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

Samuel Mercer, Yves van Gennip

TL;DR

The paper extends the classical Brezis–Pazy semigroup convergence to a novel Banach-stacking framework that handles operators acting on different Banach spaces embedded in a common metric space. It introduces TL^p as a key stacking example and proves a generalized Brezis–Pazy theorem guaranteeing uniform semigroup convergence from resolvent convergence, plus three main gradient-flow convergence results under Gamma-convergence: (i) in Hilbert-stackings with lambda-convex functionals, (ii) in TL^p(Omega) with P0-convex functionals, and (iii) corresponding to Gamma-converging functionals with relaxed coercivity. A central tool is the Moreau envelope energy bound, which enables convergence without well-prepared initial data. Collectively, the results facilitate discrete-to-continuum gradient-flow convergence in graph-based and TL^p settings, broadening the applicability of nonlinear semigroup theory to multi-space evolutions.

Abstract

A 1972 theorem by Brezis and Pazy establishes the uniform convergence of nonlinear semigroups generated by $ω$-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 $Γ$-converging $λ$-convex functions on a Banach stacking of Hilbert spaces; the third a proof of uniform convergence for gradient flows of $Γ$-converging functions that satisfy a convexity condition that was formulated by Bénilan 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 García Trillos and Slepčev in a 2016 paper.

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

TL;DR

The paper extends the classical Brezis–Pazy semigroup convergence to a novel Banach-stacking framework that handles operators acting on different Banach spaces embedded in a common metric space. It introduces TL^p as a key stacking example and proves a generalized Brezis–Pazy theorem guaranteeing uniform semigroup convergence from resolvent convergence, plus three main gradient-flow convergence results under Gamma-convergence: (i) in Hilbert-stackings with lambda-convex functionals, (ii) in TL^p(Omega) with P0-convex functionals, and (iii) corresponding to Gamma-converging functionals with relaxed coercivity. A central tool is the Moreau envelope energy bound, which enables convergence without well-prepared initial data. Collectively, the results facilitate discrete-to-continuum gradient-flow convergence in graph-based and TL^p settings, broadening the applicability of nonlinear semigroup theory to multi-space evolutions.

Abstract

A 1972 theorem by Brezis and Pazy establishes the uniform convergence of nonlinear semigroups generated by -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 -converging -convex functions on a Banach stacking of Hilbert spaces; the third a proof of uniform convergence for gradient flows of -converging functions that satisfy a convexity condition that was formulated by Bénilan and Crandall in a 1991 publication (and which we term `-convexity') on a Banach stacking of spaces, corresponding to the space introduced by García Trillos and Slepčev in a 2016 paper.

Paper Structure

This paper contains 27 sections, 66 theorems, 238 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Setting and main results
  3. Convex analysis
  4. Convergence of semigroups and gradient flows
  5. Relations to the literature
  6. Section breakdown
  7. Semigroups and notation
  8. Operators
  9. Notions of accretivity
  10. Semigroups
  11. Properties of lambda-convex functions and gradient flows
  12. Definition and properties of lambda-convex functions
  13. Definition and properties of gradient flows
  14. Subdifferentials over Lp and P0-convexity
  15. Banach stackings
  16. ...and 12 more sections

Key Result

Lemma 2.13

Suppose that $X$ is a Hilbert space with an inner product $\langle \cdot,\cdot \rangle$. Then $A$ is accretive on $X$ if and only if, for all $(x,y),(\hat{x},\hat{y})\in A$,

Figures (4)

  • Figure 1: Schematic illustration of a Banach stacking
  • Figure 2: Non-$\lambda$-convex
  • Figure 3: $\lambda$-convex
  • Figure 5: Moreau-envelope bound for $\Phi(x):=|x|$

Theorems & Definitions (216)

  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Definition 2.4
  • Definition 2.5
  • Definition 2.6
  • Definition 2.7
  • Definition 2.8
  • Definition 2.9
  • Remark 2.10
  • ...and 206 more