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Hamilton--Jacobi equations for controlled gradient flows: cylindrical test functions

Giovanni Conforti, Richard C. Kraaij, Daniela Tonon

TL;DR

This work advances a rigorous, intrinsic well-posedness framework for Hamilton-Jacobi equations tied to controlled gradient flows in metric spaces, extending the program initiated in CoKrTo21. It introduces smooth cylindrical test functions and proves that viscosity sub/supersolutions for the cylindrical Hamiltonians $H_{\dagger},H_{\ddagger}$ correspond to viscosity solutions for the non-smooth operators $\widetilde{H}_{\dagger},\widetilde{H}_{\ddagger}$, thereby enabling a robust comparison principle and opening the door to an existence theory. The core strategy runs a six-step approximation chain from the cylindrical to Tataru-distance–inclusive Hamiltonians, using EVI structure, Laplace-type smoothing of the Tataru distance, large-deviation tools (Varadhan), and a viscosity-pushing toolbox to preserve sub/supersolution properties across steps. The results apply in general geodesic metric spaces with energy functionals satisfying realistic regularity/compactness assumptions, and particularly illuminate the Wasserstein space context with various entropies. Collectively, the paper lays the groundwork for a comprehensive well-posedness and existence theory for infinite-dimensional Hamilton-Jacobi equations arising from gradient-flow control problems.

Abstract

This work is the second part of a program initiated in arXiv:2111.13258 aiming at the development of an intrinsic geometric well-posedness theory for Hamilton-Jacobi equations related to controlled gradient flow problems in metric spaces. Our main contribution is that of showing that the comparison principle proven therein implies a comparison principle for viscosity solutions relative to smoother Hamiltonians, acting on test functions that are mere cylindrical functions of the underling squared metric distance and whose rigorous definition is achieved from the Evolutional Variational Inequality formulation of gradient flows (EVI). In particular, the new Hamiltonians no longer require to work with test functions containing Tataru's distance. This substantial simplification paves the way for the development of a comprehensive existence theory.

Hamilton--Jacobi equations for controlled gradient flows: cylindrical test functions

TL;DR

This work advances a rigorous, intrinsic well-posedness framework for Hamilton-Jacobi equations tied to controlled gradient flows in metric spaces, extending the program initiated in CoKrTo21. It introduces smooth cylindrical test functions and proves that viscosity sub/supersolutions for the cylindrical Hamiltonians correspond to viscosity solutions for the non-smooth operators , thereby enabling a robust comparison principle and opening the door to an existence theory. The core strategy runs a six-step approximation chain from the cylindrical to Tataru-distance–inclusive Hamiltonians, using EVI structure, Laplace-type smoothing of the Tataru distance, large-deviation tools (Varadhan), and a viscosity-pushing toolbox to preserve sub/supersolution properties across steps. The results apply in general geodesic metric spaces with energy functionals satisfying realistic regularity/compactness assumptions, and particularly illuminate the Wasserstein space context with various entropies. Collectively, the paper lays the groundwork for a comprehensive well-posedness and existence theory for infinite-dimensional Hamilton-Jacobi equations arising from gradient-flow control problems.

Abstract

This work is the second part of a program initiated in arXiv:2111.13258 aiming at the development of an intrinsic geometric well-posedness theory for Hamilton-Jacobi equations related to controlled gradient flow problems in metric spaces. Our main contribution is that of showing that the comparison principle proven therein implies a comparison principle for viscosity solutions relative to smoother Hamiltonians, acting on test functions that are mere cylindrical functions of the underling squared metric distance and whose rigorous definition is achieved from the Evolutional Variational Inequality formulation of gradient flows (EVI). In particular, the new Hamiltonians no longer require to work with test functions containing Tataru's distance. This substantial simplification paves the way for the development of a comprehensive existence theory.
Paper Structure (33 sections, 27 theorems, 188 equations, 1 figure)

This paper contains 33 sections, 27 theorems, 188 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Hamilton--Jacobi equations in infinite dimensional spaces
  3. Contribution of this work and perspectives
  4. Organization
  5. Acknowledgements
  6. Introduction to the technical aspects of the paper
  7. Bounds via the evolutional variational inequality
  8. Test functions: Comparison via Ekeland and Tataru's distance
  9. Test functions: Existence via smooth cylindrical test functions
  10. Relating the sets of test functions: a stepping stone towards well-posedness theory
  11. EVI-gradient flows and main results
  12. Set-up
  13. Two collections of Hamiltonians
  14. Main results
  15. Preparations for the proof of Theorem \ref{['theorem:carry_over_solutions']}
  16. ...and 18 more sections

Key Result

Theorem 3.15

Let Assumptions assumption:distance_and_energy, assumption:weak_topology and assumption:gradientflow be satisfied. Let $\lambda > 0$ and $h \in C_b(E)$ be weakly continuous. We then have

Figures (1)

  • Figure 1: Carrying over viscosity sub- and supersolutions. In this diagram, an arrow connecting an operator $A$ with operator $B$ with subscript 'sub' means that viscosity subsolutions of $f - \lambda A f = h$ are also viscosity subsolutions of $f - \lambda B f = h$. If a line is dashed, the result holds under the additional assumption that the subsolution is weakly upper semi-continuous. Similarly for arrows with a subscript 'super'. The loop equipped with 'weak reg. sub' means that the weak upper semi-continuous regularization of any viscosity subsolution is also a viscosity subsolution. Similarly for the loop decorated with 'weak reg. super'.

Theorems & Definitions (66)

  • Definition 3.1
  • Definition 3.2
  • Remark 3.5
  • Definition 3.6
  • Definition 3.8
  • Definition 3.10
  • Remark 3.11
  • Definition 3.12
  • Definition 3.13
  • Definition 3.14
  • ...and 56 more