We consider a class of nonlocal conservation laws with a second-order viscous regularization term which finds an application in modelling macroscopic traffic flow. The velocity function depends on a weighted average of the density ahead, where the averaging kernel is of exponential type. We show that, as the nonlocal impact and the viscosity parameter simultaneously tend to zero (under a suitable balance condition), the solution of the nonlocal problem converges to the entropy solution of the corresponding local conservation law. The key ideas of our proof are to observe that the nonlocal term satisfies a third-order equation with diffusive and dispersive effects and to deduce a suitable energy estimate on the nonlocal term. The convergence result is then based on the compensated compactness theory.

我们考虑一类具有二阶粘性正则化项的非局部守恒定律，它在宏观交通流建模中得到了应用。速度函数取决于前面密度的加权平均值，其中平均内核是指数类型的。我们表明，由于非局部影响和粘度参数同时趋于零（在合适的平衡条件下），非局部问题的解会收敛到相应局部守恒定律的熵解。我们证明的关键思想是观察非局部项满足具有扩散和色散效应的三阶方程，并推导出对非局部项的合适能量估计。然后收敛结果基于补偿紧凑性理论。