3D Magnetohydrodynamic Equations Research Articles

Existence and uniqueness of low-energy weak solutions to the compressible 3D magnetohydrodynamics equations

We prove the existence and uniqueness of weak solutions of the three dimensional compressible magnetohydrodynamics (MHD) equations. We first obtain the existence of weak solutions with small L2-norm which may display codimension-one discontinuities in density, pressure, magnetic field and velocity gradient. The weak solutions we consider here exhibit just enough regularity and structure which allow us to develop uniqueness and continuous dependence theory for the compressible MHD equations. Our results generalise and extend those for the intermediate weak solutions of compressible Navier-Stokes equations.

Discontinuity of weak solutions to the 3D NSE and MHD equations in critical and supercritical spaces

We demonstrate that the three dimensional incompressible magneto-hydrodynamics (MHD) system is ill-posed due to the discontinuity of weak solutions in a wide range of spaces. Specifically, we construct initial data which has finite energy and is small in certain spaces, such that any Leray-Hopf type of weak solution to the MHD system starting from this initial data is discontinuous at time t=0 in such spaces. An analogous result is also obtained for the Navier-Stokes equation which extends the previous result of ill-posedness in B˙∞,∞−1 by Cheskidov and Shvydkoy to spaces that are not necessarily critical. The region of the spaces where the norm inflation occurs almost touches L2.

Plasmoid-mediated reconnection in solar UV bursts

Context. Ultraviolet bursts are transients in the solar atmosphere with an increased impulsive emission in the extreme UV lasting for one to several tens of minutes. They often show spectral profiles indicative of a bi-directional outflow in response to magnetic reconnection. Aims. To understand UV bursts, we study how motions of magnetic elements at the surface can drive the self-consistent formation of a current sheet resulting in plasmoid-mediated reconnection. In particular, we want to study the role of the height of the reconnection in the atmosphere. Methods. We conducted numerical experiments solving the 2D magnetohydrodynamic equations from the solar surface to the upper atmosphere. Motivated by observations, we drove a small magnetic patch embedded in a larger system of magnetic field of opposite polarity. This type of configuration creates an X-type neutral point in the initial potential field. The models are characterized by the (average) plasma-β at the height of this X point. Results. The driving at the surface stretches the X-point into a thin current sheet, where plasmoids appear, accelerating the reconnection, and a bi-directional jet forms. This is consistent with what is expected for UV bursts or explosive events, and we provide a self-consistent model of the formation of the reconnection region in such events. The gravitational stratification gives a natural explanation for why explosive events are restricted to a temperature range around a few 0.1 MK, and the presence of plasmoids in the reconnection process provides an understanding of the observed variability during the transient events on a timescale of minutes. Conclusions. Our numerical experiments provide a comprehensive understanding of UV bursts and explosive events, in particular of how the atmospheric response changes if the reconnection happens at different plasma-β, that is, at different heights in the atmosphere. This analysis also gives new insight into how UV bursts might be related to the photospheric Ellerman bombs.

Remarks on the global regularity and time decay of the 2D MHD equations with partial dissipation

This paper focuses on a system of the two‐dimensional (2D) magnetohydrodynamic (MHD) equations with the partial kinematic dissipation (∂yyu1,∂xxu2) and the partial magnetic diffusion (∂yyb1,∂xxb2). Based on the basic energy estimates only, we are able to show that this system always possesses a unique global smooth solution when the initial data are sufficiently smooth. Moreover, we obtain optimal large‐time decay rates of both solutions and their higher order derivatives by developing the classic Fourier splitting methods together with the auxiliary decay estimates of the first derivative of solutions and induction technique.

Stability for a system of the 2D magnetohydrodynamic equations with partial dissipation

This paper examines the stability problem on perturbations near a physically important steady state solution of the 2D magnetohydrodynamic (MHD) system with only partial dissipation. We obtain two main results. The first assesses the asymptotic linear stability with explicit decay rates while the second affirms the global stability in the Sobolev space H1 setting.

Global regularity for 3D magneto-hydrodynamics equations with only horizontal dissipation

We investigate the Cauchy problem for the 3D magneto-hydrodynamics equations with only horizontal dissipation for the small initial data. With the help of the dissipation in the horizontal direction and the structure of the system, we analyze the properties of the decay of the solution and apply these decay properties to get the global regularity of the solution. In the process, we mainly use the frequency decomposition in Green's function method and energy method.

