Narrow Results

In this paper we investigate the Cauchy problem for the three-dimensional incompressible magnetohydrodynamic equations. A logarithmal improved blow-up criterion of smooth solutions is obtained.

A set of reduced MHD equations is derived using the equation of state including plasma compressibility. By applying assumption of pressure, i.e., R²P = const., a set of reduced magnetohydrodynamic equations for toroidal plasmas are obtained for large aspect ratio, high β tokamaks. These equations include all terms of the same order as the toroidal effect and only involve three variables, namely the flux, stream function and pressure.

The asymptotic stability of the steady state with the strictly positive constant density and the vanishing velocity and magnetic field to the Cauchy problem of the three-dimensional compressible viscous, heat-conducting magnetohydrodynamic equations with Coulomb force is established under small initial perturbations. Using a general energy method, we obtain the optimal time decay rates of the solution and its higher-order spatial derivatives by introducing the negative Sobolev and Besov spaces. As a corollary, the $L^p$–$L^2$$(1\le p\le 2)$ type of the decay rates follows without requiring that the $L^p$ norm of initial data is small.

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.

This paper focuses on the stability and convergence analysis of the first-order Euler implicit/explicit scheme based on mixed finite element approximation for three-dimensional (3D) time-dependent MHD equations. Firstly, for initial data $u_0,B_0\in H^{\alpha }$ with $\alpha =1,2$, the regularity results of the continuous solution $(u,p,B)$ and the spatial semi-discretization solution $(u_h,p_h,B_h)$ are obtained, and $L^2$-error estimate of $(u_h,B_h)$ is deduced by using the negative norm technique. Next, through the use of mathematic induction, the $H^2$-stability of the fully discrete first-order scheme is proved under the stability condition depending on the smoothness of initial data. Here, the stability condition is $\mathrm{\Delta }t\le C_0$ for $\alpha =2$ and $\mathrm{\Delta }{\mathrm{\text{th}}}^{-\frac{1}{2}}\le C_0$ for $\alpha =1$ where $C_0$ is some positive constant. Then, under the stability condition, the optimal $H^1$-$L^2$ error estimate of the fully discrete solution $(u_h^n,B_h^n)$ and optimal $L^2$-error estimate of the fully discrete solution $p_h^n$ are established by using the parabolic dual argument.

In this paper, we attempt to introduce a new approach to determine the magnetic helicity and energy dissipation of the normal electromagnetic vortex filaments in magnetohydrodynamics (MHD). We successfully compute the vector potential and magnetic helicity of Frenet–Serret vectors and associated electromagnetic fields in the existence of the Lorentz force. We further describe a set of differential equations relating the magnetic vector potential of the electromagnetic vector fields and the Godbillon–Vey helicity model. Then, we focus on the time evolution of the normal electromagnetic field lines in MHD to evaluate the evolution of the magnetic helicity and energy dissipation theoretically. As a result, we obtain analytical solutions and numerical simulations of the time evolution equations of both geometric quantities as well as the magnetic and electric field vectors.

We consider the two-dimensional (2D) Magneto-HydroDynamics (MHD) equations describing the evolution of a viscous incompressible electrically conducting fluid. In this paper, adopting the vorticity-current formulation and assuming that the initial fluid vorticity and magnetic current are in $L^1({ℝ}^2)$, we prove the local in-time existence and regularity of the solutions.

This paper is concerned with the zero-viscosity limit of the three-dimensional (3D) incompressible stationary magnetohydrodynamics (MHD) equations in the 3D unbounded domain $\mathrm{\Omega }:={ℝ}^{+}\times {ℝ}^2$. The main result of this paper establishes that the solution of 3D incompressible stationary MHD equations converges to the solution of the 3D incompressible stationary Euler equations as the viscosity coefficient goes to zero.

In this paper, we consider regularity criteria for solutions to the 3D generalized MHD equations with fractional dissipative term -(-Δ)^αu for the velocity field and -(-Δ)^βb for the magnetic field. For the case α = β, it is proved that if the velocity field belongs to with 0 ≤ r < α or the gradient of velocity field belongs to with 0 ≤ r ≤ α on [0, T], then the solution remains smooth on [0, T].