Deterministic particle approximation for scalar - - PowerPoint PPT Presentation

Deterministic particle approximation for scalar aggregation-diffusion equations with nonlinear mobility

Emanuela Radici

joint work with M. Di Francesco & S. Fagioli

Universit` a degli Studi dell’Aquila

Crowds: models and control 4th June 2019, CIRM Marseille

E.Radici (UNIVAQ) Deterministic Particle... CIRM, 04/06/2019 1 / 29

Aggregation case Introduction

Introduction

N particles located at positions x1(t), . . . , xN(t) xi(t) energetical setting:

nonlocal interaction potential W depending on the relative distance

• f the particles

no inertia (negligible in many socio-biological aggregation phenomena)

E.Radici (UNIVAQ) Deterministic Particle... CIRM, 04/06/2019 2 / 29

Aggregation case Introduction

Introduction

N particles located at positions x1(t), . . . , xN(t) xi(t) energetical setting:

nonlocal interaction potential W depending on the relative distance

• f the particles

no inertia (negligible in many socio-biological aggregation phenomena)

˙ xi(t) = − 1

N

• j=i ∇W (xi(t) − xj(t))

E.Radici (UNIVAQ) Deterministic Particle... CIRM, 04/06/2019 2 / 29

Aggregation case Introduction

˙ xi = − 1

N

• j=i ∇W (xi − xj)
• ∂tρ = ∇ · (ρ∇W ∗ ρ)

Bertozzi, Carrillo, Laurent, Rosado and Brandman: Lp theory Ambrosio, Gigli and Savar´ e: optimal transport with smooth potentials Carrillo, Choi, Di Francesco, Figalli, Hauray, Laurent and Slepˇ cev:

• ptimal transport with

midly-singular potentials

E.Radici (UNIVAQ) Deterministic Particle... CIRM, 04/06/2019 3 / 29

Aggregation case Introduction

˙ xi = − 1

N

• j=i ∇W (xi − xj)
• ∂tρ = ∇ · (ρ∇W ∗ ρ)

If the potential W is attractive then the particles tend to concentrate

E.Radici (UNIVAQ) Deterministic Particle... CIRM, 04/06/2019 4 / 29

Aggregation case Introduction

˙ xi = − 1

N

• j=i ∇W (xi − xj)
• ∂tρ = ∇ · (ρ∇W ∗ ρ)

If the potential W is attractive then the particles tend to concentrate (the density ρ blows up)

E.Radici (UNIVAQ) Deterministic Particle... CIRM, 04/06/2019 4 / 29

Aggregation case Introduction

Non linear mobility

• ne way to prevent the overcrowing

effect is to let the mobility depend also on a velocity term that decreases where the concentration is higher

E.Radici (UNIVAQ) Deterministic Particle... CIRM, 04/06/2019 5 / 29

Aggregation case Introduction

Non linear mobility

• ne way to prevent the overcrowing

effect is to let the mobility depend also on a velocity term that decreases where the concentration is higher not a new idea (see Hillen, Painter, Dolak, Schmeiser, Burger, Di Francesco . . .)

E.Radici (UNIVAQ) Deterministic Particle... CIRM, 04/06/2019 5 / 29

Aggregation case Introduction

Non linear mobility

• ne way to prevent the overcrowing

effect is to let the mobility depend also on a velocity term that decreases where the concentration is higher not a new idea (see Hillen, Painter, Dolak, Schmeiser, Burger, Di Francesco . . .) the continuum model becomes ∂tρ = ∇ · (ρv(ρ)∇W ∗ ρ)

E.Radici (UNIVAQ) Deterministic Particle... CIRM, 04/06/2019 5 / 29

Aggregation case Introduction

Non linear mobility

• ne way to prevent the overcrowing

effect is to let the mobility depend also on a velocity term that decreases where the concentration is higher not a new idea (see Hillen, Painter, Dolak, Schmeiser, Burger, Di Francesco . . .) the continuum model becomes ∂tρ = ∇ · (ρv(ρ)∇W ∗ ρ) Is there a microscopic counterpart of this PDE?