On the Gevrey regularity of solutions to the 3D ideal MHD equations

In this paper, similar to the incompressible Euler equation, we prove the propagation of the Gevrey regularity of solutions to the three-dimensional incompressible ideal magnetohydrodynamics (MHD) equations. We also obtain an uniform estimate of Gevery radius for the solution of MHD equation.

Numerical efficacy study of data assimilation for the 2D magnetohydrodynamic equations

We study the computational efficiency of several nudging data assimilation algorithms for the 2D magnetohydrodynamic equations, using varying amounts and types of data. We find that the algorithms work with much less resolution in the data than required by the rigorous estimates in [7]. We also test other abridged nudging algorithms to which the analytic techniques in [7] do not seem to apply. These latter tests indicate, in particular, that velocity data alone is sufficient for synchronization with a chaotic reference solution, while magnetic data alone is not. We demonstrate that a new nonlinear nudging algorithm, which is adaptive in both time and space, synchronizes at a super exponential rate.

Global well‐posedness of classical solutions to 3D compressible magnetohydrodynamic equations with large potential force

We consider the Cauchy problem of isentropic compressible magnetohydrodynamic equations with large potential force in . When the initial data (ρ0,u0,H0) is of small energy, we investigate the global well‐posedness of classical solutions where the flow density is allowed to contain vacuum states.

Local existence and blow-up criterion of 3D ideal magnetohydrodynamics equations

We investigate the local existence of smooth solutions of a 3D ideal magnetohydrodynamics (MHD) equations in a bounded domain and give a blow-up criteria to this equations with respect to vorticists.

A 3D cell-centered Lagrangian scheme for the ideal Magnetohydrodynamics equations on unstructured meshes

We propose a cell-centered Lagrangian numerical scheme for solving the 3D ideal Magnetohydrodynamics (MHD) equations on unstructured meshes. All the variables are cell-centered and the Lagrangian conservative system of ideal MHD equations is compatibly discretized in this scheme. The geometric conservation law (GCL) requirement is satisfied by using the discrete compatible Lagrangian framework of polyhedral meshes. A nodal solver for MHD equations is then constructed through the nodal pressure flux and magnetic field invoking total energy conservation and thermodynamic consistency. Besides, the magnetic divergence free constraint is fulfilled by a projection method after each time step. Various numerical tests are presented to assert the robustness and accuracy of our scheme.

Analyticity of Mild Solution for the 3D Incompressible Magneto-hydrodynamics Equations in Critical Spaces

The Cauchy problem for the 3D incompressible magneto-hydrodynamics equations in critcal spaces is considered. We first prove the global well-posedness of mild solution for the system in some time dependent spaces. Furthermore, we obtain analyticity of the solution.

Global regularity of the 2D magnetohydrodynamic equations with fractional anisotropic dissipation

This paper is dedicated to establishing the global regularity for the two dimensional magnetohydrodynamic equations with fractional anisotropic dissipation when the fractional powers are restricted to some certain ranges. In addition, the global regularity results for the two dimensional magnetohydrodynamic equations with partial dissipation are also obtained. Consequently, these results bring us more closer to the resolution of the global regularity problem on the two dimensional magnetohydrodynamic equations with standard Laplacian magnetic diffusion.

The 3D incompressible magnetohydrodynamic equations with fractional partial dissipation

The magnetohydrodynamic (MHD) equations have played pivotal roles in the study of many phenomena in geophysics, astrophysics, cosmology and engineering. The fundamental problem of whether or not classical solutions of the 3D MHD equations can develop finite-time singularities remains an outstanding open problem. Mathematically this problem is supercritical in the sense that the 3D MHD equations do not have enough dissipation. If we replace the standard velocity dissipation Δu and the magnetic diffusion Δb by −(−Δ)αu and −(−Δ)βb, respectively, the resulting equations with α≥54 and α+β≥52 then always have global classical solutions. An immediate issue is whether or not the hyperdissipation can be further reduced. This paper shows that the global regularity still holds even if there is only directional velocity dissipation and horizontal magnetic diffusion −(−Δh)54b, where Δh=∂12+∂22.

Finite element analysis of effects of Joule heating, electro-osmosis and slip conditions on the performance of magnetohydrodynamic (MHD) micropumps

Magnetohydrodynamic (MHD) micropump which is energy saving and less pollution has been widely investigated by researchers in recent years. Based on the finite element analysis (FEA) of numerical model of 3D MHD micropump, we investigated the comprehensive effects of Joule heating, electro-osmosis and slip condition at walls on flow performance. This paper presents a new method for solving the full 3D MHD equations by coupling multi-physics fields, and gives a more comprehensive analysis of fundamental principles of MHD micropump. The temperature distribution under different flow channel geometries, the electro-osmotic velocity in MHD micropump, and the influence of slip condition at walls on flow velocity are analyzed in detail. The flow velocity slightly increases when taking the effect of Joule heating into account. In addition, the flow velocity is hardly affected by electron-osmosis in this work. It should be pointed out that the mean temperature of MHD micropump decreases with driven voltage increasing. With the advantage of fast and accurate analysis of MHD micropump performance, this study holds promising potential for the optimal design and fundamental research of MHD micropump.

