非局部守恒律系统：非局部广义Aw-Rascle-Zhang模型的适定性与奇异极限
Systems of nonlocal conservation laws: well-posedness and the singular limit for a nonlocal generalized Aw-Rascle-Zhang model
AI总结：
研究受交通流启发的非局部守恒律系统（非局部广义Aw-Rascle-Zhang模型），通过不动点论证证明弱解相关性质，包括存在唯一性、稳定性等，还研究了奇异极限，最后给出数值模拟及开放问题讨论。
AI中文摘要：
本文研究受交通流启发的非局部守恒律系统，即广义Aw-Rascle-Zhang（GARZ）模型的非局部版本。非局部性源于通过单边核进行的速度下游空间平均。通过在非局部速度中使用不动点论证，证明了有界变差初始数据的弱解的存在性和唯一性。还建立了关于初始数据的稳定性以及弱解在\(L^1\)中由强解的逼近。在关于速度和初始数据的额外物理意义假设下，得到了密度的最大值原理或不变区域估计。最后研究了非局部核收敛到狄拉克分布时的奇异极限，在额外假设下可证明收敛到唯一的局部熵解。提供了一些数值模拟，并以开放问题的讨论结束本文。
英文摘要：
In this paper, we study a system of nonlocal conservation laws motivated by traffic flow: a nonlocal version of the generalized Aw-Rascle-Zhang (GARZ) model. The nonlocality arises from downstream spatial averaging of the velocity by a one-sided kernel. We prove the existence and uniqueness of weak solutions for initial data of bounded variation via a fixed-point argument in the nonlocal velocity. We also establish stability with respect to the initial datum and an approximation of weak solutions by strong solutions in $L^1$. Under additional, physically meaningful assumptions on the velocity and the initial datum, we obtain either a maximum principle for the density or invariant-region estimates. Finally, we study the singular limit as the nonlocal kernel converges to a Dirac distribution. Indeed, under additional assumptions, convergence to the unique local entropy solution can be proved. Some numerical simulations are provided, and the paper concludes with a discussion of open problems.
