Remarkable Localized Integral Identities for 3D Compressible Euler Flow and the Double-Null Framework

Abstract

We derive new, localized geometric integral identities for solutions to the 3D compressible Euler equations under an arbitrary equation of state when the sound speed is positive. The integral identities are coercive in the derivatives of the specific vorticity (defined to be vorticity divided by density) and the derivatives of the entropy gradient vectorfield, and the error terms exhibit remarkable regularity and null structures. Our framework plays a fundamental role in our companion works (Abbrescia L, Speck J. The emergence of the singular boundary from the crease in 3D compressible Euler flow, 2022; Abbrescia and Speck, The emergence of the Cauchy horizon from the crease in 3D compressible Euler flow (in preparation)) on the structure of the maximal classical development for shock-forming solutions. It allows one to simultaneously unleash the full power of the geometric vectorfield method for both the wave- and transport- parts of the flow on compact regions, and our approach reveals fundamental new coordinate-invariant structural features of the flow. In particular, the integral identities yield localized control over one additional derivative of the vorticity and entropy compared to standard results, assuming that the initial data enjoy the same gain. Similar results hold for the solution’s higher derivatives. We derive the identities in detail for two classes of spacetime regions that frequently arise in PDE applications: (i) compact spacetime regions that are globally hyperbolic with respect to the acoustical metric, where the top and bottom boundaries are acoustically spacelike—but not necessarily equal to portions of constant Cartesian-time hypersurfaces; and (ii) compact regions covered by double-acoustically null foliations. Our results have implications for the geometry and regularity of solutions, the formation of shocks, the structure of the maximal classical development of the data, and for controlling solutions whose state along a pair of intersecting characteristic hypersurfaces is known. Our analysis relies on a recent new formulation of the compressible Euler equations that splits the flow into a geometric wave-part coupled to a div-curl-transport part. The main new contribution of the present article is our analysis of the spacelike, co-dimension one and two boundary integrals that arise in the div-curl identities. By exploiting interplay between the elliptic and hyperbolic parts of the new formulation and using careful geometric decompositions, we observe several crucial cancellations, which in total show that after a further integration with respect to an acoustical time function, the boundary integrals have a good sign, up to error terms that can be controlled due to their good null structure and regularity properties.