Blowup of regular solutions for the relativistic Euler–Poisson equations

Abstract

In this paper, we study the blowup phenomena for the regular solutions of the isentropic relativistic Euler-Poisson equations with a vacuum state in spherical symmetry. Using a general family of testing functions, we obtain new blowup conditions for the relativistic Euler-Poisson equations. We also show that the proposed blowup conditions are valid regardless of the speed requirement, which was one of the key constraints stated in "Y. Geng, Singularity Formation for Relativistic Euler and Euler-Poisson Equations with Repulsive Force, Commun.

Singularity formation for the relativistic Euler‐Poisson equations with repulsive force and damping

“…where the unknowns n, 𝜌, and v in system (1) represent the charge density, the proper mass-energy density, and the velocity of an electro-fluid, respectively. c is the speed of light; −Φ is the electrostatic potential in the inertial frame; p is the pressure which satisfies p = 𝜌 𝛾 , where 𝛾 > 1 is the adiabatic index [5]. In addition, n and 𝜌 satisfy 𝜌 = n…”

“…where e ≥ 0 is the special internal energy [5]. According to the first thermodynamics law [1], we have…”

“…Definition 1 (The regular solutions [5]). The solution (𝜌, v) of system ( 6) is called a regular solution, if…”

“…$$ where the unknowns $$$ n,\rho $$$, and $$$ v $$$ in system () represent the charge density, the proper mass‐energy density, and the velocity of an electro‐fluid, respectively. $$$ c $$$ is the speed of light; $$$ -\Phi $$$ is the electrostatic potential in the inertial frame; $$$ p $$$ is the pressure which satisfies $$$ p={\rho}^{\gamma } $$$, where $$$ \gamma >1 $$$ is the adiabatic index [5]. In addition, $$$ n $$$ and $$$ \rho $$$ satisfy $$$ \rho =n\left(1+\frac{e}{c^2}\right) $$$, where $$$ e\ge 0 $$$ is the special internal energy [5].…”

Space-charge effect of the time-varying electron injection in a diode: classical and relativistic regimes

“…There can be many relativistic versions of our problem, for instance see Ref. 32 . We choose to formulate in a simple way to be reducible to classical regime when v c.…”