Structural Compactness and Stability of Semi-Monotone Flows

Fitzpatrick's variational representation of maximal monotone operators is here extended to a class of pseudo-monotone operators in Banach spaces. On this basis, the associated first-order flow is here reformulated as a minimization principle, extending a method that was pioneered by Brezis, Ekeland, and Nayroles for gradient flows. This formulation is used to prove that this family of problems is compact and stable with respect to arbitrary perturbations not only of data but also of operators. This is achieved via evolutionary $\Gamma$-convergence with respect to a nonlinear topology of weak type. These results are applied to the Cauchy problem for quasilinear parabolic PDEs. This provides the structural compactness and stability of models of several physical phenomena: nonlinear diffusion, incompressible viscous flow, phase transitions, and so on.