E.Radici (UNIVAQ) Deterministic Particle... CIRM, 04/06/2019 5 / 29

Aggregation case Introduction

Non linear mobility

• ne way to prevent the overcrowing

effect is to let the mobility depend also on a velocity term that decreases where the concentration is higher not a new idea (see Hillen, Painter, Dolak, Schmeiser, Burger, Di Francesco . . .) the continuum model becomes ∂tρ = ∂x(ρv(ρ)W ′ ∗ ρ) Is there a microscopic counterpart of this PDE?

E.Radici (UNIVAQ) Deterministic Particle... CIRM, 04/06/2019 5 / 29

Aggregation case Introduction

Setting

initial density ¯ ρ ∈ BV (R, [0, 1]) with compact support and ¯ ρL1 = 1

E.Radici (UNIVAQ) Deterministic Particle... CIRM, 04/06/2019 6 / 29

Aggregation case Introduction

Setting

initial density ¯ ρ ∈ BV (R, [0, 1]) with compact support and ¯ ρL1 = 1 velocity v ∈ C1([0, ∞)) monotone decreasing and such that it takes the value 0, assume v(ρ) = (1 − ρ)+

E.Radici (UNIVAQ) Deterministic Particle... CIRM, 04/06/2019 6 / 29

Aggregation case Introduction

Setting

initial density ¯ ρ ∈ BV (R, [0, 1]) with compact support and ¯ ρL1 = 1 velocity v ∈ C1([0, ∞)) monotone decreasing and such that it takes the value 0, assume v(ρ) = (1 − ρ)+ potential W ∈ C2(R) attractive, radially symmetric with W ′′ locally Lipschitz

E.Radici (UNIVAQ) Deterministic Particle... CIRM, 04/06/2019 6 / 29

Aggregation case Introduction

Setting

initial density ¯ ρ ∈ BV (R, [0, 1]) with compact support and ¯ ρL1 = 1 velocity v ∈ C1([0, ∞)) monotone decreasing and such that it takes the value 0, assume v(ρ) = (1 − ρ)+ potential W ∈ C2(R) attractive, radially symmetric with W ′′ locally Lipschitz Continuum Problem ∂tρ = ∂x(ρv(ρ)W ′ ∗ ρ) ρ(0, ·) = ¯ ρ(·)

E.Radici (UNIVAQ) Deterministic Particle... CIRM, 04/06/2019 6 / 29

Aggregation case Introduction

Plan of the talk

1 Find a candidate discrete model E.Radici (UNIVAQ) Deterministic Particle... CIRM, 04/06/2019 7 / 29

Aggregation case Introduction

Plan of the talk

1 Find a candidate discrete model 2 Study the deterministic many particle limit E.Radici (UNIVAQ) Deterministic Particle... CIRM, 04/06/2019 7 / 29

Aggregation case Introduction

Plan of the talk

1 Find a candidate discrete model 2 Study the deterministic many particle limit 3 Aggregation-Diffusion case E.Radici (UNIVAQ) Deterministic Particle... CIRM, 04/06/2019 7 / 29

Aggregation case Introduction

Plan of the talk

1 Find a candidate discrete model 2 Study the deterministic many particle limit 3 Aggregation-Diffusion case 4 Numerical simulations E.Radici (UNIVAQ) Deterministic Particle... CIRM, 04/06/2019 7 / 29

Aggregation case Discrete model

Lagrangian description via pseudo inverse function

distribution function R(t, x) = x

−∞ ρ(t, y)dy

pseudo inverse function X(t, z) = inf{x ∈ R : R(t, x) > z}

E.Radici (UNIVAQ) Deterministic Particle... CIRM, 04/06/2019 8 / 29

Aggregation case Discrete model

Lagrangian description via pseudo inverse function

distribution function R(t, x) = x

−∞ ρ(t, y)dy

pseudo inverse function X(t, z) = inf{x ∈ R : R(t, x) > z}

(see Gosse, Toscani, Russo, Matthes, Osberger, Di Francesco, Rosini, Fagioli . . .)

