MHD System Research Articles

Exact boundary controllability for the ideal magneto-hydrodynamic equations

We address the problem of controllability of the MHD system in a rectangular domain with a control prescribed on the side boundary. We identify a necessary and sufficient condition on the data to be null-controllable, i.e., can be driven to the zero state. We also show that the validity of this condition allows the states to be stirred to each other. If the condition is not satisfied, one can move from one state to another with the help of a simple shear external magnetic force.

Global Well-Posedness for the Three-Dimensional Full Compressible Viscous Non-resistive MHD System

In this paper, we consider the three-dimensional full compressible viscous non-resistive MHD system. Global well-posedness is proved for an initial-boundary value problem around a strong background magnetic field. It is also shown that the unique solution converges to the steady state at an almost exponential rate as time tends to infinity. The proof is based on the celebrated two-tier energy method, due to Guo and Tice [\emph{Arch. Ration. Mech. Anal.}, 207 (2013), pp. 459--531; \emph{Anal. PDE.}, 6 (2013), pp. 287--369], reformulated in Lagrangian coordinates. The obtained result may be viewed as an extension of Tan and Wang [\emph{SIAM J. Math. Anal.}, 50 (2018), pp. 1432--1470] to the context of heat-conductive fluids. This in particular verifies the stabilization effects of vertical magnetic field in the full compressible non-resistive fluids.

Optimal large-time behavior of the full 3D compressible MHD system

We investigate optimal large-time behavior for higher-order spatial derivatives of strong solutions to the 3D full compressible MHD system. More precisely, we employ low-frequency and high-frequency decomposition and delicate energy estimates to show that

Study on the Gas-Liquid Two-Phase Flow Regime in the Power Generation Channel of a Liquid Metal Mhd System

Study on the Gas-Liquid Two-Phase Flow Regime in the Power Generation Channel of a Liquid Metal Mhd System

Global well-posedness of 2D incompressible Magnetohydrodynamic equations with horizontal dissipation

<p style='text-indent:20px;'>This paper focuses on two-dimensional incompressible non-resistive MHD equations with only horizontal dissipation in <inline-formula><tex-math id="M1">\begin{document}$ \mathbb{T}\times\mathbb{R} $\end{document}</tex-math></inline-formula>. Invoking three Poincaré-type inequalities about the horizontal derivative, we study the global well-posedness of the system near a background magnetic via the structure of the perturbation MHD system and the symmetry condition imposed on the initial data. By a precise time-weighted energy estimate, we also establish the global well-posedness of the system with only horizontal magnetic damping. Here we overcome the difficulties brought by the absence of magnetic diffusion and the appearance of the boundary. We note that the stability of MHD equations with one-directional dissipation in <inline-formula><tex-math id="M2">\begin{document}$ \mathbb{R}^2 $\end{document}</tex-math></inline-formula> or a bounded domain appears to be unknown.</p>

Hydrostatic approximation of the 2D MHD system in a thin strip with a small analytic data

In this paper, we study the global well-posedness of the two-dimensional rescaled anisotropic MHD system and also the hydrostatic MHD with no-slip boundary condition in a strip, for small initial data, which are analytic in the horizontal variable. We also justify the limit of the MHD system (when ϵ→0 where ϵ denotes the height of the initial strip) is given by a couple of a Prandtl like a system for the velocity and a magnetic field equation when the initial data are small and analytic.

Note on energy equality of MHD system

Note on energy equality of MHD system

The analytical solution of the MHD equations via Minkowski 2-D space continuum of liquid metals through a square duct under multiple transport effects

The problem of mass, heat transfer due the steady, laminar for mercury and gallium liquids flow in to a square duct in the presence of transverse magnetic field and ultrasonic fields, is tackled analytically resorting to the wave-position ansatz as the unique independent variable in the 2-D Minkowskiinner product space. The MHD system of partial differential equations is solved by assuming the absence of slip boundary conditions. Each of duct wall is electrically insulated, and have a constant temperature and, the fluid is under a constant pressure gradient. The resulting solutions are visualized graphically for the velocity, temperature distribution, induced magnetic field, entropy generation rate and Bejan numbers, which are illustrated according to different suggested values corresponding to the affected fields. Good agreements are found with previous studies.

Global strong solution to the 2D inhomogeneous incompressible magnetohydrodynamic fluids with density-dependent viscosity and vacuum

In this paper, we investigate an initial boundary value problem for two-dimensional inhomogeneous incompressible MHD system with density-dependent viscosity. First, we establish a blow-up criterion for strong solutions with vacuum. Precisely, the strong solution exists globally if |nabla mu (rho )|_{L^{infty }(0, T; L^{p})} is bounded. Second, we prove the strong solution exists globally (in time) only if |nabla mu (rho _{0})|_{L^{p}} is suitably small, even the presence of vacuum is permitted.

