Zero-relaxation and vanishing-damping limits of pressureless Euler system

TL;DR

This paper investigates the behavior of the pressureless Euler system under different relaxation limits, showing convergence to static or undamped solutions depending on the relaxation time.

Contribution

It provides a rigorous analysis of the zero-relaxation and vanishing-damping limits for the pressureless Euler system in the measure space.

Findings

Entropy solution converges to a static solution as relaxation time approaches zero.

Entropy solution converges to the pressureless Euler system as damping vanishes.

Results are established in the Radon measure space.

Abstract

We are concerned with the one-dimensional pressureless Euler system with relaxation in the Radon measure space. As the relaxation time tends to zero, the entropy solution converges to a static solution with the density converging to its initial value. As the relaxation time tends to infinity, which means the damping vanishes, the entropy solution of damped pressureless Euler system converges to that of pressureless Euler system.