LAUSR

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.