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

TL;DR

This paper demonstrates that for the p-system with general pressure laws, if Bakhvalov's condition fails, then approximate solutions can have unbounded total variation, indicating limitations in existing BV bounds.

Contribution

It extends previous results by showing the non-existence of BV bounds for approximate solutions under general pressure laws when Bakhvalov's condition is not satisfied.

Findings

Existence of front tracking solutions with unbounded total variation

Extension of previous arguments to a broader class of pressure laws

Failure of BV bounds without Bakhvalov's condition

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.