  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 $\gamma \in (1,3]$.
