The instant-response limit in Whitham's nonlinear traffic-flow model: uniform well-posedness and global existence

Solutions of Whitham's traffic-flow equations are shown to exist globally in time for small values of the response-time parameter, provided that parameter vanishes more rapidly than the cube of the diffusion parameter. In the hyperbolic-hyperbolic singular limit that occurs as the response-time vanishes, the solution converges to the entropy weak solution of the simplified equations.