Convergence of the Follow-The-Leader scheme for scalar conservation - - PowerPoint PPT Presentation

Macroscopic model Microscopic models Convergence

Convergence of the Follow-The-Leader scheme for scalar conservation laws with space dependent flux

Graziano Stivaletta

University of L’Aquila Work in collaboration with M. Di Francesco (University of L’Aquila)

CROWDS: models and control

CIRM Marseille, June 3-7, 2019

Macroscopic model Microscopic models Convergence

1

Macroscopic model A general overview Assumptions

2

Microscopic models The FTL approximation Main properties

3

Convergence The uniform control of TV and W1 Convergence to the entropy solution

Macroscopic model Microscopic models Convergence A general overview

Let us consider the Cauchy problem

• ρt +

ρv(ρ)φ(x)

x = 0,

x ∈ R, t > 0, ρ(x, 0) = ρ(x), x ∈ R. (1) ρ(x, t) is a density; v(ρ) is a velocity map; φ(x) is a given external potential; ρ(x) is the initial datum.

Macroscopic model Microscopic models Convergence A general overview

Let us consider the Cauchy problem

• ρt +

ρv(ρ)φ(x)

x = 0,

x ∈ R, t > 0, ρ(x, 0) = ρ(x), x ∈ R. (1) ρ(x, t) is a density; v(ρ) is a velocity map; φ(x) is a given external potential; ρ(x) is the initial datum. Possible applications: Traffic flows; Sedimentation processes; Flow of glaciers; Formation of Bose-Einstein condensates.

Macroscopic model Microscopic models Convergence A general overview

Goal: We want to derive a weak (eventually entropic) solution to (1) from systems of deterministic particles of Follow-The-Leader type.

Macroscopic model Microscopic models Convergence A general overview

Goal: We want to derive a weak (eventually entropic) solution to (1) from systems of deterministic particles of Follow-The-Leader type. The FTL particle approach has been applied for other models, as: LWR model.

• M. Di Francesco and M.D. Rosini (2015).
• M. Di Francesco, S. Fagioli and M.D. Rosini (2017).
• H. Holden and N.H. Risebro (2018).

ARZ model.

• A. Aw, A. Klar, T. Materne and M. Rascle (2002).
• F. Berthelin, P. Degond, M. Delitala and M. Rascle (2008).
• M. Di Francesco S. Fagioli and M.D. Rosini (2016).
• F. Berthelin and P. Goatin (2017).

Hughes model.

• M. Di Francesco, S. Fagioli, M.D. Rosini and G. Russo (2017).

IBVP with Dirichlet type conditions.

• M. Di Francesco, S. Fagioli, M.D. Rosini and G. Russo (2017).

Nonlocal transport equations.

• S. Fagioli and E. Radici (2018).
• M. Di Francesco, S. Fagioli and E. Radici (2019).

Macroscopic model Microscopic models Convergence A general overview

We now introduce our concept of entropy solution to (1). Definition Let ρ ∈ BV (R). We say that ρ ∈ L∞ [0, +∞) ; BV (R) is an entropy solution to (1) if: ρ(·, t) → ρ strongly in L1(R) as t ց 0. ρ satisfies the entropy condition, that is

T

• R
• |ρ(x, t) − k|ϕt(x, t) + sign(ρ(x, t) − k)

f(ρ(x, t)) − f(k) φ(x)ϕx(x, t) − sign(ρ(x, t) − k)f(k)φ′(x)ϕ(x, t)

• dxdt ≥ 0,

for all k ≥ 0 and for all ϕ ∈ Cc(R × (0, +∞)) with ϕ ≥ 0.

Macroscopic model Microscopic models Convergence Assumptions

Assumptions on v and ρ: (V) v ∈ Lip(R+) is a non-negative and non-increasing function with v(0) := vmax < +∞.

Macroscopic model Microscopic models Convergence Assumptions

Assumptions on v and ρ: (V) v ∈ Lip(R+) is a non-negative and non-increasing function with v(0) := vmax < +∞. (I) ρ ∈ L∞(R) ∩ BV (R) is a non-negative compactly supported function with [xmin, xmax] := Conv(supp(ρ)).

