Global well-posedness of first-order mean field games and master equations with nonlinear dynamics

This paper addresses the generic first-order mean field game problem with nonlinear dynamics. A major contribution here is the provision of new crucial a priori estimates for the associated forward-backward ordinary differential equation (FBODE) system. In addition, we require monotonicity conditions intimately on the coefficient functions but not on the Hamiltonians to handle their non-separable nature and nonlinear dynamics; as tackling Hamiltonians directly, it potentially dissolves much useful information. Moreover, we are able to deal with the non-separable Hamiltonian under the hypothesis that the first-order derivative of the drift function in the measure variable is not too large relative to the convexity of the running cost function. Our approach involves demonstrating the local existence of a solution, providing new a priori estimates for the sensitivity of the backward equation with respect to the forward initial condition, and gluing these into a global solution. Furthermore, we establish the local and global existence as well as uniqueness of classical solutions for the mean field game and its master equation. To illustrate the effectiveness of our proposed general theory, we apply it to resolve two non-trivial non-linear-quadratic examples with a non-separable Hamiltonian and nonlinear dynamics, which remains unexplained in contemporary literature.