📝 SummarXiv

Abstract

We rigorously construct non-isentropic and self-similar multi-d Euler flows in which a central cavity (vacuum region) collapses. While isentropic flows of this type have been analyzed earlier by Hunter \cite{hun_60} and others, the non-isentropic setting introduces additional complications, in particular with respect to the behavior along the fluid-vacuum interface. The flows we construct satisfy the physical boundary conditions: the interface is a material surface along which the pressure vanishes, and it propagates with a non-vanishing and finite acceleration until collapse. We introduce a number of algebraic conditions on the parameters in the problem (spatial dimension, adiabatic index, similarity parameters). With these conditions satisfied, a simple argument based on trapping regions for the associated similarity ODEs yields the existence of non-isentropic cavity flows. We finally verify that the conditions are all met for several physically relevant cases in both two and three dimensions.