We consider a conservation law model of traffic flow, where the velocity of each car depends on a weighted average of the traffic density $$\rho $$ ρ ahead. The averaging kernel is of exponential type: $$w_\varepsilon (s)=\varepsilon ^{-1} e^{-s/\varepsilon }$$ w ε ( s ) = ε - 1 e - s / ε . By a transformation of coordinates, the problem can be reformulated as a $$2\times 2$$ 2 × 2 hyperbolic system with relaxation. Uniform BV bounds on the solution are thus obtained, independent of the scaling parameter $$\varepsilon $$ ε . Letting $$\varepsilon \rightarrow 0$$ ε → 0 , the limit yields a weak solution to the corresponding conservation law $$\rho _t + ( \rho v(\rho ))_x=0$$ ρ t + ( ρ v ( ρ ) ) x = 0 . In the case where the velocity $$v(\rho )= a-b\rho $$ v ( ρ ) = a - b ρ is affine, using the Hardy–Littlewood rearrangement inequality we prove that the limit is the unique entropy-admissible solution to the scalar conservation law.

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关于具有非局部通量的交通流：松弛表示

我们考虑交通流的守恒定律模型，其中每辆车的速度取决于前方交通密度 $$\rho $$ ρ 的加权平均值。平均核是指数型的：$$w_\varepsilon (s)=\varepsilon ^{-1} e^{-s/\varepsilon }$$ w ε ( s ) = ε - 1 e - s / ε 。通过坐标变换，问题可以重新表述为 $$2\times 2$$ 2 × 2 双曲线系统。因此获得了解的统一 BV 边界，与缩放参数 $$\varepsilon $$ ε 无关。让 $$\varepsilon \rightarrow 0$$ ε → 0 ，极限产生对应守恒定律 $$\rho _t + ( \rho v(\rho ))_x=0$$ ρ t + ( ρ v ( ρ ) ) x = 0。在速度 $$v(\rho )= ab\rho $$ v ( ρ ) = a - b ρ 是仿射的情况下，