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Well-posedness and mean-field limit of discontinuous weighted dynamics via the relative entropy method

Immanuel Ben Porat, José A. Carrillo, Alexandra Holzinger

Abstract

We consider deterministic particle dynamics with time evolving weights and their associated Kolmogorov equation and mean-field equation. We prove existence and unique- ness for the limit PDE alongside estimates on the growth of the logarithmic gradient as well as existence of weak solutions for the Kolmogorov equation satisfying an appropriate entropy inequality. We then apply these estimates and the relative entropy method as developed in [17], in order to derive the associated equation as a mean field limit. Our results cover both interactions and influence kernels with mild regularity assumptions.

Well-posedness and mean-field limit of discontinuous weighted dynamics via the relative entropy method

Abstract

We consider deterministic particle dynamics with time evolving weights and their associated Kolmogorov equation and mean-field equation. We prove existence and unique- ness for the limit PDE alongside estimates on the growth of the logarithmic gradient as well as existence of weak solutions for the Kolmogorov equation satisfying an appropriate entropy inequality. We then apply these estimates and the relative entropy method as developed in [17], in order to derive the associated equation as a mean field limit. Our results cover both interactions and influence kernels with mild regularity assumptions.
Paper Structure (11 sections, 20 theorems, 227 equations)

This paper contains 11 sections, 20 theorems, 227 equations.

Table of Contents

  1. Introduction
  2. Previous results
  3. Main results and assumptions
  4. Well-posedness of limiting PDE and existence of solutions to Kolmogorov equation
  5. Notation and preliminaries
  6. Well-posedness of limiting PDE
  7. Existence of weak solutions for the $N$-particle hierarchy \ref{['Kolmogorov eq']}
  8. The mean field limit
  9. Evolution in time of the relative entropy
  10. Proof of Theorem \ref{['main thm']}
  11. Proof of Theorem \ref{['final exp reminder']}

Key Result

Theorem 1.4

Let assumptions H1-H2 hold.

Theorems & Definitions (43)

  • Remark 1.1
  • Remark 1.2
  • Definition 1.3: Strong solution to \ref{['limit PDE Intro']}
  • Theorem 1.4: Limiting PDE \ref{['limit PDE Intro']}
  • Definition 1.5: Weak solution to \ref{['Kolmogorov eq']}
  • Proposition 1.6
  • Theorem 1.7: Propagation of chaos
  • Proposition 2.1
  • proof
  • Lemma 2.2
  • ...and 33 more