## 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 C13pre-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 [12, Chapter IX, §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 C1,12 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 C1,12 one-sided cusps while the pressure and normal velocity remain C^{2}; as such, we term this surface a weak contact discontinuity.

Original language | English (US) |
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Article number | 26 |

Journal | Annals of PDE |

Volume | 8 |

Issue number | 2 |

DOIs | |

State | Published - Dec 2022 |

## Keywords

- Compressible euler
- Pre-shock
- Shock development
- Shock formation
- Weak contact
- Weak rarefaction

## ASJC Scopus subject areas

- Analysis
- Mathematical Physics
- Physics and Astronomy(all)
- Geometry and Topology
- Applied Mathematics