Uniform global existence and convergence of Euler-Maxwell systems with small parameters

Abstract

The Euler-Maxwell system with small parameters arises in the modeling of magnetized plasmas and semiconductors. For initial data close to constant equilibrium states, we prove uniform energy estimates with respect to the parameters, which imply the global existence of smooth solutions. Under reasonable assumptions on the convergence of initial conditions, this allows to show the global-in-time convergence of the Euler-Maxwell system as each of the parameters goes to zero.

Convergence rates in zero-relaxation limits for Euler-Maxwell and Euler-Poisson systems

“…For smooth initial data, the global existence of smooth solutions, which are sufficiently close to the equilibrium state, to (1.1)-(1.2) was proved in [20,4,24]. For the relaxation limit problems, the local-in-time and the globalin-time convergence of (1.1) as ε → 0 were obtained by Hajjej-Peng [6] and Wasiolek [21],…”

Global relaxation and nonrelativistic limit of nonisentropic Euler‐Maxwell systems

“…In particular, we do not assume any relation between and in the third limit. In comparison with the isentropic case in Wasiolek, the proofs in our paper are more complicated due to the essential difficulties from the presence of energy equation. In the energy equation, the energy relaxation time is a function of the momentum relaxation time with , which can be used to describe more physical phenomena.…”

The rigorous derivation of unipolar Euler–Maxwell system for electrons from bipolar Euler–Maxwell system by infinity‐ion‐mass limit

“…Together with the Poisson equation, we have This implies the solvability of ρ e in the expression of ρ i , which leads to the unipolar Euler–Poisson model for ions that does not contain any information of electrons. By now the decoupling is successful (see details in Xi and Zhao 22 ). Contrarily, when the zero‐electron‐mass limit is applied to the bipolar Euler–Maxwell system (), formally, the momentum equation for electrons becomes Due to the Lorentz force on the right hand side, it is obvious that the decoupling is not successful.…”

“…For the local‐in‐time convergence of small parameters, we refer to other studies 14‐20 and the references therein. By establishing the uniform global estimates with respect to small parameters, the global‐in‐time convergence of small parameters for Equations () are studied in previous studies 15,21,22 …”

“…In Li et al, 29 they studied the periodic case when the doping profile b is not a constant but a function of the space variable x . However, as pointed out in Xi and Zhao, 22 when the zero‐electron‐mass limit is applied in bipolar models, it is not proper to simply ignore the effect of ions and only make the asymptotic analysis to the equations for electrons. In fact, rather than staying the same, the equations for ions have a limiting process, of which the key point is actually decoupling, but not the vanishing of equations.…”