Continuous time approximation of Nash equilibria

We consider the problem of approximating Nash equilibria of $N$ functions $f_1,\dots, f_N$ of $N$ variables. In particular, we deduce conditions under which systems of the form $$\dot u_j(t)=-\nabla_{x_j}f_j(u(t))$$ $(j=1,\dots, N)$ are well posed and in which the large time limits of their solutions $u(t)=(u_1(t),\dots, u_N(t))$ are Nash equilibria for $f_1,\dots, f_N$. To this end, we will invoke the theory of maximal monotone operators. We will also identify an application of these ideas in game theory and show how to approximate equilibria in some game theoretic problems in function spaces.