# Formation and development of singularities for the compressible Euler equations

### Tristan Buckmaster

Department of Mathematics, Princeton University, Princeton, NJ 08544, USA### Theodore D. Drivas

Department of Mathematics, Stony Brook University, Stony Brook, NY 11794, USA### Steve Shkoller

Department of Mathematics, University of California Davis, Davis, CA 95616, USA### Vlad Vicol

Courant Institute of Mathematical Sciences, New York University, New York, NY 10012, USA

This book chapter is published *open access.*

## Abstract

In this paper we review the authors’ recent work (Buckmaster, Drivas, Shkoller, Vicol, Preprint 2021), which gives a complete description of the formation and development of singularities for the compressible Euler equations in two space dimensions, under azimuthal symmetry. This solves an open problem posed by Landau and Lifshitz, which was previously open even in one space dimension. Our proof applies mutatis mutandis in the drastically simpler situations of one-dimensional flows, or multidimensional flows with radial symmetry. We prove that for smooth and generic initial data with azimuthal symmetry, the 2D compressible Euler equations yield a local in time smooth solution, which in finite time forms a first gradient singularity, the so-called $C_{1/3}$ *preshock*. We then show that a discontinuous entropy producing *shock wave* instantaneously develops from the preshock. Simultaneous to the development of the shock, two other characteristic surfaces of higher-order cusp-type singularities emerge from the preshock. These surfaces have been termed *weak discontinuities* by Landau and Lifshitz (1987), Chapter IX, §96, who conjectured their existence. We prove that along the characteristic surface moving with the fluid, a *weak contact discontinuity* is formed, while along the slowest surface in the problem, a *weak rarefaction wave* emerges. The constructed solution is the *unique* solution of the Euler equations in a certain class of entropy-producing weak solutions with azimuthal symmetry and with regularity determined by the fact that it arises from a generic preshock.