Strichartz estimates and low regularity solutions of 3D relativistic Euler equations

Abstract

We study the low regularity well-posedness for Cauchy problem of 3D relativistic Euler equations. Firstly, we introduce a new decomposition for relativistic velocity and derive new transport equations for vorticity, which both play a crucial role in energy and Strichartz estimates. According to Smith-Tataru's approach, we then establish a Strichartz estimate of linear wave equations endowed with the acoustic metric. This leads us to prove a complete local well-posedness result if the initial logarithmic enthalpy, velocity, and modified vorticity $(h_0, \bu_0, \bw_0) \in H^s \times H^s \times H^{s_0} (2<s_0<s)$. Therefore, we give an affirmative answer to "Open Problem D" proposed by Disconzi. Moreover, for $(h_0,{\bu}_0,\bw_0) \in H^{2+} \times H^{2+} \times H^2$, by frequency truncation, there is a stronger Strichartz estimate for solutions on a short-time-interval. By semi-classical analysis and induction method, these solutions can be extended from short time intervals to a regular time interval, and a uniform Strichartz estimate with loss of derivatives can be obtained. This allows us to prove the local well-posedness of 3D relativistic equations if $(h_0,{\bu}_0,\bw_0) \in H^{2+} \times H^{2+} \times H^2$.