NONUNIQUENESS OF BV SOLUTIONS OF QUASILINEAR HYPERBOLIC SYSTEMS

Abstract

We construct examples of one-dimensional quasilinear hyperbolic systems for which the constant state and Riemann problem are unstable under O(1) perturbations in the space BV. We generate a system and solution consisting of three compressions which focus at the same point, resulting in a nonlinear interaction from which no waves emerge. Since this solution is time-reversible, this leads to nontrivial solutions of the system with constant initial data. These solutions are classical in the sense that they do not contain shocks and are discontinuous only at the origin. We then find some elementary necessary conditions for a system in conservative form to exhibit this type of behavior.