A priori estimates and large population limits for some nonsymmetric Nash systems with semimonotonicity

We address the problem of regularity of solutions (Formula presented.) to a family of semilinear parabolic systems of (Formula presented.) equations, which describe closed-loop equilibria of some (Formula presented.) -player differential games with Lagrangian having quadratic behaviour in the velocity variable, running costs (Formula presented.) and final costs (Formula presented.). By global (semi)monotonicity assumptions on the data (Formula presented.) and (Formula presented.), and assuming that derivatives of (Formula presented.) in directions (Formula presented.) are of order (Formula presented.) for (Formula presented.), we prove that derivatives of (Formula presented.) enjoy the same property. The estimates are uniform in the number of players (Formula presented.). Such a behaviour of the derivatives of (Formula presented.) arise in the theory of Mean Field Games, though here we do not make any symmetry assumption on the data. Then, by the estimates obtained we address the convergence problem (Formula presented.) in a ‘heterogeneous’ Mean Field framework, where players all observe the empirical measure of the whole population, but may react differently from one another. We also discuss some results on the joint (Formula presented.) and vanishing viscosity limit.