Macroscopic model Microscopic models Convergence Assumptions

Assumptions on v and ρ: (V) v ∈ Lip(R+) is a non-negative and non-increasing function with v(0) := vmax < +∞. (I) ρ ∈ L∞(R) ∩ BV (R) is a non-negative compactly supported function with [xmin, xmax] := Conv(supp(ρ)). Concerning φ, we deal with four different cases: (P1) φ(x) ≥ 0 for all x ∈ R (forward movement). (P2) φ(x) ≤ 0 for all x ∈ R (backward movement). (P3) xφ(x) ≥ 0 for all x ∈ R (repulsive movement). (P4) xφ(x) ≤ 0 for all x ∈ R (attractive movement).

Macroscopic model Microscopic models Convergence Assumptions

Assumptions on v and ρ: (V) v ∈ Lip(R+) is a non-negative and non-increasing function with v(0) := vmax < +∞. (I) ρ ∈ L∞(R) ∩ BV (R) is a non-negative compactly supported function with [xmin, xmax] := Conv(supp(ρ)). Concerning φ, we deal with four different cases: (P1) φ(x) ≥ 0 for all x ∈ R (forward movement). (P2) φ(x) ≤ 0 for all x ∈ R (backward movement). (P3) xφ(x) ≥ 0 for all x ∈ R (repulsive movement). (P4) xφ(x) ≤ 0 for all x ∈ R (attractive movement). Assumption on φ: (P) φ ∈ W 2,∞(R).

Macroscopic model Microscopic models Convergence Assumptions

Assumptions on v and ρ: (V) v ∈ Lip(R+) is a non-negative and non-increasing function with v(0) := vmax < +∞. (I) ρ ∈ L∞(R) ∩ BV (R) is a non-negative compactly supported function with [xmin, xmax] := Conv(supp(ρ)). Concerning φ, we deal with four different cases: (P1) φ(x) ≥ 0 for all x ∈ R (forward movement). (P2) φ(x) ≤ 0 for all x ∈ R (backward movement). (P3) xφ(x) ≥ 0 for all x ∈ R (repulsive movement). (P4) xφ(x) ≤ 0 for all x ∈ R (attractive movement). Assumption on φ: (P) φ ∈ W 2,∞(R). Extra assumption on v in case (P4): (V∗) There exists Rmax > 0 such that R := ||ρ||L∞(R) ≤ Rmax, v(ρ) > 0 for ρ < Rmax and v(ρ) ≡ 0 for ρ ≥ Rmax.

Macroscopic model Microscopic models Convergence The FTL approximation

Let ρ a fixed initial datum satisfying (I) and such that ||ρ||L1(R) = 1. We consider, for a fixed n ∈ N, n + 1 particles having equal mass ℓn := 1/n.

Macroscopic model Microscopic models Convergence The FTL approximation

Let ρ a fixed initial datum satisfying (I) and such that ||ρ||L1(R) = 1. We consider, for a fixed n ∈ N, n + 1 particles having equal mass ℓn := 1/n. Initial conditions of the particles Suitable atomization of ρ: we split [xmin, xmax] into n sub-intervals such that the mass of ρ is equal to ℓn on each of them and we set xn

0 := xmin,

xn

n := xmax,

xn

i := sup

• x ∈ R :

x

xn

i−1

ρ(x)dx < ln

• for i ∈ {1, . . . , n − 1}.

Macroscopic model Microscopic models Convergence The FTL approximation

Evolution of the particles Let us define Rn

i (t) :=

ℓn xn

i+1(t) − xn i (t) and kn := max

i ∈ {0, . . . , n} : xn

i ≤ 0

.

