One-Dimensional Carrollian Fluids III: Global Existence and Weak Continuity in $L^\infty$

TL;DR

This paper establishes the global existence of bounded entropy solutions for one-dimensional Carrollian fluid equations with a specific constitutive law, using vanishing viscosity and compensated compactness methods, advancing the mathematical understanding of these equations.

Contribution

It provides the first rigorous proof of global-in-time solutions in $L^ Infty$ for Carrollian fluids, extending previous blow-up criteria results.

Findings

Proved global existence of entropy solutions in $L^ Infty$

Developed a compensated compactness framework for the equations

Established a kinetic formulation of the Carrollian fluid equations

Abstract

The Carrollian fluid equations arise as the $c \to 0$ limit of the relativistic fluid equations and have recently experienced a surge of activity in the flat-space holography community. However, the rigorous mathematical well-posedness theory for these equations does not appear to have been previously studied. This paper is the third in a series in which we initiate the systematic analysis of the Carrollian fluid equations. In the present work we prove the global-in-time existence of bounded entropy solutions to the isentropic Carrollian fluid equations in one spatial dimension for a particular constitutive law ($\gamma = 3$). Our method is to use a vanishing viscosity approximation for which we establish a compensated compactness framework. Using this framework we also prove the compactness of entropy solutions in $L^\infty$, and establish a kinetic formulation of the problem. This global existence result in $L^\infty$ extends the $C^1$ theory presented in our companion paper ``One-Dimensional Carrollian Fluids II: $C^1$ Blow-up Criteria''.