No BV bounds for approximate solutions to p-system with general pressure law
Alberto Bressan, Geng Chen, Qingtian Zhang, and Shengguo Zhu
Abstract: For the p-system with large BV initial data, an assumption introduced in  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  to a general class of pressure laws.
 N. S. Bakhvalov, The existence in the large of a regular solution of a quasilinear hyperbolic system, Ž. Vyčisl. Mat. i Mat. Fiz. 10 (1970), 969–980 (Russian).
 A. Bressan, G. Chen, and Q. Zhang, Lack of BV bounds for approximate solutions to the p-system with large data, J. Differential Equations 256 (2014), no. 8, 3067–3085 [MR3199757].