Macroscopic model Microscopic models Convergence The FTL approximation

Evolution of the particles Let us define Rn

i (t) :=

ℓn xn

i+1(t) − xn i (t) and kn := max

i ∈ {0, . . . , n} : xn

i ≤ 0

. The microscopic models we study are (Case P1)

˙

xn

i (t) = v(Rn i (t))φ(xn i (t)),

i ∈ {0, . . . , n − 1}, ˙ xn

n(t) = vmaxφ(xn n(t)),

xn

i (0) = xn i ,

i ∈ {0, . . . , n}, (2)

Macroscopic model Microscopic models Convergence The FTL approximation

Evolution of the particles Let us define Rn

i (t) :=

ℓn xn

i+1(t) − xn i (t) and kn := max

i ∈ {0, . . . , n} : xn

i ≤ 0

. The microscopic models we study are (Case P1)

˙

xn

i (t) = v(Rn i (t))φ(xn i (t)),

i ∈ {0, . . . , n − 1}, ˙ xn

n(t) = vmaxφ(xn n(t)),

xn

i (0) = xn i ,

i ∈ {0, . . . , n}, (2) (Case P2)

˙

xn

0 (t) = vmaxφ(xn 0 (t)),

˙ xn

i (t) = v(Rn i−1(t))φ(xn i (t)),

i ∈ {1, . . . , n}, xn

i (0) = xn i ,

i ∈ {0, . . . , n}, (3)

Macroscopic model Microscopic models Convergence The FTL approximation

Evolution of the particles Let us define Rn

i (t) :=

ℓn xn

i+1(t) − xn i (t) and kn := max

i ∈ {0, . . . , n} : xn

i ≤ 0

. The microscopic models we study are (Case P1)

˙

xn

i (t) = v(Rn i (t))φ(xn i (t)),

i ∈ {0, . . . , n − 1}, ˙ xn

n(t) = vmaxφ(xn n(t)),

xn

i (0) = xn i ,

i ∈ {0, . . . , n}, (2) (Case P2)

˙

xn

0 (t) = vmaxφ(xn 0 (t)),

˙ xn

i (t) = v(Rn i−1(t))φ(xn i (t)),

i ∈ {1, . . . , n}, xn

i (0) = xn i ,

i ∈ {0, . . . , n}, (3) (Case P3)

        

˙ xn

0 (t) = vmaxφ(xn 0 (t)),

˙ xn

i (t) = v(Rn i−1(t))φ(xn i (t)),

i ∈ {1, . . . , kn}, ˙ xn

i (t) = v(Rn i (t))φ(xn i (t)),

i ∈ {kn + 1, . . . , n − 1}, ˙ xn

n(t) = vmaxφ(xn n(t)),

xn

i (0) = xn i

i ∈ {0, . . . , n}, (4)

Macroscopic model Microscopic models Convergence The FTL approximation

Evolution of the particles Let us define Rn

i (t) :=

ℓn xn

i+1(t) − xn i (t) and kn := max

i ∈ {0, . . . , n} : xn

i ≤ 0

. The microscopic models we study are (Case P1)

˙

xn

i (t) = v(Rn i (t))φ(xn i (t)),

i ∈ {0, . . . , n − 1}, ˙ xn

n(t) = vmaxφ(xn n(t)),

xn

i (0) = xn i ,

i ∈ {0, . . . , n}, (2) (Case P2)

˙

xn

0 (t) = vmaxφ(xn 0 (t)),

˙ xn

i (t) = v(Rn i−1(t))φ(xn i (t)),

i ∈ {1, . . . , n}, xn

i (0) = xn i ,

i ∈ {0, . . . , n}, (3) (Case P3)

        

˙ xn

0 (t) = vmaxφ(xn 0 (t)),

˙ xn

i (t) = v(Rn i−1(t))φ(xn i (t)),

i ∈ {1, . . . , kn}, ˙ xn

i (t) = v(Rn i (t))φ(xn i (t)),

i ∈ {kn + 1, . . . , n − 1}, ˙ xn

n(t) = vmaxφ(xn n(t)),

xn

i (0) = xn i

i ∈ {0, . . . , n}, (4) (Case P4)

