Newtonian limit for the relativistic Euler-Poisson equations with vacuum
The authors of this interesting paper investigate Newtonian limit of the relativistic Euler-Poisson equation. Let us denote the ratio \(\beta =|\mathbf{v}|/c\). Then the relativistic Euler-Poisson (EP) system has the form \begin{align*} &\partial_t(\rho /\sqrt{1-\beta^2})+ \operatorname{div}(\rho |\mathbf{v}| /\sqrt{1-\beta^2})=0,\\ &\partial_t\biggl(\frac{ (n(\rho )c^2+p(n(\rho ))) \mathbf{v}}{c^2(1-\beta^2)}\biggr)+\operatorname{div} \biggl(\frac{(n(\rho )c^2+ p(n(\rho ))) \mathbf{v}\otimes\mathbf{v}} {c^2(1-\beta^2)}\biggr) +\nabla p(\rho )= \frac{\rho \nabla\varphi} {\sqrt{1-\beta^2}},\\ &\triangle \varphi = \frac{4\pi\rho } {\sqrt{1-\beta^2}}, \end{align*} where \(n\), \(\rho \), \(\mathbf{v}\), \(c\) and \(\varphi \) represent the proper density of charge, the mass-energy density of the current, the velocity of electric fluid, the speed of light and the electrostatic potential, respectively. This system is supplemented with a free boundary condition and initial data in \((0, R(t)) \times [0, T ]\). Under free boundary conditions and vacuum, it is shown existence and uniqueness of local smooth solutions. These solutions converge to the solutions of the classical Euler-Poisson equation at the rate of \(c^{-2}\). From the mathematical standpoint, the authors overcome the strong nonlinearity caused by the Lorentz factor \(\sqrt{1-u^2/c^2}\), the vacuum occurring on the moving boundary and the singularity at the center point by applying the weighted Sobolev space, respectively. Actually the relativistic EP system is reduced to an evolution PDE under free boundary condition. It is supposed that initial data is sufficiently smooth, \(n_0(x)\in C^2([0,1])\), and \(u_0\) satisfies certain differential conditions. Then the equation under consideration that is a PDE has a unique smooth solution satisfying a certain estimate.
scientific article; zbMATH DE number 7471736
scientific article