Quasi-neutral limit of the Euler-Poisson and Euler-Monge-Ampère systems
Grégoire Loeper
TL;DR
The paper studies the quasi-neutral limit $\epsilon \to 0$ for two collisionless-fluid models, the pressureless Euler–Poisson system $EP_{\epsilon}$ and its fully nonlinear counterpart the Euler–Monge–Ampère system $EMA_{\epsilon}$, showing convergence to the incompressible Euler equations on the torus $\mathbb{T}^d$ ($d=2,3$) at large scales. For $EP_{\epsilon}$, using energy estimates and a reformulation with new unknowns $\beta_1$, $\rho_1$, and $\omega_1$ defined by $\nabla\cdot v = \epsilon \beta_1$, $\rho=1+\epsilon^2 \rho_1 = 1+\epsilon \Delta \phi$, $\nabla\times v = \omega = \bar{\omega} + \epsilon \omega_1$, they prove uniform bounds on $\epsilon^{-1}(v^{\epsilon}-\bar{v})$ and $\epsilon^{-2}(\rho^{\epsilon}-1)$ and convergence of the scaled perturbations to the incompressible limit; non-prepared data also converge. For $EMA_{\epsilon}$, they frame it as a penalized approximate geodesic problem on the group of volume-preserving maps, analyze the linearized Monge–Ampère operator and establish that small perturbations in density lead to controlled Hessians and quadratic remainder terms, and obtain energy estimates via a div–curl decomposition to prove convergence toward the Euler flow with non-prepared data also handled. The study highlights how fast oscillations from the electric coupling and the Monge–Ampère constraint interact with slow, incompressible modes to yield the quasi-neutral limit, providing rigorous justification for using incompressible Euler dynamics as large-scale limits of these models.
Abstract
This paper studies the pressureless Euler-Poisson system and its fully non-linear counterpart, the Euler-Monge-Amp\`ere system, where the fully non-linear Monge-Amp\`ere equation substitutes for the linear Poisson equation. While the first is a model of plasma physics, the second is derived as a geometric approximation to the Euler incompressible equations. Using energy estimates, convergence of both systems to the Euler incompressible equations is proved.