This book chapter is published open access.
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 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.