Non-Uniqueness and BV Blowup for Vacuously Liu-Admissible Solutions to P-System Via Computer-Assisted Proof

Abstract

In this paper, we consider non-uniqueness and finite time blowup of the BV-norm for exact solutions to genuinely nonlinear hyperbolic systems in one space dimension, in particular the p-system. The recent Bressan-De Lellis result [Arch. Ration. Mech. Anal., 247(6):Paper No. 106, 12, 2023] shows that whenever a BV solution exists, with finite (but possibly very large total variation), it is unique if each shock verifies the Liu E-condition. We show non-uniqueness of solutions by convex integration. The solutions we construct are Liu-admissible for a trivial reason: there are no shocks, so the Liu E-condition is vacuously satisfied. But our construction shows exactly that there is an issue, a qualitative difference between small data solutions (for example in the BV class) and large data solutions, where this exact type of phenomenon might occur. Our result can be interpreted as a cautionary example to show that the Liu E-condition is satisfactory for small oscillations but not for large oscillations where typically these type of constructions appear. In particular, we present Riemann initial data which admits infinitely many bounded solutions, each of which experience, not just finite time, but in fact instantaneous blowup of the BV norm. The Riemann initial data is allowed to come from an open set in state space. Our method provably does not admit the natural strictly convex entropy. The proof of our theorem is computer-assisted. Our code is available on the GitHub.