We consider a conservation law model of traffic flow, where the velocity of each car depends on a weighted average of the traffic density ρ ahead. The averaging kernel is of exponential type: wε(s) = ε- 1e-s/ε. By a transformation of coordinates, the problem can be reformulated as a 2 × 2 hyperbolic system with relaxation. Uniform BV bounds on the solution are thus obtained, independent of the scaling parameter ε. Letting ε→ 0 , the limit yields a weak solution to the corresponding conservation law ρt+ (ρv(ρ)) x= 0. In the case where the velocity v(ρ) = a- bρ is affine, using the Hardy–Littlewood rearrangement inequality we prove that the limit is the unique entropy-admissible solution to the scalar conservation law.
All Science Journal Classification (ASJC) codes
- Mathematics (miscellaneous)
- Mechanical Engineering