The Vanishing Power-Like Coupling Term Limits of Riemann Solutions for the Mean-Field Games

Construction of Riemann solutions is offered at length for a hyperbolic system in the one-dimensional setting and conservative form derived from the one-dimensional backward–forward mean-field games in the anti-monotone case with the quadratic Hamiltonian and the power-like coupling term. The formation of singular delta shock wave and vacuum state is concretely explored in the vanishing power-like coupling term limits of Riemann solutions consisting of either two shock waves or two rarefaction waves, where the remarkable phenomena of concentration and cavitation can be nearly observed and analyzed. In addition, the theoretical analysis results are validated through corresponding numerical experiments.