Compressible Flow and Euler's Equations

TL;DR

This paper analyzes the behavior and structure of solutions to the three-dimensional compressible Euler's equations with arbitrary equations of state, focusing on the formation of shocks and singularities from near-constant initial states.

Contribution

It provides a complete description of the maximal development of solutions, including the geometry of the singular boundary where shocks form, under small initial perturbations.

Findings

Identification of the singular boundary where wave inverse density vanishes

Detailed geometric description of shock formation boundary

Analysis of solution behavior near singularities

Abstract

We consider the classical compressible Euler's Equations in three space dimensions with an arbitrary equation of state, and whose initial data corresponds to a constant state outside a sphere. Under suitable restriction on the size of the initial departure from the constant state, we establish theorems which give a complete description of the maximal development. In particular, the boundary of the domain of the maximal solution contains a singular part where the inverse density of the wave fronts vanishes and the shocks form. We obtain a detailed description of the geometry of this singular boundary and a detailed analysis of the behavior of the solution there.