Long-Time Behavior of Alfvén Waves in a Flowing Plasma: Generation of the Magnetic Island

Magnetic islands are the regions enclosed by magnetic field lines and separated by reconnection points. In this paper, we study the long-time behavior of the solution for the linearized MHD system around the linearly stable, steady flowing plasma with sheared velocity and magnetic field. As a consequence, we prove that for a class of initial data, the magnetic islands appear at the final state.

Existence in critical spaces for the magnetohydrodynamical system in 3D bounded Lipschitz domains

Existence of mild solutions for the 3D MHD system in bounded Lipschitz domains is established in critical spaces with the absolute boundary conditions.

Existence of Solutions to Steady MHD System with Multiply Connected Boundary

Existence of Solutions to Steady MHD System with Multiply Connected Boundary

MHD Equations in a Bounded Domain

Abstract We consider the MHD system in a bounded domain Ω ⊂ ℝ N , N = 2; 3, with Dirichlet boundary conditions. Using Dan Henry’s semigroup approach and Giga–Miyakawa estimates we construct global in time, unique solutions to fractional approximations of the MHD system in the base space (L 2(Ω )) N × (L 2(Ω )) N . Solutions to MHD system are obtained next as a limits of that fractional approximations.

Global small solutions to the 3D MHD system with a velocity damping term

To our knowledge, how to construct the global small solutions to the 3D MHD system with only one damping term is still an open problem. Motivated by Chen et al. [1], we give a positive answer in T3 when the initial magnetic field is close to a background magnetic field satisfying the Diophantine condition.

Large-time Behavior of Magnetohydrodynamics with Temperature-Dependent Heat-Conductivity

For the strong solutions to the equations of a planar magnetohydrodynamic compressible flow with the heat conductivity proportional to a nonnegative power of the temperature, we first prove that both the specific volume and the temperature are proved to be bounded from below and above independently of time. Then, we also show that the global strong solution is nonlinearly exponentially stable as time tends to infinity. This is the first result obtaining the exponential stability behavior of strong solutions to the equations of a planar magnetohydrodynamic compressible flow without any smallness conditions on the data. Our result can be regarded as a natural generalization of the previous ones for the compressible Navier-Stokes system to MHD system with either constant heat-conductivity or nonlinear and temperature-depending heat-conductivity. As a direct consequence, it is shown that the global strong solution to the constant heat-conductivity MHD system whose existence is obtained by Kazhikhov in 1987 is nonlinearly exponentially stable.

Group properties and solutions for the 1D Hall MHD system in the cold plasma approximation

We study the Lie point symmetries and the similarity transformations for the partial differential equations of the nonlinear one-dimensional magnetohydrodynamic system with the Hall term known as HMHD system. For this 1+1 system of partial differential equations we find that is invariant under the action of a seventh dimensional Lie algebra. Furthermore, the one-dimensional optimal system is derived while the Lie invariants are applied for the derivation of similarity transformations. We present different kinds of oscillating solutions.

Local well-posedness for the isentropic compressible MHD system with vacuum

Abstract

Global solutions to the ideal MHD system with a strong magnetic background

We consider the Cauchy problem of the ideal MHD system with a constant non-zero background magnetic field e. We provide a new proof of global existence for small perturbations of an initial stable state (0,e). The proof is based on the null structure of the system and some new weighted energy estimates established along the characteristics.

Global weak solutions to a 2D compressible non-resistivity MHD system with non-monotone pressure law and nonconstant viscosity

In this paper, we consider a two-dimensional non-resistivity MHD system describing the evolution of viscous compressible and electrically conducting fluids under the action of a vertical magnetic field, with non-monotone pressure law and density-depending viscosity λ=λ(ρ). Using an approximate scheme and the compactness method which Bresch and Jabin proposed in (Bresch and Jabin, 2018 [3]), we prove the global existence of weak solutions. This improves Li and Sun's work (Li and Sun, 2019 [32]) to more general pressure law and viscosity.

Decay estimates of the smooth solution to the compressible magnetohydrodynamic equations on [formula omitted

Consider the time decay rates of the smooth solution to the compressible MHD system on T3. It is shown that if the density ρ is uniformly bounded from up and below, and the L6 norm of the magnetic H is uniformly bounded, then all derivatives of the smooth solution to the compressible MHD system decay exponentially fast to the equilibrium. Furthermore, we construct a global smooth solution obeying these decay rates with small energy but possibly large oscillations.