Multi-Dimensional Stability of Rarefaction Waves for the Compressible Euler-Poisson Equations of Plasma.

This research proposal addresses a fundamental problem in the theory of multidimensional hyperbolic conservation laws: the structural stability of planar rarefaction waves for the compressible Euler-Poisson equations. This system serves as a key hydrodynamical model for a warm, collisional plasma, where the charged particle fluid interacts with a self-consistent electrostatic field. While the stability of multidimensional rarefaction waves has been recently resolved for the compressible Euler equations of gas dynamics, the incorporation of the electric field introduces a profound analytical challenge that this project seeks to overcome.The central difficulty stems from the non-local nature of the electric potential, determined instantaneously across space by the Poisson equation. This non-locality destroys the finite speed of propagation (causality), a foundational property of the homogeneous Euler system. Consequently, the spacetime geometry of the characteristic surfaces that define the rarefaction wave regions becomes globally coupled. The boundaries of these regions are no longer predetermined by the initial data but become "free boundaries," interacting non-locally with the entire solution. This leads to a highly intricate and rich nonlinear coupling that is absent in classical gas dynamics.The primary objective is to establish the local-in-time existence and uniqueness of continuous, piecewise smooth solutions featuring two-dimensional rarefaction waves emerging from discontinuous initial data. We will first tackle the case of irrotational flow, where the vorticity vanishes. Subsequently, we will confront the significantly more complex case where vorticity is present, which deforms the characteristic surfaces and introduces further analytic obstacles. The research plan involves a novel nonlinear iterative scheme that decouples the non-local problem into a sequence of local hyperbolic problems with sources. The convergence of this scheme will be established using a sophisticated acoustic geometric framework pioneered by Christodoulou and adopted by Luo-Yu in the study of multi-dimensional compressible Euler equations. This involves working within a characteristic coordinate system, deriving precise energy estimates for a wave-transport system, and controlling the geometry of the characteristic surfaces to obtain uniform estimates without loss of derivatives.Success in this project will provide the first rigorous mathematical proof of the stability of multidimensional rarefaction waves in a plasma context, significantly advancing our understanding of how long-range forces influence nonlinear wave phenomena. The results will have broad implications for the analysis of non-local hyperbolic systems and contribute to the foundational mathematics of plasma physics.