We study a class of nonlinear nonlocal conservation laws with discontinuous flux, modeling crowd dynamics and traffic flow, without any additional conditions on finiteness/discreteness of the set of discontinuities or on the monotonicity of the kernel/the discontinuous coefficient. Strong compactness of the Godunov and Lax-Friedrichs type approximations is proved, providing the existence of entropy solutions. A proof of the uniqueness of the adapted entropy solutions is provided, establishing the convergence of the entire sequence of finite volume approximations to the adapted entropy solution. As per the current literature, this is the first well-posedness result for the aforesaid class and connects the theory of nonlocal conservation laws (with discontinuous flux), with its local counterpart in a generic setup. Some numerical examples are presented to display the performance of the schemes and explore the limiting behavior of these nonlocal conservation laws to their local counterparts.

翻译：本文研究一类具有不连续通量的非线性非局部守恒律，该方程可模拟人群动力学与交通流问题，且无需对间断点集的有限性/离散性或核函数/不连续系数的单调性施加额外条件。证明了Godunov型和Lax-Friedrichs型逼近的强紧性，从而确保熵解的存在性。通过建立适应熵解的唯一性证明，论证了整个有限体积逼近序列收敛于适应熵解。据现有文献而言，这是针对上述方程类别的首个适定性结果，并在一般框架下建立了非局部守恒律（含不连续通量）与其局部对应理论的联系。文中还给出了数值算例以展示格式的性能，并探索这些非局部守恒律向局部对应方程的极限行为。