Simultaneous development of shocks and cusps for 2D Euler with azimuthal symmetry from smooth data

TL;DR

This paper rigorously analyzes the formation of shocks and cusp singularities in 2D Euler equations with azimuthal symmetry, revealing the precise structure of pre-shocks and the simultaneous emergence of characteristic cusp surfaces from smooth initial data.

Contribution

It provides a detailed mathematical description of shock and cusp formation from smooth data in 2D Euler equations with azimuthal symmetry, including the structure of pre-shocks and singularity surfaces.

Findings

Pre-shock forms as a $C^{1/3}$ singularity with fractional series expansion.

Discontinuous shock develops immediately after the pre-shock.

Two cusp-type singularity surfaces emerge, with specific regularity properties.

Abstract

A fundamental question in fluid dynamics concerns the formation of discontinuous shock waves from smooth initial data. We prove that from smooth initial data, smooth solutions to the 2d Euler equations in azimuthal symmetry form a first singularity, the so-called $C^{\frac{1}{3}} $ pre-shock. The solution in the vicinity of this pre-shock is shown to have a fractional series expansion with coefficients computed from the data. Using this precise description of the pre-shock, we prove that a discontinuous shock instantaneously develops after the pre-shock. This regular shock solution is shown to be unique in a class of entropy solutions with azimuthal symmetry and regularity determined by the pre-shock expansion. Simultaneous to the development of the shock front, two other characteristic surfaces of cusp-type singularities emerge from the pre-shock. These surfaces have been termed weak discontinuities by Landau & Lifschitz [Chapter IX, \S 96], who conjectured some type of singular behavior of derivatives along such surfaces. We prove that along the slowest surface, all fluid variables except the entropy have $C^{1, {\frac{1}{2}} }$ one-sided cusps from the shock side, and that the normal velocity is decreasing in the direction of its motion; we thus term this surface a weak rarefaction wave. Along the surface moving with the fluid velocity, density and entropy form $C^{1, {\frac{1}{2}} }$ one-sided cusps while the pressure and normal velocity remain $C^2$; as such, we term this surface a weak contact discontinuity.