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

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 $L^{\mathrm{\infty }}$, and establish a kinetic formulation of the problem. This global existence result in $L^{\mathrm{\infty }}$ extends the $C^1$ theory presented in our companion paper ``One-Dimensional Carrollian Fluids II: $C^1$ Blow-up Criteria’'.