Global solutions of the Euler--Maxwell two-fluid system in 3D

Abstract

The fundamental "two-fluid" model for describing plasma dynamics is given by the Euler-Maxwell system, in which compressible ion and electron fluids interact with their own self-consistent electromagnetic field. We prove global stability of a constant neutral background, in the sense that irrotational, smooth and localized perturbations of a constant background with small amplitude lead to global smooth solutions in three space dimensions for the Euler-Maxwell system. Our construction applies equally well to other plasma models such as the Euler-Poisson system for two-fluids and a relativistic Euler-Maxwell system for two fluids. Our solutions appear to be the first nontrivial global smooth solutions in all of these models.Contents 1 4 Indeed, the ratio me/M i is no bigger than the ratio of the electron mass to the proton mass which equals 1/1836. 5 This is called the electrostatic approximation. 6 YAN GUO, ALEXANDRU D. IONESCU, AND BENOIT PAUSADER Klein-Gordon equations (faster time decay of linear waves like t −3/2 , absence of quadratic resonances), global smooth irrotational flows were constructed in [22] via the normal form method of Shatah [41]: Theorem 1.3 (Stability of a neutral equilibrium [22]). Solutions of (1.15) with initial data (n 0 , v 0 ) small, smooth, neutral and irrotational in the sense thatremain globally smooth and decay to 0 in L ∞ as t → +∞.The neutral assumption was later removed in [15] and this result was extended to two spatial dimensions independently in [28,37] (see also [30,31]). Theorem 1.3 was the first positive result indicating that the dispersive effect alone in the two-fluid theory may prevent shock formation 6 and it started an investigation to understand to which extent the introduction of electromagnetic forces could stabilize the full Euler-Maxwell system.Recently, further progress was made in this direction in the study of another simplified model: the Euler-Poisson equation for the ions 7 :(1.17)Here the electron dynamics with constant temperature is decoupled from the ion dynamics via the Boltzmann relation. The model equation then becomesThis system has intermediate behavior between (1.14) and (1.16). The linearized solutions decay slowly (like t −4/3 ) and create many strong degeneracies near the zero frequency, where the dispersion relation is similar to the wave dispersion up to third order (see λ i in Lemma A.4). Nevertheless, the first and third authors were able to obtain an analogue of Theorem 1.3 for perturbations of a neutral equilibrium by using a variation on the normal form method, controlling bilinear multipliers with rough coefficients using arguments inspired by [26]. Here, a crucial property is the fact that the nonlinearity is an exact derivative, which helps compensate for the degeneracy at the 0 frequency.1.2.2. Two-fluid models with different speeds. Both systems (1.15) and (1.17) can be reduced (under the irrotational assumption) to a (complex) scalar quasilinear equation with one speed. This is no longer the case for more complicated two-fluid mo...

Large-time behavior for fluid and kinetic plasmas with collisions

“…We first mention that in the non damped case, the dispersive Euler-Maxwell system near constant steady states has been studied by Germain-Masmoudi [13] and Guo-Ionescu-Pausader [18] by using the tools in the harmonic analysis. In the damped situation, it is easy to obtain the global existence of small amplitude classical solutions near the constant steady state only basing on the energy method.…”

The combined non-relativistic and quasi-neutral limit of two-fluid Euler–Maxwell equations

“…Then the local existence of smooth solutions holds according to the result of Kato [7]. The global existence is considered recently in [5,6]. We mention also a result in [4] on the global existence of weak solutions for a simplified one-dimensional model of (1.4).…”

Absence of Shocks for One Dimensional Euler–Poisson System

“…Instead, we will see that solutions approach to a nonlinear asymptotic state. To show this phenomenon, we set 8) then system (1.6) is equivalent to the following complex-valued Klein-Gordon equation…”