| dc.description.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 a low-regularity Sobolev space and ``transport-part" in higher regularity Sobolev space and H\"older spaces. The solutions are allowed to have nontrivial vorticity and entropy. We use the geometric framework from [Disconzi-Speck,19], where the relativistic Euler flow is decomposed into a ``wave-part", that is, geometric wave equations for the velocity components, density and enthalpy, and a ``transport-part", that is, transport-div-curl systems for the vorticity and entropy gradient. 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. Due to lack of ellipticity, one can not immediately rely on the approach taken in [Disconzi-Speck,19] to control these terms. 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. | |
