Stability and Asymptotics in Electrohydrodynamics

Abstract

This dissertation focuses on the relative energy analysis of two-species fluids composed of charged particles. It presents a formal derivation of the relative energy identity for both the bipolar Euler-Maxwell system and the unmagnetized case of the bipolar Euler-Poisson system. Furthermore, the dissertation explores several applications of the relative energy method to Euler-Poisson systems, enabling a comprehensive stability analysis of these systems. The first application establishes the high-friction limit of a bipolar Euler-Poisson system with friction, converging towards a bipolar drift-diffusion system. Moreover, the second application investigates the limits of zero-electron-mass and quasi-neutrality in a bipolar Euler-Poisson system. In the former limit, a non-linear adiabatic electron system is obtained, while the combined limit yields an Euler system. A weak-strong uniqueness principle for a single-species Euler-Poisson system in the whole space is also established. This principle is further extended to an Euler-Riesz system, considering a more general interaction potential. The theory of Riesz potentials, along with representation formulas for the potentials, is employed to overcome the technical challenges in these studies.