This paper focuses on a 2D magnetohydrodynamic system with only horizontal dissipation in the domain $\Omega = \mathbb {T}\times \mathbb {R}$ with $\mathbb {T}=[0,\,1]$ being a periodic box. The goal here is to understand the stability problem on perturbations near the background magnetic field $(1,\,0)$. Due to the lack of vertical dissipation, this stability problem is difficult. This paper solves the desired stability problem by simultaneously exploiting two smoothing and stabilizing mechanisms: the enhanced dissipation due to the coupling between the velocity and the magnetic fields, and the strong Poincaré type inequalities for the oscillation part of the solution, namely the difference between the solution and its horizontal average. In addition, the oscillation part of the solution is shown to converge exponentially to zero in $H^{1}$ as $t\to \infty$. As a consequence, the solution converges to its horizontal average asymptotically.

This paper is concerned with numerical solutions of the LDG method for 1D wave equations. Superconvergence and energy conserving properties have been studied. We first study the superconvergence phenomenon for linear problems when alternating fluxes are used. We prove that, under some proper initial discretization, the numerical trace of the LDG approximation at nodes, as well as the cell average, converge with an order 2k+1. In addition, we establish k+2-th order and k+1-th order superconvergence rates for the function value error and the derivative error at Radau points, respectively. As a byproduct, we prove that the LDG solution is superconvergent with an order k+2 towards the Radau projection of the exact solution. Numerical experiments demonstrate that in most cases, our error estimates are optimal, i.e., the error bounds are sharp. In the second part, we propose a fully discrete numerical scheme that conserves the discrete energy. Due to the energy conserving property, after long time integration, our method still stays accurate when applied to nonlinear Klein-Gordon and Sine-Gordon equations.

The out-of-equilibrium situation in which an initially sharp-edged cylindrical electron beam, that could, e.g., model electrons flowing within a wire, is injected into a plasma is considered. A detailed computation of the subsequently produced magnetic field is presented. The control parameter of the problem is shown to be the ratio of the beam radius to the electron skin depth. Two alternative ways to address analytically the problem are considered: one uses the usual Laplace transform approach, the other one involves Riemann's method in which causality conditions manifest through some integrals of triple products of Bessel functions.

We consider abstract second order evolution equations with unboundedfeedback with delay. Existence results are obtained under somerealistic assumptions. Sufficient and explicit conditions are derived that guarantee the exponential or polynomial stability. Some new examples that enter into our abstract framework are presented.

When it is usually using a bigger algebra system to formulate the Maxwell equations, in this paper we consider a real four-dimensional algebra to express the Maxwell equations without appealing to the imaginary number and higher dimensional algebras. In terms of g-based Jordan algebra formulation the Lorentz gauge condition is found to be a necessary and sufficient condition to render the second pair of Maxwell equations, while the first pair of Maxwell equations is proved to be an intrinsic algebraic property. Then, we transform the g-based Jordan algebra to a Lie algebra of the dilation proper orthochronous Lorentz group, which gives us an incentive to consider a linear matrix operator of the Lie type, rendering more easy to derive the Maxwell equations and the wave equations. The new formulations fully match the requirements for the classical electrodynamic equations and the Lorentz gauge condition. The mathematical advantage of our formulations is that they are irreducible in the sense that, when compared to the formulations which using other bigger algebras (e.g., biquaternions and Clifford algebras), the number of explicit components and operations is minimal. From this aspect, the g-based Jordan algebra and Lie algebra are the most suitable algebraic systems to implement the Maxwell equations into a more compact form.