The geometry of maximal development for the Euler equations

Steve Shkoller, Vlad Vicol

Published 2023-10-12Version 1

We establish the maximal hyperbolic development of Cauchy data for the multi-dimensional compressible Euler equations. For an open set of compressive and generic $H^7$ initial data, we construct unique $H^7$ solutions to the Euler equations in the maximal spacetime region such that at any point in this spacetime, the solution can be smoothly and uniquely computed by tracing both the fast and slow acoustic characteristic surfaces backward-in-time, until reaching the Cauchy data prescribed along the initial time-slice. The future temporal boundary of this spacetime region is a singular hypersurface, consisting of the union of three sets: first, a co-dimension-$2$ surface of ``first singularities'' called the pre-shock; second, a downstream hypersurface emanating from the pre-shock, on which the Euler solution experiences a continuum of gradient catastrophes; third, an upstream hypersurface consisting of a Cauchy horizon emanating from the pre-shock, which the Euler solution cannot reach. We develop a new geometric framework for the description of the acoustic characteristic surfaces which is based on the Arbitrary Lagrangian Eulerian (ALE) framework, and combine this with a new type of differentiated Riemann-type variables which are linear combinations of gradients of velocity and sound speed and the curvature of the fast acoustic characteristic surfaces. With these new variables, we establish uniform $H^7$ Sobolev bounds for solutions to the Euler equations without derivative loss and with optimal regularity. This is the first result on the maximal hyperbolic development of compressive Cauchy data in all regions of spacetime.