Shock formation in the compressible Euler equations and related systems

TL;DR

This paper proves shock formation in the compressible Euler equations and related systems, providing bounds and extending results to magnetohydrodynamics and variable area duct flows in one and three dimensions.

Contribution

It establishes new shock formation results for the Euler equations, MHD, and variable area duct flows, improving upon previous work and covering more complex systems.

Findings

Established $L^ abla$ bounds for solutions

Proved shock formation in 1D and spherically symmetric 3D flows

Extended results to MHD and variable area ducts

Abstract

We prove shock formation results for the compressible Euler equations and related systems of conservation laws in one space dimension, or three dimensions with spherical symmetry. We establish an $L^\infty$ bound for $C^1$ solutions of the one-D Euler equations, and use this to improve recent shock formation results of the authors. We prove analogous shock formation results for one-D MHD with orthogonal magnetic field, and for compressible flow in a variable area duct, which has as a special case spherically symmetric three dimensional flow on the exterior of a ball