General Relativity Seminar

Speaker: Sifan Yu, Vanderbilt University

Title: Rough solutions of the relativistic Euler equations

Abstract: I will discuss recent works on the relativistic Euler equations with dynamic vorticity and entropy. We use a new formulation of the equations, which has geo-analytic structures. In this geometric formulation, we decompose the flow into geometric “sound-wave part” and “transport-div-curl part”. This allows us to derive sharp results about the dynamics, including the existence of low-regularity solutions. Then, I will discuss the results of rough solutions of the relativistic Euler equations and the role that nonlinear geometric optics plays in the framework. 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.” This talk will include the main ideas of the proof, as well as a comparison of the relativistic and non-relativistic scenarios.