No BV bounds for approximate solutions to p-system with general pressure law

For the p-system with large BV initial data, an assumption introduced in [N. S. Bakhvalov, Z. Vycisl. Mat. i Mat. Fiz. (Russian) 10 (1970) 969-980] by Bakhvalov guarantees the global existence of entropy weak solutions with uniformly bounded total variation. The present paper provides a partial converse to this result. Whenever Bakhvalov's condition does not hold, we show that there exist front tracking approximate solutions, with uniformly positive density, whose total variation becomes arbitrarily large. The construction extends the arguments in [A. Bressan, G. Chen and Q. Zhang, J. Diff. Eqs. 256(8) (2014) 3067-3085] to a general class of pressure laws.