Rough sound waves in $3D$ compressible Euler flow with vorticity

Abstract

We prove a series of results tied to the regularity and geometry of solutions to the $3D$ compressible Euler equations with vorticity and entropy. Our 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: shocks 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, i.e., the geometry of 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.

This version of our article closely matches the published version, which features some corrections and an abbreviated introduction compared to the first version we posted on arXiv.org on September 5, 2019. As in the published version, here we refer readers to the first arXiv version for an extended introduction that features some additional background material

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• Low regularity ill-posedness for elastic waves driven by shock formation
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• On the rough solutions of 3D compressible Euler equations: an alternative proof
• Low regularity ill-posedness and shock formation for 3D ideal compressible MHD
• Strongly hyperbolic quasilinear systems revisited, with applications to relativistic fluid dynamics
• Rough Solutions of the Relativistic Euler Equations