Variation Instability Near Vacuum in One-Dimensional Isentropic Flow

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 <= 3. When the initial velocity field is unbounded, variation blowup in. can occur for any gamma > 1.