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

We address the problem of regularity of solutions $u^i(t, x^1, \dots, x^N)$ to a family of semilinear parabolic systems of $N$ equations, which describe closed-loop equilibria of some $N$-player differential games with Lagrangian having quadratic behaviour in the velocity variable, running costs $f^i(x)$ and final costs $g^i(x)$. By global (semi)monotonicity assumptions on the data $f=(f^i)_{1 \leq i \leq N}$ and $g=(g^i)_{1 \leq i \leq N}$, and assuming that derivatives of $f^i, g^i$ in directions $x^j$ are of order $1/N$ for $j \neq i$, we prove that derivatives of $u^i$ enjoy the same property. The estimates are uniform in the number of players $N$. Such a behaviour of the derivatives of $f^i, g^i$ 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 $N \to \infty$ 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 $N \to \infty$ and vanishing viscosity limit.