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A system of continuity equations with nonlocal interactions of Morse type

Marco Di Francesco, Valeria Iorio

Abstract

We study a system of two continuity equations with nonlocal velocity fields using interaction potentials of both attractive and repulsive Morse type. Such a system is of interest in many contexts in multi-population modelling. We prove existence, uniqueness and stability in the 2-Wasserstein spaces of probability measures via Jordan-Kinderlehrer-Otto scheme and gradient flow solutions in the spirit of the Ambrosio-Gigli-Savaré theory. We then formulate a deterministic particle scheme for this model and prove that gradient flow solutions are obtained in the many particle limit by discrete densities constructed out of moving particles satisfying a suitable system of ODEs. The ODE system is formulated in a non standard way in order to bypass the Lipschitz singularity of the kernel, with difference quotients of the kernel replacing its derivative.

A system of continuity equations with nonlocal interactions of Morse type

Abstract

We study a system of two continuity equations with nonlocal velocity fields using interaction potentials of both attractive and repulsive Morse type. Such a system is of interest in many contexts in multi-population modelling. We prove existence, uniqueness and stability in the 2-Wasserstein spaces of probability measures via Jordan-Kinderlehrer-Otto scheme and gradient flow solutions in the spirit of the Ambrosio-Gigli-Savaré theory. We then formulate a deterministic particle scheme for this model and prove that gradient flow solutions are obtained in the many particle limit by discrete densities constructed out of moving particles satisfying a suitable system of ODEs. The ODE system is formulated in a non standard way in order to bypass the Lipschitz singularity of the kernel, with difference quotients of the kernel replacing its derivative.

Paper Structure

This paper contains 12 sections, 12 theorems, 161 equations.

Table of Contents

  1. Introduction
  2. Existence and uniqueness of gradient flow solutions
  3. Preliminaries and statement of the well-posedness result
  4. Proof of Theorem \ref{['thm:main']}
  5. $\lambda$-convexity of the functional
  6. JKO scheme
  7. Flow interchange
  8. Conclusion of the existence and uniqueness proof
  9. Deterministic particle approximation
  10. Analysis of particles collisions
  11. $L^p$ estimates on the particle scheme
  12. Many particle limit

Key Result

Theorem 2.1

Let $T>0$ be fixed and $m \in (1, \infty]$. Assume $(\rho_0, \eta_0) \in (\mathcal{P}_2 (\mathbb{R}) \cap L^m (\mathbb{R}))^2$. Then, there exists a unique gradient flow solution $(\rho,\eta)\in AC([0,T]; (\mathcal{P}_2 (\mathbb{R}) \cap L^m (\mathbb{R}) )^2 )$ to system eq:macroscopic_model in the for $t \in [0,T]$, and for some constant $C>0$ independent of $t$. Finally, if $\bm\gamma^1=(\rho^1

Theorems & Definitions (24)

  • Definition 2.1: $\lambda$-convexity along geodesics
  • Definition 2.2: $k$-flow
  • Definition 2.3: Fréchet sub-differential in $\mathcal{P}_2(\mathbb{R})^2$f
  • Definition 2.4: Gradient flow solution
  • Theorem 2.1
  • Proposition 2.1
  • proof
  • Proposition 2.2
  • proof
  • Proposition 2.3
  • ...and 14 more