E.Radici (UNIVAQ) Deterministic Particle... CIRM, 04/06/2019 8 / 29

Aggregation case Discrete model

Lagrangian description via pseudo inverse function

distribution function R(t, x) = x

−∞ ρ(t, y)dy

pseudo inverse function X(t, z) = inf{x ∈ R : R(t, x) > z}

(see Gosse, Toscani, Russo, Matthes, Osberger, Di Francesco, Rosini, Fagioli . . .)

formal change of variables in the PDE:

density ∂tρ = ∂x(ρv(ρ)W ′ ∗ ρ) pseudo inverse ∂tX = −v(

1 ∂zX )

1

0 W ′(X(z) − X(ξ))dξ

E.Radici (UNIVAQ) Deterministic Particle... CIRM, 04/06/2019 8 / 29

Aggregation case Discrete model

Lagrangian description via pseudo inverse function

∂tX = −v 1 ∂zX z W ′(X(z) − X(ξ))dξ − v 1 ∂zX 1

z

W ′(X(z) − X(ξ))dξ

discretization of ∂zX(t, z) via forward and backward finite differences

∂zX(t, z) ≈ ±X(t, z ± 1

N ) − X(t, z) 1 N

take X(t, z) piecewise constant

X(t, z) =

N−1

• i=0

xi(t)χ[ i

N , i+1 N )(z)

then, for every i = 0, . . . , N, it is immediate to obtain

˙ xi(t) = −v

• 1/N

xi+1 − xi

j>i

W ′(xi − xj) N − v

• 1/N

xi − xi−1

j<i

W ′(xi − xj) N

E.Radici (UNIVAQ) Deterministic Particle... CIRM, 04/06/2019 9 / 29

Aggregation case Discrete model

Discretization of the initial condition

Let us call [¯ xmin, ¯ xmax] the minimal interval containing supp[¯ ρ], then we discretize the initial condition in N intervals of measure 1/N

¯ x0 ¯ xN−1 ¯ xN

R ¯ ρ

¯ x1

E.Radici (UNIVAQ) Deterministic Particle... CIRM, 04/06/2019 10 / 29

Aggregation case Discrete model

Discretization of the initial condition

Let us call [¯ xmin, ¯ xmax] the minimal interval containing supp[¯ ρ], then we discretize the initial condition in N intervals of measure 1/N

¯ x0 ¯ xN−1 ¯ xN

R ¯ ρ

¯ x1

¯ x0 = ¯ xmin, ¯ xN = ¯ xmax

E.Radici (UNIVAQ) Deterministic Particle... CIRM, 04/06/2019 10 / 29

Aggregation case Discrete model

Discretization of the initial condition

Let us call [¯ xmin, ¯ xmax] the minimal interval containing supp[¯ ρ], then we discretize the initial condition in N intervals of measure 1/N

¯ x0 ¯ xN−1 ¯ xN

R ¯ ρ

¯ x1

¯ x0 = ¯ xmin, ¯ xN = ¯ xmax for i = 1, . . . , N − 1 ¯ xi = sup

• x > ¯

xi−1 : x

¯ xi−1

¯ ρ(x)dx < 1 N

• E.Radici (UNIVAQ)

Deterministic Particle... CIRM, 04/06/2019 10 / 29

Aggregation case Discrete model

Discrete Problem

Let the N + 1 particles xi evolve accordingly to

• ˙

xi(t) = − v(Ri)

N

• j>i W ′(xi − xj) − v(Ri−1)

N

• j<i W ′(xi − xj),

xi(0) = ¯ xi, where Ri(t) =

1 N(xi+1(t)−xi(t)) for i = 0, . . . , N − 1

E.Radici (UNIVAQ) Deterministic Particle... CIRM, 04/06/2019 11 / 29

Aggregation case Discrete model

Discrete Problem

Let the N + 1 particles xi evolve accordingly to

• ˙

xi(t) = − v(Ri)

N

• j>i W ′(xi − xj) − v(Ri−1)

N

• j<i W ′(xi − xj),

xi(0) = ¯ xi, where Ri(t) =

1 N(xi+1(t)−xi(t)) for i = 0, . . . , N − 1

t

¯ x0 ¯ xN x0(t) xN(t)

R

E.Radici (UNIVAQ) Deterministic Particle... CIRM, 04/06/2019 11 / 29

Aggregation case Discrete model

Discrete density ρN

a discrete version of the maximum principle holds c1(¯ ρ) 1 N < |xi+1(t) − xi(t)| < c2(¯ ρ)