˙

xn

i (t) = v(Rn i (t))φ(xn i (t)),

i ∈ {0, . . . , kn}, ˙ xn

i (t) = v(Rn i−1(t))φ(xn i (t)),

i ∈ {kn + 1, . . . , n}, xn

i (0) = xn i ,

i ∈ {0, . . . , n}. (5)

Macroscopic model Microscopic models Convergence The FTL approximation

The main result of our work is Theorem (Di Francesco, S. (2019)) Let T > 0 fixed arbitrarily. Assume (V), (I), (P) are satisfied and, moreover, assume (V∗) is satisfied in case (P4). If one of (P1)-(P4) holds, then the approximating sequence {ρn}n∈N, with ρn(x, t) :=

n−1

• i=0

Ri(t)✶[xi(t),xi+1(t))(x), converges, up to a subsequence, almost everywhere and in L1 on R × [0, T] to the unique entropy solution to the Cauchy problem (1).

Macroscopic model Microscopic models Convergence Main properties

(a) Particles have finite position and velocity on bounded time intervals. (b) The distance of two consecutive particles is uniformly bounded from above

• n bounded time intervals.

Macroscopic model Microscopic models Convergence Main properties

(a) Particles have finite position and velocity on bounded time intervals. (b) The distance of two consecutive particles is uniformly bounded from above

• n bounded time intervals.

(c) Particles always move in the same direction in case (P3)-(P4). Proposition (Preservation of the particles’ sign in cases (P3)-(P4)) As long as the solution to (4) and (5) exists, then xi(t) ≤ 0, i ∈ {0, . . . , kn − 1} and xi(t) ≥ 0, i ∈ {kn + 1, . . . , n}. For i = kn, it holds that (i) If xkn < 0, then xkn(t) ≤ 0; (ii) If xkn = 0, then xkn(t) ≡ 0.

Macroscopic model Microscopic models Convergence Main properties

Proof (ii, case (P3)) Let [0, τ), with τ > 0 (possibly infinite), the maximal interval of existence of the solution of (4).

Macroscopic model Microscopic models Convergence Main properties

Proof (ii, case (P3)) Let [0, τ), with τ > 0 (possibly infinite), the maximal interval of existence of the solution of (4). Since | ˙ xkn(t)| = |v(Rkn−1(t))φ(xkn(t))| ≤ vmax|φ(xkn(t))| for all t < τ, then the unique solutions to

  

˙ yi(t) = ˙ xi(t), i = kn ˙ ykn(t) = −vmax|φ(ykn(t))|, yi(0) = xi, i ∈ {0, . . . , n} and

  

˙ zi(t) = ˙ xi(t), i = kn ˙ zkn(t) = vmax|φ(zkn(t))|, zi(0) = xi, i ∈ {0, . . . , n} (6) satisfy by comparison yi(t)≤ xi(t)≤ zi(t) for all t ∈ [0, τ ∗) and i ∈ {0, . . . , n}, with τ ∗ := sup{t ≤ τ : the solutions to (6) exist}.

Macroscopic model Microscopic models Convergence Main properties

Proof (ii, case (P3)) Let [0, τ), with τ > 0 (possibly infinite), the maximal interval of existence of the solution of (4). Since | ˙ xkn(t)| = |v(Rkn−1(t))φ(xkn(t))| ≤ vmax|φ(xkn(t))| for all t < τ, then the unique solutions to

  

˙ yi(t) = ˙ xi(t), i = kn ˙ ykn(t) = −vmax|φ(ykn(t))|, yi(0) = xi, i ∈ {0, . . . , n} and

  

˙ zi(t) = ˙ xi(t), i = kn ˙ zkn(t) = vmax|φ(zkn(t))|, zi(0) = xi, i ∈ {0, . . . , n} (6) satisfy by comparison yi(t)≤ xi(t)≤ zi(t) for all t ∈ [0, τ ∗) and i ∈ {0, . . . , n}, with τ ∗ := sup{t ≤ τ : the solutions to (6) exist}. In particular yi(t) = xi(t) = zi(t) for i = kn and t ∈ [0, τ ∗), since the (kn)−th particle does not affect the evolution (which remains unchanged) of the other n particles.

