The isentropic Euler system admits some plane wave superpositions

TL;DR

This paper constructs explicit differentiable solutions to the isentropic Euler equations in multiple dimensions using solutions of Burgers equations, highlighting the importance of directional constraints and nonlinear pressure interactions.

Contribution

It introduces a new class of explicit, differentiable solutions for the multidimensional isentropic Euler system based on Burgers solutions, emphasizing directional dependencies.

Findings

Solutions are explicit and differentiable within certain directional constraints.

Superpositions of Burgers solutions lead to complex Euler solutions.

Solutions break down at shock formation in Burgers equations.

Abstract

A class of differentiable solutions is proved for the isentropic Euler equations in two and three space dimensions. The solutions are explicitly given in terms of solutions to inviscid Burgers equations, and several directions of propagation. The relative orientation of the directions is critical. Within the directional constraints, the Burgers solutions are arbitrary. The several velocities add, and the pressures combine nonlinearly. These solutions cannot exist beyond the time when shocks develop in any of the Burgers solutions.