E.Radici (UNIVAQ) Deterministic Particle... CIRM, 04/06/2019 12 / 29

Aggregation case Discrete model

Discrete density ρN

a discrete version of the maximum principle holds c1(¯ ρ) 1 N < |xi+1(t) − xi(t)| < c2(¯ ρ) x0(t) < . . . < xN(t) for all t ∈ [0, T]

E.Radici (UNIVAQ) Deterministic Particle... CIRM, 04/06/2019 12 / 29

Aggregation case Discrete model

Discrete density ρN

a discrete version of the maximum principle holds c1(¯ ρ) 1 N < |xi+1(t) − xi(t)| < c2(¯ ρ) x0(t) < . . . < xN(t) for all t ∈ [0, T] We can define the N-discrete density ρN(t, x) : =

N−1

• i=0

Ri(t)1[xi(t),xi+1(t))(x) =

N−1

• i=0

1 N(xi+1(t) − xi(t))1[xi(t),xi+1(t))(x),

E.Radici (UNIVAQ) Deterministic Particle... CIRM, 04/06/2019 12 / 29

Aggregation case Discrete model

Discrete density ρN

a discrete version of the maximum principle holds c1(¯ ρ) 1 N < |xi+1(t) − xi(t)| < c2(¯ ρ) x0(t) < . . . < xN(t) for all t ∈ [0, T] We can define the N-discrete density ρN(t, x) : =

N−1

• i=0

Ri(t)1[xi(t),xi+1(t))(x) =

N−1

• i=0

1 N(xi+1(t) − xi(t))1[xi(t),xi+1(t))(x), ρNL1(R) = 1 and supp[ρN(t, ·)] ⊆ supp[¯ ρ(·)] for every N

E.Radici (UNIVAQ) Deterministic Particle... CIRM, 04/06/2019 12 / 29

Aggregation case Many Particle Limit

Aggregation regime

Theorem (Di Francesco, Fagioli, R. 2019) Let ¯ ρ, v, W be as in the assumptions and T > 0 be a fixed time, then ρN converges in L1([0, T] × R) to a L∞-measure ρ that is the unique entropy solution of the Cauchy problem ∂tρ = ∂x(ρv(ρ)W ′ ∗ ρ) , ρ(0, ·) = ¯ ρ(·) .

E.Radici (UNIVAQ) Deterministic Particle... CIRM, 04/06/2019 13 / 29

Aggregation case Many Particle Limit

Sketch of the proof

The argument consists of two main steps

1 prove that (ρN) converges strongly in L1([0, T] × supp[¯

ρ])

2 show that the limit is an entropy solution of the PDE E.Radici (UNIVAQ) Deterministic Particle... CIRM, 04/06/2019 14 / 29

Aggregation case Many Particle Limit

Convergence of ρN

Generalized Aubin Lions Lemma If the following conditions hold I) supN

T

• ρN(t, ·)L1(supp[ρ(t)]) + TV [ρN(t, ·)]
• + |supp[ρN(t, ·)|]dt < ∞

II) ∃ C > 0 independent on N, such that dW 1(ρN(t, ·), ρN(s, ·)) < C|t − s| for all t, s ∈ (0, T) = ⇒ then ρN is strongly relatively compact in L1([0, T] × supp[¯ ρ]).

E.Radici (UNIVAQ) Deterministic Particle... CIRM, 04/06/2019 15 / 29

Aggregation case Many Particle Limit

Convergence of ρN

Condition I) is a direct consequence of the properties of v and of the following contraction inequality d dt TV [ρN(t, ·)] ≤ c TV [ρ(t, ·)] hence TV [ρN(t, ·)] ≤ ecTTV [¯ ρ] .

