On finite time BV blow-up for the p-system

TL;DR

This paper investigates conditions under which the total variation of entropy weak solutions to the p-system can blow up in finite time, revealing the role of shock interactions and constructing approximate solutions with finite-time blow-up.

Contribution

It establishes that finite-time blow-up of total variation requires infinitely many large shocks and constructs approximate solutions exhibiting such blow-up.

Findings

Finite-time BV blow-up implies infinitely many large shocks.

Constructed approximate solutions with finite-time total variation blow-up.

Wave strengths from interactions are exactly characterized.

Abstract

The paper studies the possible blowup of the total variation for entropy weak solutions of the p-system, modeling isentropic gas dynamics. It is assumed that the density remains uniformly positive, while the initial data can have arbitrarily large total variation (measured in terms of Riemann invariants). Two main results are proved. (I) If the total variation blows up in finite time, then the solution must contain an infinite number of large shocks in a neighborhood of some point in the $t$-$x$ plane. (II) Piecewise smooth approximate solutions can be constructed whose total variation blows up in finite time. For these solutions the strength of waves emerging from each interaction is exact, while rarefaction waves satisfy the natural decay estimates stemming from the assumption of genuine nonlinearity.