The Characteristic Initial Value Problem for the 3D Compressible Euler Equations

Abstract

We prove local existence and uniqueness, propagation of Sobolev regularity, and continuation criteria results for the characteristic initial value problem for the 3D compressible Euler equations. This is the Cauchy problem with initial data given on a pair of transversally intersecting (acoustically) null hypersurfaces. The region of existence in our main theorem is a future-neighborhood of the union of the data null hypersurfaces, where the size of the neighborhood depends on various norms of the data as well as the structure of the data null hypersurfaces. We allow for any equation of state with positive sound speed, and we do not make any symmetry, irrotationality, or isentropicity assumptions. Our analysis relies on the geometric formulation of the flow derived in [Speck19], which decomposes the equations into a "wave-part" coupled to a "transport-div-curl" part. Compared to Rendall's celebrated solution of the characteristic initial value problem for wave equations [Rendall90], substantial new ideas are needed to handle the fact that the compressible Euler equations are more complicated than pure wave equations.