E.Radici (UNIVAQ) Deterministic Particle... CIRM, 04/06/2019 16 / 29

Aggregation case Many Particle Limit

Convergence of ρN

In the scalar case we can take advantage of the following identity: if µ, ν are probability measures and Xµ(·), Xν(·) their pseudo inverse functions, then dW 1(µ, ν) = Xµ − XνL1([0,1]) , We can explicitly compute XρN(t, ·)(z) =

N−1

• i=0
• xN

i (t) +

• z − i

N

• 1

RN

i (t)

• 1[ i

N , i+1 N )(z). E.Radici (UNIVAQ) Deterministic Particle... CIRM, 04/06/2019 17 / 29

Aggregation case Many Particle Limit

Convergence of ρN

In the scalar case we can take advantage of the following identity: if µ, ν are probability measures and Xµ(·), Xν(·) their pseudo inverse functions, then dW 1(µ, ν) = Xµ − XνL1([0,1]) , We can explicitly compute XρN(t, ·)(z) =

N−1

• i=0
• xN

i (t) +

• z − i

N

• 1

RN

i (t)

• 1[ i

N , i+1 N )(z).

Condition II) is a consequence of the following estimates XρN(t,·) − XρN(s,·)L1 ≤ c N

N

• i=0

t

s

• ˙

xN

i (τ)

• dτ,

sup

i

| ˙ xN

i (τ)| ≤ 2c.

E.Radici (UNIVAQ) Deterministic Particle... CIRM, 04/06/2019 17 / 29

Aggregation case Many Particle Limit

Consistence of the scheme

If we knew for every N and every ϕ ∈ C∞

c ((0, T) × R) that

T

• R

ρNϕt − ρNv(ρN)W ′ ∗ ρNϕxdxdt = 0, then the convergence result of the previous step would directly provide a weak solution for the continuum problem.

E.Radici (UNIVAQ) Deterministic Particle... CIRM, 04/06/2019 18 / 29

Aggregation case Many Particle Limit

Consistence of the scheme

If we knew for every N and every ϕ ∈ C∞

c ((0, T) × R) that

T

• R

ρNϕt − ρNv(ρN)W ′ ∗ ρNϕxdxdt = 0, then the convergence result of the previous step would directly provide a weak solution for the continuum problem. This is not true in general. However, in our case we can prove that T

• R

ρNϕt − ρNv(ρN)W ′ ∗ ρNϕxdxdt − → 0 as N − → ∞.

E.Radici (UNIVAQ) Deterministic Particle... CIRM, 04/06/2019 18 / 29

Aggregation case Many Particle Limit

Entropy Solution

Denoting f (ρ) = ρv(ρ), we get that ρ satisfies the following entropy condition

0 ≤

• R

∞ |ρ−c|ϕt−sign(ρ−c)[(f (ρ)−f (c))W ′∗ρϕx−f (c)W ′′∗ρϕ]+

• R

|¯ ρ−c|ϕ(0)

for all constants c ≥ 0 and all test function ϕ ∈ C∞

c ((0, ∞) × R), ϕ ≥ 0

Uniqueness Uniqueness of the entropy solution is a consequence of a more general result of Karlsen and Risebro. Weak solutions of the problem are not unique

E.Radici (UNIVAQ) Deterministic Particle... CIRM, 04/06/2019 19 / 29

Aggregation-Diffusion case

The aggregation-diffusion model

A first interesting development of this result concerns the generalization of the deterministic particle approximation approach to the case of aggregation-diffusion equations of the form    ∂tρ = ∂x(ρv(ρ)∂x(a(ρ) + W ∗ ρ)) [0, T] × [0, ℓ] ρ(0, ·) = ¯ ρ(·) {0} × [0, ℓ] v(ρ)∂x(a(ρ) + W ∗ ρ) = 0 [0, T] × ({0} ∪ {ℓ}) where the PDE can be rephrased as ∂tρ = ∂xxφ(ρ) + ∂x(ρv(ρ)W ′ ∗ ρ) through the relation φ(ρ) = ρ ξv(ξ)a′(ξ)dξ

E.Radici (UNIVAQ) Deterministic Particle... CIRM, 04/06/2019 20 / 29

Aggregation-Diffusion case

The aggregation-diffusion model

A first interesting development of this result concerns the generalization of the deterministic particle approximation approach to the case of aggregation-diffusion equations of the form    ∂tρ = ∂xxφ(ρ) + ∂x(ρv(ρ)W ′ ∗ ρ) [0, T] × [0, ℓ] ρ(0, ·) = ¯ ρ(·) {0} × [0, ℓ] v(ρ)∂x(a(ρ) + W ∗ ρ) = 0 [0, T] × ({0} ∪ {ℓ}) φ Lipschitz, non-decreasing and φ(0) = 0 ¯ ρ ∈ L∞ ∩ BV ([0, ℓ], R+) far from vacuum W , v satisfy the same assumption as before

