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Preprint 2014-014

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 [3] 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 [4] to a general class of pressure laws.

References
[3] 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).
[4] 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].
Paper:
Available as PDF (375 Kbytes).
Author(s):
Alberto Bressan
Geng Chen
Qingtian Zhang
Shengguo Zhu
Submitted by:
; 2014-10-03.