Rough Solutions of the Relativistic Euler Equations

Abstract: We prove that the time of classical existence of smooth solutions to the relativistic Euler equations can be bounded from below in terms of norms that measure the "(sound) wave-part" of the data in Sobolev space and "transport-part" in higher regularity Sobolev space and H\"older spaces. The solutions are allowed to have nontrivial vorticity and entropy. Our main result is that the Sobolev norm $H^{2+}$ of the variables in the "wave-part" and the H\"older norm $C^{0,0+}$ of the variables in the "transport-part" can be controlled in terms of initial data for short times. We note that the Sobolev norm assumption $H^{2+}$ is the optimal result for the variables in the "wave-part." Compared to low regularity results for quasilinear wave equations and the 3D non-relativistic compressible Euler equations, the main new challenge of the paper is that when controlling the acoustic geometry and bounding the wave equation energies, we must deal with the difficulty that the vorticity and entropy gradient are four-dimensional space-time vectors satisfying a space-time div-curl-transport system, where the space-time div-curl part is not elliptic. To overcome this difficulty, we show that the space-time div-curl systems imply elliptic div-curl-transport systems on constant-time hypersurfaces plus error terms that involve favorable differentiations and contractions with respect to the four-velocity. By using these structures, we are able to adequately control the vorticity and entropy gradient with the help of energy estimates for transport equations, elliptic estimates, Schauder estimates, and Littlewood-Paley theory.