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Transporting a Dirac mass in a mean field planning problem

Pierre Cardaliaguet, Sebastian Munoz, Alessio Porretta

TL;DR

This work analyzes a 1D mean field planning problem with an initial Dirac mass, proving existence of a solution for all $\theta>0$ and showing that the evolving density concentrates toward a Barenblatt-type self-similar profile $\phi$ under a continuous space-time rescaling. The authors introduce a Lyapunov functional in rescaled variables to prove convergence of the density to $\phi$ as $t\to0^+$, with a full treatment of the noncritical case $\theta\neq2$ and the critical case $\theta=2$ via a Lasry–Lions monotonicity argument; they also derive quantitative convergence rates for $\theta>2$. Uniqueness is established in a class of solutions that converge to $\phi$ in the rescaled frame, with a precise rate-driven argument validating the case $\theta>2$ and highlighting the threshold role of $\theta=2$. The results reveal finite-speed propagation due to congestion and connect singular initial data to a self-similar profile, providing a rigorous framework for singular data in mean field planning problems and linking to porous-media-type scaling phenomena.

Abstract

We study a mean field planning problem in which the initial density is a Dirac mass. We show that there exists a unique solution which converges to a self-similar profile as time tends to $0$. We proceed by studying a continuous rescaling of the solution, and characterizing its behavior near the initial time through an appropriate Lyapunov functional.

Transporting a Dirac mass in a mean field planning problem

TL;DR

This work analyzes a 1D mean field planning problem with an initial Dirac mass, proving existence of a solution for all and showing that the evolving density concentrates toward a Barenblatt-type self-similar profile under a continuous space-time rescaling. The authors introduce a Lyapunov functional in rescaled variables to prove convergence of the density to as , with a full treatment of the noncritical case and the critical case via a Lasry–Lions monotonicity argument; they also derive quantitative convergence rates for . Uniqueness is established in a class of solutions that converge to in the rescaled frame, with a precise rate-driven argument validating the case and highlighting the threshold role of . The results reveal finite-speed propagation due to congestion and connect singular initial data to a self-similar profile, providing a rigorous framework for singular data in mean field planning problems and linking to porous-media-type scaling phenomena.

Abstract

We study a mean field planning problem in which the initial density is a Dirac mass. We show that there exists a unique solution which converges to a self-similar profile as time tends to . We proceed by studying a continuous rescaling of the solution, and characterizing its behavior near the initial time through an appropriate Lyapunov functional.

Paper Structure

This paper contains 9 sections, 25 theorems, 202 equations.

Table of Contents

  1. Introduction
  2. Existence of a solution
  3. Estimates on the flow and the density
  4. Estimates on $u^{(\varepsilon)}$
  5. Existence of a solution
  6. Local behavior near the singularity
  7. Convergence to the self-similar profile
  8. Convergence rate in the case $\theta>2$
  9. Uniqueness in the case $\theta>2$

Key Result

Theorem 1.1

Fix $\theta>0$, and assume that $m_T:\mathbb{R} \to [0,\infty)$ is supported in an interval $[a,b]$ and satisfies $m_T^\theta\in C^{1,\sigma}(a,b)$ for some $\sigma>0$ and the compatibility condition for some constant $C>1$. Then there exists a unique solution to eq.planning such that the rescaled densities $x\mapsto t^\alpha m(t, t^\alpha x)$ converge, in a suitable sense, to the self-similar pr

Theorems & Definitions (51)

  • Theorem 1.1
  • Lemma 2.1
  • proof
  • Lemma 2.2
  • proof
  • Corollary 2.3
  • proof
  • Lemma 2.4
  • proof
  • Lemma 2.5
  • ...and 41 more