通过计算机辅助证明的 $p$-系统 Liu 可解解的有限时间 BV 爆破
Finite time BV blowup for Liu-admissible solutions to $p$-system via computer-assisted proof
AI总结：
本文通过计算机辅助证明，针对一维空间真正非线性双曲系统（特别是 $p$-系统）的 Liu 可解解，构造了从状态空间开集出发的 Riemann 初值，使得存在无穷多个有界解，其 $BV$ 范数在有限时间内甚至瞬时爆破，并证明了该结果与 Bressan-De Lellis 关于 Liu E-条件适定性的结果的最优性。
摘要：
本文考虑一维空间真正非线性双曲系统（特别是 $p$-系统）精确解的 $BV$-范数有限时间爆破。我们考虑满足激波可接受准则（如 Lax E-条件和 Liu E-条件）的解。特别地，我们给出 Riemann 初值，它允许无穷多个有界解，每个解不仅经历有限时间，而且实际上瞬时 $BV$ 范数爆破。该 Riemann 初值允许来自状态空间中的开集。我们的方法可证明不承认严格凸熵。 本文的主要结果与 Jenssen [SIAM J. Math. Anal., 31(4):894--908, 2000] 比较，后者展示了人工 $3\times 3$ 系统（非真正非线性）有界解的 $BV$ 爆破，或 $L^\infty$ 爆破。Baiti-Jenssen [Discrete Contin. Dynam. Systems, 7(4):837--853, 2001] 改进了 Jenssen 的结果，可以考虑真正非线性系统，但爆破仅在 $L^\infty$ 中，且他们无法构造在 $BV$ 中爆破的有界解。此外，他们的系统是非物理的，且可证明不承认全局严格凸熵。我们的结果也显示了近期 Bressan-De Lellis 结果 [Arch. Ration. Mech. Anal., 247(6):Paper No. 106, 12, 2023] 关于 Liu E-条件适定性的最优性。我们的定理证明是计算机辅助的，遵循 Székelyhidi [Arch. Ration. Mech. Anal., 172(1):133--152, 2004] 的框架。我们的代码可在 GitHub 上获取。
英文摘要：
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 Székelyhidi [Arch. Ration. Mech. Anal., 172(1):133--152, 2004]. Our code is available on the GitHub.
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