Macroscopic model Microscopic models Convergence Main properties

Proof (ii, case (P3)) Let [0, τ), with τ > 0 (possibly infinite), the maximal interval of existence of the solution of (4). Since | ˙ xkn(t)| = |v(Rkn−1(t))φ(xkn(t))| ≤ vmax|φ(xkn(t))| for all t < τ, then the unique solutions to

  

˙ yi(t) = ˙ xi(t), i = kn ˙ ykn(t) = −vmax|φ(ykn(t))|, yi(0) = xi, i ∈ {0, . . . , n} and

  

˙ zi(t) = ˙ xi(t), i = kn ˙ zkn(t) = vmax|φ(zkn(t))|, zi(0) = xi, i ∈ {0, . . . , n} (6) satisfy by comparison yi(t)≤ xi(t)≤ zi(t) for all t ∈ [0, τ ∗) and i ∈ {0, . . . , n}, with τ ∗ := sup{t ≤ τ : the solutions to (6) exist}. In particular yi(t) = xi(t) = zi(t) for i = kn and t ∈ [0, τ ∗), since the (kn)−th particle does not affect the evolution (which remains unchanged) of the other n particles. ykn(t) ≡ xkn(t) ≡ zkn(t) ≡ 0 for t ∈ [0, τ ∗), since ykn(0) = zkn(0) = 0 and the ODEs ˙ ykn(t) = −vmax|φ(zkn(t))| and ˙ zkn(t) = vmax|φ(zkn(t))| have as stationary solutions y = z = 0.

Macroscopic model Microscopic models Convergence Main properties

Proof (ii, case (P3)) Let [0, τ), with τ > 0 (possibly infinite), the maximal interval of existence of the solution of (4). Since | ˙ xkn(t)| = |v(Rkn−1(t))φ(xkn(t))| ≤ vmax|φ(xkn(t))| for all t < τ, then the unique solutions to

  

˙ yi(t) = ˙ xi(t), i = kn ˙ ykn(t) = −vmax|φ(ykn(t))|, yi(0) = xi, i ∈ {0, . . . , n} and

  

˙ zi(t) = ˙ xi(t), i = kn ˙ zkn(t) = vmax|φ(zkn(t))|, zi(0) = xi, i ∈ {0, . . . , n} (6) satisfy by comparison yi(t)≤ xi(t)≤ zi(t) for all t ∈ [0, τ ∗) and i ∈ {0, . . . , n}, with τ ∗ := sup{t ≤ τ : the solutions to (6) exist}. In particular yi(t) = xi(t) = zi(t) for i = kn and t ∈ [0, τ ∗), since the (kn)−th particle does not affect the evolution (which remains unchanged) of the other n particles. ykn(t) ≡ xkn(t) ≡ zkn(t) ≡ 0 for t ∈ [0, τ ∗), since ykn(0) = zkn(0) = 0 and the ODEs ˙ ykn(t) = −vmax|φ(zkn(t))| and ˙ zkn(t) = vmax|φ(zkn(t))| have as stationary solutions y = z = 0. Finally, we notice that τ ∗ ≥ τ, since neither ykn or zkn hits ykn−1 or zkn+1 respectively.

Macroscopic model Microscopic models Convergence Main properties

(d) Particles never collide, consequently they always maintain the same order: this gives the global existence of the solutions to (2)-(5). Proposition (Discrete maximum principle) As long as the solution to one of (2), (3) or (4) exists, then xn

i+1(t) − xn i (t) ≥ ℓn

R e−L′t with L′ := vmax||φ′(x)||L∞(R). As long as the solution to (5) exists, then xn

i+1(t) − xn i (t) ≥

ℓn Rmax .

Macroscopic model Microscopic models Convergence Main properties

Proof (Case (P1)) We first notice that xi+1 − xi ≥ ℓn R for all i ∈ {0, . . . , n − 1}.

Macroscopic model Microscopic models Convergence Main properties

Proof (Case (P1)) We first notice that xi+1 − xi ≥ ℓn R for all i ∈ {0, . . . , n − 1}. Now we use a backwards recursive argument.

