare situation in the solar corona, obtained using a numerical solution for a complete set of three-dimensional MHD equations, are reviewed. Any assumptions concerning the flare development character or the active region’s behavior before a flare are not introduced. The initial and boundary conditions on the photosphere are specified from magnetic field measurements before a flare. The photospheric field sources are approximated by magnetic dipoles. The usage of the PERESVET program indicated that a current sheet is formed in the vicinity of a singular magnetic field line in the corona. The sheet is formed due to disturbances coming from the photosphere. The energy necessary for a flare is stored in the current sheet magnetic field during 2–3 days. The main construction principles of the PERESVET program, which makes it possible to use the maps of a measured photospheric field as boundary conditions, are presented.

Regularity criteria for the three-dimensional MHD equations

In this paper, we consider regularity criteria for solutions to the 3D MHD equations with incompressible conditions. By using some classical inequalities, we obtain the regularity of strong solutions to the three-dimensional MHD equations under certain sufficient conditions in terms of one component of the velocity field and the magnetic field respectively.

On the self-similar solutions of the magneto-hydro-dynamic equations

In this paper, we show that, for the three dimensional incompressible magneto-hydro-dynamic equations, there exists only trivial backward self-similar solution in Lp(M.3) for p ≥ 3, under some smallness assumption on either the kinetic energy of the self-similar solution related to the velocity field, or the magnetic field. Second, we construct a class of global unique forward self-similar solutions to the three-dimensional MHD equations with small initial data in some sense, being homogeneous of degree −1 and belonging to some Besov space, or the Lorentz space or pseudo-measure space, as motivated by the work in [5].

Recovering Photospheric Velocities from Vector Magnetograms by Using a Three-dimensional, Fully Magnetohydrodynamic Model

We introduce a numerical simulation method for recovering the photospheric velocity field from the vector magnetograms. The traditional method is local correlation tracking (LCT) which is based on measuring the relative displacements of features in blocks of pixels between successive white-light images or magnetograms. Within this method, there are a variety of implementations. One of recently developed implementations is induction local correlation tracking (ILCT) as described by Welsch et al. (2004). They employ the normal component of magnetic induction equation as a constraint to assure consistent solutions. Our numerical method uses the fully three-dimensional MHD equations to recover the photospheric velocity field with individual vector magnetograms. We compare our method to the ILCT method using NOAA AR8210 as an example. The differences and similarities are discussed in detail.

Proton acceleration in a single-loop disrupted during collision of two moving solitary magnetic kinks

We investigate a new model of single-loop flare based on the fact that a magnetic flux tube with axial current can be explosively disrupted during collision of two moving solitary magnetic kinks (MoSMaK), particularly paying attention to an acceleration mechanism of high energy protons. By using three-dimensional electromagnetic fields obtained from a resistive three-dimensional MHD equations during the single-loop flare, we investigate the orbit of many protons to obtain their energy spectra. We found that the protons can be accelerated to γ-ray-emitting energies (>1 MeV) with double power-law spectra up to about 25 MeV. There appears a breaking point of the index of the power-law spectra from 1.8 to 2.1 near 10 MeV. The protons are accelerated mainly in one direction along the loop by the electric field produced near the three-dimensional localized current associated with the magnetic reconnection process in the disrupted loop. We study the loop disruption for two different plasma beta in the center of the loop; and . In a high beta case (), the disruption of the loop is more violent than that of . We found that the proton acceleration for the case of occurs mainly at two different regions; one is inside the loop and the other is in the upper part of the disrupting loop. However, the energy spectra of protons are very similar for both cases.

Test Particle Acceleration in Three-dimensional Magnetohydrodynamic Turbulence

We perform numerical experiments of test particle acceleration on turbulent magnetic and electric fields obtained from pseudospectral direct numerical solutions of the compressible three-dimensional MHD equations. We find consistent acceleration of the particles to many times the plasma characteristic (Alfvén) speed and extended power laws in the density distribution of energies. Scaling laws of maximum and mean energy of particles with the nominal gyrofrequency and the MHD electric field are observed and a simple estimate is presented.