发表机构
Département de mathématiques et applications École normale supérieure, Université PSL, CNRS（巴黎高等师范学院，巴黎文理研究大学，法国国家科学研究中心）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文通过计算ansatz和计算机辅助搜索，找到特定压力律的p-系统，在满足强迹性质时获得黎曼解唯一性，同时利用T₆构型构造非唯一解，解决强迹猜想，并证明T∞和T_N结构不存在。
AI 中文摘要
本文考虑一维空间中的2×2双曲守恒律系统。我们使用作者与Székelyhidi在双曲理论中引入的计算ansatz来研究与偏微分方程对应的本构集。该集合的秩一凸几何与非唯一性和低正则解的存在性相关。通过计算机辅助搜索，我们找到一个压力律p，使得具有该压力律的p-系统及其自然严格凸熵满足大数据L²稳定性和“带平移的a-收缩”技术所需的所有条件，从而在满足强迹性质的解类中获得了某些黎曼解的唯一性。同时，本构集包含一个T₆构型，我们利用它构造了不具有强迹性质的非唯一解。这解决了强迹的尖锐性问题。我们还给出了证明，表明所有真正非线性系统不存在T∞结构，且对于所有N，p''>0的p-系统不存在T_N结构，从而阻断了这些通往凸积分的路径。
英文摘要
In this paper, we consider $2\times2$ hyperbolic systems of conservation laws in one spatial dimension. We use the computational ansatz introduced in the hyperbolic theory by the author and Székelyhidi to study the constitutive set corresponding to the PDE. The rank-one convex geometry of this set relates to non-uniqueness and the existence of low-regularity solutions. Through a computer-assisted search, we find a pressure law $p$ such that the $p$-system with this pressure law, and its natural strictly convex entropy, verifies all of the conditions necessary for the large data $L^2$ stability and the technique of ``$a$-contraction with shifts'' and thus we have uniqueness of certain Riemann solutions in the class of solutions verifying the Strong Trace Property. At the same time, the constitutive set contains a $T_6$ configuration and we use it to construct non-unique solutions without the Strong Trace Property. This resolves the question of sharpness of strong traces. We also present proofs which show nonexistence of $T_\infty$ structures for all genuinely nonlinear systems and nonexistence of $T_N$ structures for all $N$ for the $p$-system with $p''>0$, thus blocking these routes towards convex integration.
Comments93 pages, 9 figures, 849 lines of symbolic and numerical MATLAB code (not including comments). The associated GitHub repository is at https://github.com/sammykrupa/StrongTraceConjecture
