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The Filippov characteristic flow for the aggregation equation with mildly singular potentials

José Antonio Carrillo, Francois James, Frédéric Lagoutière, Nicolas Vauchelet

TL;DR

This work studies the aggregation equation with mildly singular potentials in multiple dimensions, addressing existence and uniqueness of global measure solutions. It develops a Filippov characteristic flow framework to define a nonlinear velocity field and proves a Wasserstein contraction, establishing a unique distributional solution and its representation as a pushforward by the Filippov flow. It also shows the equivalence of this Filippov-flow solution with the gradient-flow formulation and provides a convergent numerical scheme (particle and finite-volume) applicable to general measure initial data, including post-blow-up dynamics. The Erratum clarifies that Filippov-flow-defined solutions may be non-unique, but the distributional solution is unique and given by the Filippov flow associated to that solution, together with a contraction estimate ensuring stability with respect to initial data.

Abstract

Existence and uniqueness of global in time measure solution for the multidimensional aggregation equation is analyzed. Such a system can be written as a continuity equation with a velocity field computed through a self-consistent interaction potential. In Carrillo et al. (Duke Math J (2011)), a well-posedness theory based on the geometric approach of gradient flows in measure metric spaces has been developed for mildly singular potentials at the origin under the basic assumption of being lambda-convex. We propose here an alternative method using classical tools from PDEs. We show the existence of a characteristic flow based on Filippov's theory of discontinuous dynamical systems such that the weak measure solution is the pushforward measure with this flow. Uniqueness is obtained thanks to a contraction argument in transport distances using the lambda-convexity of the potential. Moreover, we show the equivalence of this solution with the gradient flow solution. Finally, we show the convergence of a numerical scheme for general measure solutions in this framework allowing for the simulation of solutions for initial smooth densities after their first blow-up time in Lp-norms.

The Filippov characteristic flow for the aggregation equation with mildly singular potentials

TL;DR

This work studies the aggregation equation with mildly singular potentials in multiple dimensions, addressing existence and uniqueness of global measure solutions. It develops a Filippov characteristic flow framework to define a nonlinear velocity field and proves a Wasserstein contraction, establishing a unique distributional solution and its representation as a pushforward by the Filippov flow. It also shows the equivalence of this Filippov-flow solution with the gradient-flow formulation and provides a convergent numerical scheme (particle and finite-volume) applicable to general measure initial data, including post-blow-up dynamics. The Erratum clarifies that Filippov-flow-defined solutions may be non-unique, but the distributional solution is unique and given by the Filippov flow associated to that solution, together with a contraction estimate ensuring stability with respect to initial data.

Abstract

Existence and uniqueness of global in time measure solution for the multidimensional aggregation equation is analyzed. Such a system can be written as a continuity equation with a velocity field computed through a self-consistent interaction potential. In Carrillo et al. (Duke Math J (2011)), a well-posedness theory based on the geometric approach of gradient flows in measure metric spaces has been developed for mildly singular potentials at the origin under the basic assumption of being lambda-convex. We propose here an alternative method using classical tools from PDEs. We show the existence of a characteristic flow based on Filippov's theory of discontinuous dynamical systems such that the weak measure solution is the pushforward measure with this flow. Uniqueness is obtained thanks to a contraction argument in transport distances using the lambda-convexity of the potential. Moreover, we show the equivalence of this solution with the gradient flow solution. Finally, we show the convergence of a numerical scheme for general measure solutions in this framework allowing for the simulation of solutions for initial smooth densities after their first blow-up time in Lp-norms.

Paper Structure

This paper contains 26 sections, 17 theorems, 214 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Weak measure solutions for the aggregation equation
  3. Weak measure solutions for conservative transport equation
  4. Solutions defined by Filippov's flow
  5. Gradient flow solutions
  6. Existence and uniqueness
  7. Macroscopic velocity and one-sided estimate
  8. Existence
  9. Uniqueness
  10. Equivalence with gradient flow solutions
  11. Numerical approximation
  12. Particle scheme
  13. Finite volume discretization
  14. Properties of the scheme
  15. Convergence of the numerical approximation
  16. ...and 11 more sections

Key Result

Theorem 2.2

Let $T>0$. Let us assume that the vector field $b\in L^1_{loc}(\mathbb{R};L^\infty(\mathbb{R}^d))$ satisfies the OSL condition, that is for all $x$ and $y$ in $\mathbb{R}^d$, for all $t\in [0,T]$, Then there exists an unique Filippov characteristic $X$ associated to this vector field.

Figures (2)

  • Figure 1: Dynamics of the cell density $\rho$ with intial data given by the sum of three bumps in the case $W_1(x)=1-e^{-5|x|}$.
  • Figure 2: Dynamics of the cell density $\rho$ with intial data given by the sum of three bumps in the case $W_2(x)=|x|$.

Theorems & Definitions (23)

  • Definition 2.1
  • Theorem 2.2: Filippov
  • Theorem 2.3: PoupaudRascle
  • Theorem 2.4
  • Theorem 2.5
  • Remark 2.6
  • Definition 2.7: Gradient flows
  • Theorem 2.8: Carrillo
  • Theorem 2.9
  • Lemma 3.1
  • ...and 13 more