Papers › Finite time BV blowup for Liu-admissible solutions to p-system via computer-assisted proof
Finite time BV blowup for Liu-admissible solutions to p-system via computer-assisted proof
Sam G. Krupa
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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^∞, for an artificial 3×3 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^∞ 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.
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