Finite time singularities for hyperbolic systems

TL;DR

This paper investigates finite time singularities in hyperbolic systems, showing how solutions can blow up with zero density but finite momentum, velocity, and energy, related to variational wave equations.

Contribution

It demonstrates the formation of finite time singularities with super norm blowup in a hyperbolic system linked to variational wave equations, extending understanding of blowup phenomena.

Findings

Solutions develop finite time singularities with zero density

Velocity and energy become infinite at blowup

Density remains zero while momentum stays finite

Abstract

In this paper, we study the formation of finite time singularities in the form of super norm blowup for a spatially inhomogeneous hyperbolic system. The system is related to the variational wave equations as those in [18]. The system posses a unique $C^1$ solution before the emergence of vacuum in finite time, for given initial data that are smooth enough, bounded and uniformly away from vacuum. At the occurrence of blowup, the density becomes zero, while the momentum stays finite, however the velocity and the energy are both infinity.