Formation of singularity and smooth wave propagation for the non-isentropic compressible Euler equations

TL;DR

This paper extends classical shock formation results to the non-isentropic compressible Euler equations with variable entropy, providing a global analysis of smooth wave propagation and singularity formation.

Contribution

It generalizes P. Lax's shock formation theory to non-isentropic flows with variable entropy and pressure laws, without restrictions on initial data size.

Findings

Derived differential equations for wave steepening in variable entropy flows

Generalized shock formation results to non-isentropic Euler equations

Established global validity of the results regardless of initial data variation

Abstract

We define compressive and rarefactive waves and give the differential equations describing smooth wave steepening for the compressible Euler equations with a varying entropy profile and general pressure laws. Using these differential equations, we directly generalize P. Lax's singularity (shock) formation results for hyperbolic systems with two variables to the compressible Euler equations for a polytropic ideal gas. Our results are valid globally without restriction on the size of the variation of initial data.