Global Existence of Solutions to the Compressible Euler Equations with Time-dependent Damping and Logarithmic State Equation

TL;DR

This paper proves the global existence of solutions for one-dimensional compressible Euler equations with time-dependent damping using a novel logarithmic pressure state equation, expanding understanding of such systems in mathematical physics.

Contribution

It introduces a new logarithmic pressure state equation that combines existing models and proves global solutions for the damped Euler equations with small initial data.

Findings

Global existence of solutions established

Logarithmic pressure derived from existing state equations

Applicable to one-dimensional compressible Euler equations

Abstract

In mathematical physics, the pressure function is determined by the equation of state. There are two existing barotropic state equations: the state equation for polytropic gas with adiabatic index greater than or equal to 1 and the state equation for generalized Chaplygin gas in cosmology. In this paper, a logarithmic pressure is derived naturally with the coexistence of the two existing state equations through an equivalent symmetric hyperbolic transformation. On the study of the logarithmic pressure, global existence of solutions with small initial data to the one-dimensional compressible Euler equations with time-dependent damping is established.