ON BV-INSTABILITY AND EXISTENCE FOR LINEARIZED RADIAL EULER FLOWS

We provide concrete examples of immediate BV-blowup from small and radially symmetric initial data for the 3-dimensional, linearized Euler system. More precisely, we exhibit data arbitrarily close to a constant state, measured in L-infinity and BV (functions of bounded variation), whose solution has unbounded BV-norm at any positive time. Furthermore, this type of BV-instability can occur in the absence of any focusing waves in the solution. We also show that the BV-norm of a solution may well remain bounded while suffering L-infinity blowup due to wave focusing. Finally, we demonstrate how an argument based on scaling of the dependent variables, together with 1-d variation estimates, yields global existence for a class of finite energy, but possibly unbounded, radial solutions.