Structural Compactness and Stability of Pseudo-Monotone Flows

TL;DR

This paper extends Fitzpatrick's variational representation to pseudo-monotone operators, reformulates associated flows as minimization problems, and proves their stability and compactness, with applications to nonlinear PDEs modeling physical phenomena.

Contribution

It introduces a variational framework for pseudo-monotone flows, extending existing methods and establishing stability under perturbations for a broad class of operators.

Findings

Proves stability of flows under data and operator perturbations.

Extends variational representation to pseudo-monotone operators.

Applies results to nonlinear PDEs in physics.

Abstract

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 initial-value problem associated with the first-order flow of such an operator 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 the problem is stable w.r.t.\ arbitrary perturbations not only of data but also of operators. This is achieved by using the notion of evolutionary $\Gamma$-convergence w.r.t.\ 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 the model of several physical phenomena: nonlinear diffusion, incompressible viscous flow, phase transitions, and so on.