Variation Instability Near Vacuum in One-Dimensional Isentropic Flow

Abstract

Abstract We consider instability of the total variation in shock-only solutions to the one-dimensional isentropic Euler system. The main results concern the possibility of immediate blowup in variation of the density field $\rho $ when the data approaches vacuum. In the case of an initially bounded velocity field, it is verified that immediate variation blowup in $\rho $ can occur whenever the adiabatic exponent satisfies $\gamma>3$, whereas no such instability occurs for $1<\gamma \leq 3$. When the initial velocity field is unbounded, variation blowup in $\rho $ can occur for any $\gamma>1$.