  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.
