An extension to Banach stackings of the Brezis--Pazy semigroup-convergence theorem, with applications to $\lambda$-convex gradient flows

TL;DR

This paper extends the Brezis--Pazy semigroup convergence theorem to a broader setting called Banach stacking, enabling new applications in graph-based gradient flows and convergence analysis across different Banach spaces.

Contribution

The paper introduces Banach stacking as a generalization of the theorem, providing new convergence results for semigroups and gradient flows in this expanded framework.

Findings

Proves uniform convergence of semigroups with pointwise generator convergence.

Establishes convergence of gradient flows for $\Gamma$-converging $\lambda$-convex functions.

Demonstrates convergence results for $P_0$-convex functions on $L^p$ spaces.

Abstract

A 1972 theorem by Brezis and Pazy establishes the uniform convergence of nonlinear semigroups generated by $\omega$-accretive operators on a Banach space. Our goal is to expand the setting of this theorem to include nonlinear semigroups that are acting on different Banach spaces. This is useful, for example, to prove discrete-to-continuum convergence for graph-based gradient flows. We name the general setting in which our theorem holds a Banach stacking. We give three main applications of the extended theorem that are of independent interest. The first establishes uniform convergence of semigroups in a Banach stacking if the generators of the semigroups converge pointwise. The second is a proof of uniform convergence for gradient flows of $\Gamma$-converging $\lambda$-convex functions on a Banach stacking of Hilbert spaces; the third a proof of uniform convergence for gradient flows of $\Gamma$-converging functions that satisfy a convexity condition that was formulated by B\'enilan and Crandall in a 1991 publication (and which we term `$P_0$-convexity') on a Banach stacking of $L^p$ spaces, corresponding to the $TL^p$ space introduced by Garc\'ia Trillos and Slep\v{c}ev in a 2016 paper.