Macroscopic model Microscopic models Convergence Main properties

Proof (Case (P1)) We first notice that xi+1 − xi ≥ ℓn R for all i ∈ {0, . . . , n − 1}. Now we use a backwards recursive argument. Base case: For i = n − 1, using a first order Taylor expansion of φ and Gronwall lemma, it follows xn(t) − xn−1(t) ≥ xn − xn−1

• e

t

vmaxφ′(˜ x(s))ds ≥ ℓn

R e−L′t.

Macroscopic model Microscopic models Convergence Main properties

Proof (Case (P1)) We first notice that xi+1 − xi ≥ ℓn R for all i ∈ {0, . . . , n − 1}. Now we use a backwards recursive argument. Base case: For i = n − 1, using a first order Taylor expansion of φ and Gronwall lemma, it follows xn(t) − xn−1(t) ≥ xn − xn−1

• e

t

vmaxφ′(˜ x(s))ds ≥ ℓn

R e−L′t. Inductive step: Suppose by contradiction the existence of j ∈ {0, . . . , n − 2} and t∗ > 0 such that the statement holds for t ≥ 0, i ∈ {j + 1, . . . , n − 1} and xj+1(t∗) − xj(t∗) < ℓn R e−L′t∗.

Macroscopic model Microscopic models Convergence Main properties

Proof (Case (P1)) We first notice that xi+1 − xi ≥ ℓn R for all i ∈ {0, . . . , n − 1}. Now we use a backwards recursive argument. Base case: For i = n − 1, using a first order Taylor expansion of φ and Gronwall lemma, it follows xn(t) − xn−1(t) ≥ xn − xn−1

• e

t

vmaxφ′(˜ x(s))ds ≥ ℓn

R e−L′t. Inductive step: Suppose by contradiction the existence of j ∈ {0, . . . , n − 2} and t∗ > 0 such that the statement holds for t ≥ 0, i ∈ {j + 1, . . . , n − 1} and xj+1(t∗) − xj(t∗) < ℓn R e−L′t∗. By continuity, there exists 0 ≤ ˜ t < t∗ such that xj+1(t) − xj(t)

• = ℓn

R e−L′˜ t

for t = ˜ t, < ℓn

R e−L′t

for ˜ t < t ≤ t∗.

Macroscopic model Microscopic models Convergence Main properties

Proof (Case (P1)) We first notice that xi+1 − xi ≥ ℓn R for all i ∈ {0, . . . , n − 1}. Now we use a backwards recursive argument. Base case: For i = n − 1, using a first order Taylor expansion of φ and Gronwall lemma, it follows xn(t) − xn−1(t) ≥ xn − xn−1

• e

t

vmaxφ′(˜ x(s))ds ≥ ℓn

R e−L′t. Inductive step: Suppose by contradiction the existence of j ∈ {0, . . . , n − 2} and t∗ > 0 such that the statement holds for t ≥ 0, i ∈ {j + 1, . . . , n − 1} and xj+1(t∗) − xj(t∗) < ℓn R e−L′t∗. By continuity, there exists 0 ≤ ˜ t < t∗ such that xj+1(t) − xj(t)

• = ℓn

R e−L′˜ t

for t = ˜ t, < ℓn

R e−L′t

for ˜ t < t ≤ t∗. Using again first order Taylor expansion of φ, the monotonicity of v and Gronwall lemma, we finally have xj+1(t) − xj(t) ≥ xj+1(˜ t) − xj(˜ t) e

t

˜ t v(Rj+1(s))φ′(˜ x(s))ds ≥ ℓn

R e−L′t. → ←

Macroscopic model Microscopic models Convergence The uniform control of TV and W1

Question: How to get the strong L1 convergence?

Macroscopic model Microscopic models Convergence The uniform control of TV and W1

Question: How to get the strong L1 convergence? Proposition (Uniform control of total variation) There exist four positive constants α, β, γ and ζ, independent on n, such that TV[ρn(·, t)] ≤ TV[ρ] + αt + βeL′t e[γt(1+t)+ζeL′t] for all t ≥ 0.

