Finite time BV blowup for Liu-admissible solutions to $p$-system via computer-assisted proof

In this paper, we consider finite time blowup of the $BV$-norm for exact solutions to genuinely nonlinear hyperbolic systems in one space dimension, in particular the $p$-system. We consider solutions verifying shock admissibility criteria such as the Lax E-condition and the Liu E-condition. 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 a strictly convex entropy. The main results in this article compare to Jenssen [SIAM J. Math. Anal., 31(4):894--908, 2000], who shows $BV$ blowup for bounded solutions, or alternatively, blowup in $L^\infty$, for an artificial $3\times 3$ system which is not genuinely nonlinear. Baiti-Jenssen [Discrete Contin. Dynam. Systems, 7(4):837--853, 2001] improves upon this Jenssen result and can consider a genuinely nonlinear system, but then the blowup is only in $L^\infty$ and they cannot construct bounded solutions which blowup in $BV$. Moreover, their system is non-physical and provably does not admit a global, strictly convex entropy. Our result also shows sharpness of the recent Bressan-De Lellis result [Arch. Ration. Mech. Anal., 247(6):Paper No. 106, 12, 2023] concerning well-posedness via the Liu E-condition. The proof of our theorem is computer-assisted, following the framework of Sz\'{e}kelyhidi [Arch. Ration. Mech. Anal., 172(1):133--152, 2004]. Our code is available on the GitHub.