Rough sound waves in 3D compressible Euler flow with vorticity

Abstract

We prove a series of intimately related results tied to the regularity and geometry of solutions to the 3D compressible Euler equations. The results concern “general” solutions, which can have nontrivial vorticity and entropy. Our geo-analytic framework exploits and reveals additional virtues of a recent new formulation of the equations, which decomposed the flow into a geometric “(sound) wave-part” coupled to a “transport-div-curl-part” (transport-part for short), with both parts exhibiting remarkable properties. Our main result is that the time of existence can be controlled in terms of the \(H^{2^+}(\mathbb {R}^3)\)-norm of the wave-part of the initial data and various Sobolev and Hölder norms of the transport-part of the initial data, the latter comprising the initial vorticity and entropy. The wave-part regularity assumptions are optimal in the scale of Sobolev spaces: Lindblad (Math Res Lett 5(5):605–622, 1998) showed that shock singularities can instantly form if one only assumes a bound for the \(H^2(\mathbb {R}^3)\)-norm of the wave-part of the initial data. Our proof relies on the assumption that the transport-part of the initial data is more regular than the wave-part, and we show that the additional regularity is propagated by the flow, even though the transport-part of the flow is deeply coupled to the rougher wave-part. To implement our approach, we derive several results of independent interest: (i) sharp estimates for the acoustic geometry, which in particular capture how the vorticity and entropy affect the Ricci curvature of the acoustical metric and therefore, via Raychaudhuri’s equation, influence the evolution of the geometry of acoustic null hypersurfaces, i.e., sound cones; (ii) Strichartz estimates for quasilinear sound waves coupled to vorticity and entropy; and (iii) Schauder estimates for the transport-div-curl-part. Compared to previous works on low regularity, the main new features of the paper are that the quasilinear PDE systems under study exhibit multiple speeds of propagation and that elliptic estimates for various components of the fluid are needed, both to avoid loss of regularity and to gain space-time integrability.