Macroscopic model Microscopic models Convergence The uniform control of TV and W1

Question: How to get the strong L1 convergence? Proposition (Uniform control of total variation) There exist four positive constants α, β, γ and ζ, independent on n, such that TV[ρn(·, t)] ≤ TV[ρ] + αt + βeL′t e[γt(1+t)+ζeL′t] for all t ≥ 0. Proof By definition TV[ρn(·, t)] = R0(t) + Rn−1(t) +

n−2

• i=0
• Ri(t) − Ri+1(t)
• .

Macroscopic model Microscopic models Convergence The uniform control of TV and W1

Question: How to get the strong L1 convergence? Proposition (Uniform control of total variation) There exist four positive constants α, β, γ and ζ, independent on n, such that TV[ρn(·, t)] ≤ TV[ρ] + αt + βeL′t e[γt(1+t)+ζeL′t] for all t ≥ 0. Proof By definition TV[ρn(·, t)] = R0(t) + Rn−1(t) +

n−2

• i=0
• Ri(t) − Ri+1(t)
• .

Let ησ(z), σ > 0, a C1 approximation of the absolute value function such that |η′

σ(z)| ≤ 1, 0 ≤ η′ σ(z)z ≤ ησ(z) for all z ∈ R and let us define

TVσ[ρn(·, t)] := R0(t) + Rn−1(t) +

n−2

• i=0

ησ

• Ri(t) − Ri+1(t)

.

Macroscopic model Microscopic models Convergence The uniform control of TV and W1

Proof After some calculation we get d dt TVσ[ρn(·, t)] = A(t) + B(t) + C(t), with A(t) := 1 + η′

σ

• R0(t) − R1(t) ˙

R0(t), B(t) := 1 − η′

σ

• Rn−2(t) − Rn−1(t) ˙

Rn−1(t), C(t) :=

n−2

• i=1
• η′

σ

• Ri(t) − Ri+1(t)

− η′

σ

• Ri−1(t) − Ri(t) ˙

Ri(t).

Macroscopic model Microscopic models Convergence The uniform control of TV and W1

Proof After some calculation we get d dt TVσ[ρn(·, t)] = A(t) + B(t) + C(t), with A(t) := 1 + η′

σ

• R0(t) − R1(t) ˙

R0(t), B(t) := 1 − η′

σ

• Rn−2(t) − Rn−1(t) ˙

Rn−1(t), C(t) :=

n−2

• i=1
• η′

σ

• Ri(t) − Ri+1(t)

− η′

σ

• Ri−1(t) − Ri(t) ˙

Ri(t). It can be shown (main tools: monotonicity of v, discrete maximum principle, second order Taylor approximation of φ) that d dt TVσ[ρn(·, t)] ≤ c0 ℓ2

n σ + c1 + c2eL′t +

c3 + c4t + c5eL′t TVσ[ρn(·, t)], where ci, i = 0, . . . , 5, are positive constants depending only on φ, v, ρ.

Macroscopic model Microscopic models Convergence The uniform control of TV and W1

Proof After some calculation we get d dt TVσ[ρn(·, t)] = A(t) + B(t) + C(t), with A(t) := 1 + η′

σ

• R0(t) − R1(t) ˙

R0(t), B(t) := 1 − η′

σ

• Rn−2(t) − Rn−1(t) ˙

Rn−1(t), C(t) :=

n−2

• i=1
• η′

σ

• Ri(t) − Ri+1(t)

− η′

σ

• Ri−1(t) − Ri(t) ˙

Ri(t). It can be shown (main tools: monotonicity of v, discrete maximum principle, second order Taylor approximation of φ) that d dt TVσ[ρn(·, t)] ≤ c0 ℓ2

n σ + c1 + c2eL′t +

c3 + c4t + c5eL′t TVσ[ρn(·, t)], where ci, i = 0, . . . , 5, are positive constants depending only on φ, v, ρ. Finally, applying Gronwall lemma and letting σ → 0, after some calculations we get the required estimate.

