One-dimensional Carrollian fluids II: $C^1$ blow-up criteria

TL;DR

This paper rigorously analyzes one-dimensional Carrollian fluid equations, establishing criteria for the existence or finite-time blow-up of solutions within a $C^1$ framework, relevant to theoretical physics and holography.

Contribution

It introduces a systematic $C^1$ analysis of Carrollian fluids, reducing the equations to conservation laws and classifying solution behaviors based on initial conditions and parameters.

Findings

Classified conditions for global existence of solutions.

Derived blow-up criteria for finite-time singularities.

Reduced complex equations to a manageable 2x2 conservation law system.

Abstract

The Carrollian fluid equations arise from the equations for relativistic fluids in the limit as the speed of light vanishes, and have recently experienced a surge of interest in the theoretical physics community in the context of asymptotic symmetries and flat-space holography. In this paper we initiate the rigorous systematic analysis of these equations by studying them in one space dimension in the $C^1$ setting. We begin by proposing a notion of isentropic Carrollian equations, and use this to reduce the Carrollian equations to a $2 \times 2$ system of conservation laws. Using the scheme of Lax, we then classify when $C^1$ solutions to the isentropic Carrollian equations exist globally, or blow up in finite time. Our analysis assumes a Carrollian analogue of a constitutive relation for the Carrollian energy density, with exponent in the range $\gamma \in (1,3]$.