One-dimensional Carrollian fluids III: Global existence and weak continuity in L ∞ $L^\infty$

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 [2].