Local smooth solutions of the relativistic Euler equation
The authors consider the relativistic Euler equation which governs the motion of a relativistic perfect fluid \[ {\partial\over \partial t} \Biggl({\rho c^2+ p\over c^2- v^2}- {p\over c^2}\Biggr)+ \sum^3_{k= 1} {\partial\over \partial x_k} \Biggl({\rho c^2+ p\over c^2- v^2} v_k\Biggr)= 0, \] \[ {\partial\over \partial t} \Biggl({\rho c^2+ p\over c^2- v^2} v_i\Biggr)+ \sum^3_{k= 1} {\partial\over \partial x_k} \Biggl({\rho c^2+ p\over c^2- v^2} v_i v_k+ p\delta_{ik}\Biggr)= 0,\;i= 1,2,3, \] in flat Minkowski space. Here \(c\) denotes the speed of light, \(p\) the pressure, \(\rho\) the mass density, and \(v\) the velocity field of the fluid. Assuming the equation of state \(p= a^2\rho\) with \(0< a< c\) the authors prove local existence and uniqueness of smooth solutions for initial data in certain uniformly local Sobolev spaces by transforming the system to a symmetric hyperbolic one. The authors also prove that the solutions of the relativistic system converge to the corresponding solutions of the nonrelativistic system as \(c\to \infty\).
scientific article; zbMATH DE number 784738