Global Regularity and Time Decay for the 2D Magnetohydrodynamic Equations with Fractional Dissipation and Partial Magnetic Diffusion

This paper focuses on a system of the 2D magnetohydrodynamic (MHD) equations with the kinematic dissipation given by the fractional operator $$(-\Delta )^\alpha $$ and the magnetic diffusion by partial Laplacian. We are able to show that this system with any $$\alpha >0$$ always possesses a unique global smooth solution when the initial data is sufficiently smooth. In addition, we make a detailed study on the large-time behavior of these smooth solutions and obtain optimal large-time decay rates. Since the magnetic diffusion is only partial here, some classical tools such as the maximal regularity property for the 2D heat operator can no longer be applied. A key observation on the structure of the MHD equations allows us to get around the difficulties due to the lack of full Laplacian magnetic diffusion. The results presented here are the sharpest on the global regularity problem for the 2D MHD equations with only partial magnetic diffusion.

Global well-posedness for the 3D viscous nonhomogeneous incompressible magnetohydrodynamic equations

In this paper, we investigate the Cauchy problem for the 3D viscous nonhomogeneous incompressible magnetohydrodynamic equations in critical Besov spaces. We aim at proving the local and global well-posedness for respectively large and small initial data having critical Besov regularity, without assumptions of small density variation.

A regularity criterion for the 3D incompressible magneto-hydrodynamics equations

The paper is dedicated to study of the regularity criterion for weak solutions to the 3D incompressible MHD equations. Employing the Littlewood–Paley decomposition, we show that if ∇˜u˜=(∂1u˜,∂2u˜)∈Ls1([0,T);B˙r1,2r130(R3)), 2s1+3r1=2, 32<r1≤∞ and ∇˜b˜=(∂1b˜,∂2b˜)∈Ls2([0,T);B˙r2,2r230(R3)), 2s2+3r2=2, 32<r2≤∞, then the solutions to the MHD actually is smooth on (0,T).

Global Regularity for the 2D MHD Equations with Partial Hyper-resistivity

AbstractThis article establishes the global existence and regularity for a system of the two-dimensional (2D) magnetohydrodynamic (MHD) equations with only directional hyper-resistivity. More precisely, the equation of $b_1$ (the horizontal component of the magnetic field) involves only vertical hyperdiffusion (given by $\Lambda_2^{2\beta} b_1$) while the equation of $b_2$ (the vertical component) has only horizontal hyperdiffusion (given by $\Lambda_1^{2\beta} b_2$), where $\Lambda_1$ and $\Lambda_2$ are directional Fourier multiplier operators with the symbols being $|\xi_1|$ and $|\xi_2|$, respectively. We prove that, for $\beta&amp;gt;1$, this system always possesses a unique global-in-time classical solution when the initial data is sufficiently smooth. The model concerned here is rooted in the MHD equations with only magnetic diffusion, which play a significant role in the study of magnetic reconnection and magnetic turbulence. In certain physical regimes and under suitable scaling, the magnetic diffusion becomes partial (given by part of the Laplacian operator). There have been considerable recent developments on the fundamental issue of whether classical solutions of these equations remain smooth for all time. The papers of Cao–Wu–Yuan [8] and of Jiu–Zhao [26] obtained the global regularity when the magnetic diffusion is given by the full fractional Laplacian $(-\Delta)^\beta$ with $\beta&amp;gt;1$. The main result presented in this article requires only directional fractional diffusion and yet we prove the regularization in all directions. The proof makes use of a key observation on the structure of the nonlinearity in the MHD equations and technical tools on Fourier multiplier operators such as the Hörmander–Mikhlin multiplier theorem. The result presented here appears to be the sharpest for the 2D MHD equations with partial magnetic diffusion.

Magneto-hydrodynamical model for plasma

Based on the Newton's second law and the Maxwell equations for the electromagnetic fields, we establish a new 3D incompressible magneto-hydrodynamics(MHD) equations for the motion of plasma under the standard Coulomb gauge. By using the Galerkin method, we prove a result on the existence of weak solutions for the new model.