E.Radici (UNIVAQ) Deterministic Particle... CIRM, 04/06/2019 21 / 29

Aggregation-Diffusion case

The aggregation-diffusion model

A first interesting development of this result concerns the generalization of the deterministic particle approximation approach to the case of aggregation-diffusion equations of the form    ∂tρ = ∂xxφ(ρ) + ∂x(ρv(ρ)W ′ ∗ ρ) [0, T] × [0, ℓ] ρ(0, ·) = ¯ ρ(·) {0} × [0, ℓ] v(ρ)∂x(a(ρ) + W ∗ ρ) = 0 [0, T] × ({0} ∪ {ℓ}) φ Lipschitz, non-decreasing and φ(0) = 0

E.Radici (UNIVAQ) Deterministic Particle... CIRM, 04/06/2019 21 / 29

Aggregation-Diffusion case

The aggregation-diffusion model

A first interesting development of this result concerns the generalization of the deterministic particle approximation approach to the case of aggregation-diffusion equations of the form    ∂tρ = ∂xxφ(ρ) + ∂x(ρv(ρ)W ′ ∗ ρ) [0, T] × [0, ℓ] ρ(0, ·) = ¯ ρ(·) {0} × [0, ℓ] v(ρ)∂x(a(ρ) + W ∗ ρ) = 0 [0, T] × ({0} ∪ {ℓ}) φ Lipschitz, non-decreasing and φ(0) = 0 this assumption includes classical nonlinear degenerate diffusions, e.g. porous medium (Vazquez), and even the so called strongly degenerate diffusions with φ′(ρ) = 0 for ρ ∈ [ρ1, ρ2] (Betancourt, B¨

urger, Karlsen, Risebro)

E.Radici (UNIVAQ) Deterministic Particle... CIRM, 04/06/2019 21 / 29

Aggregation-Diffusion case

Particle system for the aggregation-diffusion model

The particle system becomes ˙ xi(t) = ˙ xd

i (t) + ˙

xnL

i (t),

xi(0) = ¯ xi, where ˙ xd

i (t) = N(φ(Ri−1) − φ(Ri))

˙ xnL

i (t) = − v(Ri) N

• j>i W ′(xi − xj) − v(Ri−1)

N

• j<i W ′(xi − xj)

Russo, Gosse Toscani

E.Radici (UNIVAQ) Deterministic Particle... CIRM, 04/06/2019 22 / 29

Aggregation-Diffusion case

Aggregation-Diffusion regime

Theorem (Fagioli, R. 2018) Let ¯ ρ, φ, v, W be as in the assumptions and T > 0 be a fixed time, then ρN converges in L1([0, T] × [0, ℓ]) to a L∞-measure ρ that is a weak solution of the Cauchy problem    ∂tρ = ∂xxφ(ρ) + ∂x(ρv(ρ)W ′ ∗ ρ) [0, T] × [0, ℓ] ρ(0, ·) = ¯ ρ(·) {0} × [0, ℓ] v(ρ)∂x(a(ρ) + W ∗ ρ) = 0 [0, T] × ({0} ∪ {ℓ})

E.Radici (UNIVAQ) Deterministic Particle... CIRM, 04/06/2019 23 / 29

Aggregation-Diffusion case

Sketch of the proof

Discrete local in time Min-Max Problem c1(¯ ρ) 1 N < |xi+1(t) − xi(t)| < c2(¯ ρ, T) 1 N

E.Radici (UNIVAQ) Deterministic Particle... CIRM, 04/06/2019 24 / 29

Aggregation-Diffusion case

Sketch of the proof

Discrete local in time Min-Max Problem c1(¯ ρ) 1 N < |xi+1(t) − xi(t)| < c2(¯ ρ, T) 1 N in the pure diffusive regime we can prove a contraction property for the total variation d dt TV [ρN(t, x)] ≤ 0

