Evolution equations with complete irreversibility and energy conservation

TL;DR

This paper studies an evolutionary variational inequality with complete irreversibility and energy conservation, establishing well-posedness, qualitative properties, and long-term behavior without relying on regularization.

Contribution

It introduces a novel approach to analyze variational inequalities with irreversibility and energy conservation, characterizing solutions as limits of generalized gradient flows.

Findings

Proved well-posedness using a minimizing movement scheme

Established comparison principle and qualitative properties of solutions

Analyzed long-time convergence to equilibrium

Abstract

This paper is concerned with the initial-boundary value problem for an evolutionary variational inequality complying with three intrinsic properties: complete irreversibility, unilateral equilibrium of an energy and an energy conservation law, which cannot generally be realized in dissipative systems such as standard gradient flows. Main results consist of well-posedness in a strong formulation, qualitative properties of strong solutions (i.e., comparison principle and the three properties mentioned above) and long-time dynamics of strong solutions (more precisely, convergence to an equilibrium). The well-posedness will be proved based on a minimizing movement scheme without parabolic regularization, which will also play a crucial role for proving qualitative and asymptotic properties of strong solutions. Moreover, the variational inequality under consideration will be characterized as a singular limit of some (generalized) gradient flows.