Structurally stable singularities and Lipschitz stable optimal transport metrics for the compressible Euler equations
It is well known that solutions to the compressible Euler equations can develop singularities in finite time. In this paper, we carry out a detailed analysis on behaviors of solutions up to the time of the first singularity for the one-dimensional compressible Euler equations with general smooth initial data. Our main results consist of three parts. First, for an open dense set of $C^3$ initial data, we show that the solution of Euler equations is twice continuously differentiable except at most finitely many points when the first singularity happens, using Thom's Transversality Theorem. Second, for any initial data in the open dense set of $C^3$ functions given in the first result, we provide the precise asymptotic description of the solution in a semi-neighborhood in the $(x,t)$-plane of each singular point at the time of the first singularity, and verify that the solution has a cusp-type singularity with Hölder exponent $1/3$ at each singular point. The proofs of the first two results are based on the representation of the solution in terms of a semilinear system. Third, for smooth initial data with small BV norm, we construct two Finsler type optimal transport metrics, then under these metrics show that the solution depends Lipschitz continuously on the initial data up to the time of the first singularity, with uniformly bounded Lipschitz constants. In particular, the $C^{1/3}$ generic singularity is stable in this sense. On the other hand, since our first two results hold for an open and dense set of initial data, any Hölder continuous cusp singularity with exponent other than $1/3$ is unstable under initial perturbations.