E.Radici (UNIVAQ) Deterministic Particle... CIRM, 04/06/2019 24 / 29

Aggregation-Diffusion case

Sketch of the proof

Discrete local in time Min-Max Problem c1(¯ ρ) 1 N < |xi+1(t) − xi(t)| < c2(¯ ρ, T) 1 N in the pure diffusive regime we can prove a contraction property for the total variation d dt TV [ρN(t, x)] ≤ 0 dW 1(ρN(t), ρN(s)) ≤ C(¯ ρ, φ)|t − s|

E.Radici (UNIVAQ) Deterministic Particle... CIRM, 04/06/2019 24 / 29

Aggregation-Diffusion case

Sketch of the proof

Discrete local in time Min-Max Problem c1(¯ ρ) 1 N < |xi+1(t) − xi(t)| < c2(¯ ρ, T) 1 N in the pure diffusive regime we can prove a contraction property for the total variation d dt TV [ρN(t, x)] ≤ 0 dW 1(ρN(t), ρN(s)) ≤ C(¯ ρ, φ)|t − s| the strong L1-limit measure ρ satisfies T ℓ ρϕt + φ(ρ)ϕxx − ρv(ρ)W ′ ∗ ρϕx = 0 for all ϕ ∈ C∞

c ((0, T) × (0, ℓ)).

E.Radici (UNIVAQ) Deterministic Particle... CIRM, 04/06/2019 24 / 29

Numerics

Numerical simulations for nonlocal attractive potential

v(ρ) = 1 − ρ, W = N(0, 1), ¯ ρ(x) = 0.2χ[−0.5,0](x) + 0.6χ[0.5,1](x)

E.Radici (UNIVAQ) Deterministic Particle... CIRM, 04/06/2019 25 / 29

Numerics

Numerical simulations for nonlocal attractive potential

v(ρ) = 1 − ρ, W = N(0, 1), ¯ ρ(x) = χ[−0.5,0](x) + χ[0.5,1](x)

E.Radici (UNIVAQ) Deterministic Particle... CIRM, 04/06/2019 26 / 29

Numerics

numerical simulations for aggregation-diffusion case

v(ρ) = 1 − ρ, W = N(0, 1), φ(ρ) = ρ2, ¯ ρ(x) = 3 4(1 − x2)1[−1,1](x)

E.Radici (UNIVAQ) Deterministic Particle... CIRM, 04/06/2019 27 / 29

Numerics

numerical simulations for aggregation-diffusion case

v(ρ) = 1 − ρ, W = N(0, 1), ¯ ρ(x) = 0.5χ[0,0.5)(x) + 0.8χ[0.5,1](x), φ(ρ) = 1 20ρ2χ[0, 2

5 )(ρ) +

1 125χ[ 2

5 , 3 5 )(ρ) +

• 1

125 + 1 20

• ρ − 3

5 2 χ[ 3

5 ,∞)(ρ) E.Radici (UNIVAQ) Deterministic Particle... CIRM, 04/06/2019 28 / 29

Conclusions

Open Questions

1 Particle approximation for the Opinion

Formation dynamics

E.Radici (UNIVAQ) Deterministic Particle... CIRM, 04/06/2019 29 / 29

Conclusions

Open Questions

1 Particle approximation for the Opinion

Formation dynamics

2 Rigorous validation of gradient flow

structure for generalized Wasserstein distance

E.Radici (UNIVAQ) Deterministic Particle... CIRM, 04/06/2019 29 / 29

Conclusions

Open Questions

1 Particle approximation for the Opinion

Formation dynamics

2 Rigorous validation of gradient flow

structure for generalized Wasserstein distance

3 Particle approximation in higher

dimension

E.Radici (UNIVAQ) Deterministic Particle... CIRM, 04/06/2019 29 / 29

Conclusions

Open Questions

1 Particle approximation for the Opinion

Formation dynamics

2 Rigorous validation of gradient flow

structure for generalized Wasserstein distance

3 Particle approximation in higher

dimension

4 Dynamics of several species systems via

nonlocal self and cross interaction potentials

E.Radici (UNIVAQ) Deterministic Particle... CIRM, 04/06/2019 29 / 29

Conclusions

Thank you for your kind attention!

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