Dissipative structures in compressible inviscid fluids

TL;DR

This paper investigates whether total energy dissipates at shock discontinuities in ideal compressible fluids and compares this behavior with incompressible fluids, revealing fundamental differences.

Contribution

It provides a detailed analysis of energy dissipation at shocks in compressible Euler systems and contrasts it with incompressible cases, highlighting key distinctions.

Findings

Energy dissipates on shock discontinuities in compressible fluids.

Incompressible fluids do not accumulate energy on codimension one structures.

The analysis clarifies the role of singular structures in energy dissipation.

Abstract

This note aims at the following problem. In an ideal density dependent fluid system, is the total energy dissipated on shock type discontinuities? To this end, we study the local energy balance for weak solutions to the isentropic compressible Euler system and derive fine properties of the defect measure. This is done by a careful analysis of the small scale properties of the solutions at the shock discontinuity. By means of the same technique, we also consider the inhomogeneous incompressible case, and, comparing the result, we confirm the general principle that the accumulation of the total energy on codimension one singular structures is a feature that distinguishes compressible and incompressible models.