Asymptotic behavior of nonautonomous monotone and subgradient evolution equations

In a Hilbert setting $H$, we study the asymptotic behavior of the trajectories of nonautonomous evolution equations $\dot{x}(t)+A_t(x(t))\ni 0$, where for each $t\ge 0$, $A_t:H\text{tto}H$ denotes a maximal monotone operator. We provide general conditions guaranteeing the weak ergodic convergence of each trajectory $x(\cdot )$ to a zero of a limit maximal monotone operator $A_{\mathrm{\infty }}$, as the time variable $t$ tends to $+\mathrm{\infty }$. The crucial point is to use the Br'ezis-Haraux function, or equivalently the Fitzpatrick function, to express at which rate the excess of $\text{gph}{A}_{\mathrm{\infty }}$ over $\text{gph}{A}_t$ tends to zero. This approach gives a sharp and unifying view on this subject. In the case of operators $A_t=\mathrm{\partial }{\varphi }_t$ which are subdifferentials of closed convex functions ${\varphi }_t$, we show convergence results for the trajectories. Then, we specialize our results to multiscale evolution equations, and obtain asymptotic properties of hierarchical minimization, and selection of viscosity solutions. Illustrations are given in the field of coupled systems, and partial differential equations.