Blowup of smooth solutions for relativistic Euler equations

TL;DR

This paper investigates the formation of singularities in smooth solutions of relativistic Euler equations, proving finite-time blowup under various initial conditions and analyzing the effects of initial momentum and energy.

Contribution

It establishes finite-time blowup results for smooth solutions with both finite and infinite initial energy, and analyzes stability and propagation speed of perturbations.

Findings

Smooth solutions with compact support blow up in finite time.

Existence and stability of solutions with infinite initial energy in the subluminal region.

Finite propagation speed for perturbations around non-vacuum backgrounds.

Abstract

We study the singularity formation of smooth solutions of the relativistic Euler equations in $(3+1)$-dimensional spacetime for both finite initial energy and infinite initial energy. For the finite initial energy case, we prove that any smooth solution, with compactly supported non-trivial initial data, blows up in finite time. For the case of infinite initial energy, we first prove the existence, uniqueness and stability of a smooth solution if the initial data is in the subluminal region away from the vacuum. By further assuming the initial data is a smooth compactly supported perturbation around a non-vacuum constant background, we prove the property of finite propagation speed of such a perturbation. The smooth solution is shown to blow up in finite time provided that the radial component of the initial "generalized" momentum is sufficiently large.