Quasineutral limit, dispersion and oscillations for Korteweg type fluids

TL;DR

This paper rigorously analyzes the quasineutral limit of a viscous plasma model with capillarity, showing strong convergence of density fluctuations and transition to incompressible fluid behavior.

Contribution

It introduces a detailed mathematical framework incorporating capillarity effects to control high-frequency oscillations in the quasineutral limit.

Findings

Density fluctuations converge strongly to zero as   0.

The fluid dynamics become incompressible in the limit.

Capillarity terms provide additional dispersive estimates to manage oscillations.

Abstract

In the setting of general initial data and whole space we perform a rigorous analysis of the quasineutral limit for a hydrodynamical model of a viscous plasma with capillarity tensor represented by the Navier Stokes Korteweg Poisson system. We shall provide a detailed mathematical description of the convergence process by analyzing the dispersion of the fast oscillating acoustic waves. However the standard acoustic wave analysis is not sufficient to control the high frequency oscillations in the electric field but it is necessary to estimates the dispersive properties induced by the capillarity terms. Therefore by using these additional estimates we will be able to control, via compensated compactness, the quadratic nonlinearity of the stiff electric force field. In conclusion, opposite to the zero capillarity case \cite{DM12} where persistent space localized time high frequency oscillations need to be taken into account, we show that as $\lambda \to 0$, the density fluctuation $\rho^{\lambda}-1$ converges strongly to zero and the fluids behaves according to an incompressible dynamics.