Relaxation limit of pressureless Euler-Poisson equations

TL;DR

This paper studies how solutions to one-dimensional pressureless Euler--Poisson equations behave as the momentum relaxation time approaches zero, showing convergence to drift--diffusion equations and establishing uniqueness of solutions.

Contribution

It provides a direct solution formula approach to prove the relaxation limit and demonstrates the uniqueness of entropy solutions for the pressureless Euler--Poisson system.

Findings

Solutions converge to drift--diffusion equations as relaxation time tends to zero

Constructed explicit solution formulas for both systems

Proved uniqueness of entropy solutions using Oleinik condition

Abstract

We investigate the relaxation problem for the one-dimensional pressureless Euler--Poisson equations with the initial density being a finite Radon measure. The entropy solution of this linearly degenerate hyperbolic system converges to the weak solution of part of drift--diffusion equations without diffusion term when the momentum relaxation time tends to zero. Independent of compactness, our strategy is to construct the formula of solutions to both systems and directly obtain the convergent result. Furthermore, the uniqueness of entropy solution to pressureless Euler--Poisson equations is also proved by Oleinik condition and the initially weak continuity of kinetic energy measure.