Macroscopic model Microscopic models Convergence The uniform control of TV and W1

Proposition (Uniform time continuity of 1-Wasserstein distance) There exists a constant c, dependent only on v and φ, such that W1

• ρn(·, t), ρn(·, s)

≤ c|t − s| for all t, s > 0, where W1 is the 1-Wasserstein distance. ✶

Macroscopic model Microscopic models Convergence The uniform control of TV and W1

Proposition (Uniform time continuity of 1-Wasserstein distance) There exists a constant c, dependent only on v and φ, such that W1

• ρn(·, t), ρn(·, s)

≤ c|t − s| for all t, s > 0, where W1 is the 1-Wasserstein distance. Proof Let L := vmax||φ||L∞(R). We recall that W1

• ρn(·, t), ρn(·, s)

= ||Xρn(·,t) − Xρn(·,s)||L1([0,1]), where Xρn(·,t) is the pseudo-inverse of the cumulative distribution of ρn(·, t), ✶

Macroscopic model Microscopic models Convergence The uniform control of TV and W1

Proposition (Uniform time continuity of 1-Wasserstein distance) There exists a constant c, dependent only on v and φ, such that W1

• ρn(·, t), ρn(·, s)

≤ c|t − s| for all t, s > 0, where W1 is the 1-Wasserstein distance. Proof Let L := vmax||φ||L∞(R). We recall that W1

• ρn(·, t), ρn(·, s)

= ||Xρn(·,t) − Xρn(·,s)||L1([0,1]), where Xρn(·,t) is the pseudo-inverse of the cumulative distribution of ρn(·, t), that is Xρn(·,t)(z) =

n−1

• i=0
• xi(t) + (z − iℓn)Ri(t)−1

✶[iℓn,(i+1)ℓn)(z).

Macroscopic model Microscopic models Convergence The uniform control of TV and W1

Proposition (Uniform time continuity of 1-Wasserstein distance) There exists a constant c, dependent only on v and φ, such that W1

• ρn(·, t), ρn(·, s)

≤ c|t − s| for all t, s > 0, where W1 is the 1-Wasserstein distance. Proof Let L := vmax||φ||L∞(R). We recall that W1

• ρn(·, t), ρn(·, s)

= ||Xρn(·,t) − Xρn(·,s)||L1([0,1]), where Xρn(·,t) is the pseudo-inverse of the cumulative distribution of ρn(·, t), that is Xρn(·,t)(z) =

n−1

• i=0
• xi(t) + (z − iℓn)Ri(t)−1

✶[iℓn,(i+1)ℓn)(z). It can be shown that W1

• ρn(·, t), ρn(·, s)

≤ A(s, t) + B(s, t), where A(s, t) :=

n−1

• i=0
• (i+1)ℓn

iℓn

|xi(t) − xi(s)|dz ≤ L(t − s), B(s, t) :=

n−1

• i=0
• (i+1)ℓn

iℓn

(z − iℓn)

Ri(t)−1 − Ri(s)−1 dz ≤ L

2 (t − s) in cases (P1)-(P2),

L(t − s) in cases (P3)-(P4).

Macroscopic model Microscopic models Convergence Convergence to the entropy solution

Proof (Main result) Step 1: {ρn}n converges (up to a subsequence) almost everywhere and in L1

• n R × [0, T] to a certain function ρ (Generalised Aubin-Lions lemma (R. Rossi

and G. Savar´ e, 2003)).

Macroscopic model Microscopic models Convergence Convergence to the entropy solution

Proof (Main result) Step 1: {ρn}n converges (up to a subsequence) almost everywhere and in L1

• n R × [0, T] to a certain function ρ (Generalised Aubin-Lions lemma (R. Rossi

and G. Savar´ e, 2003)). Step 2: {ρn}n satisfies, for every k ≥ 0 and ϕ ∈ C∞

c (R × (0, T)) with ϕ ≥ 0,

lim inf

n→+∞

T

• R
• |ρn − k|ϕt + sign(ρn − k)

f(ρn) − f(k) φϕx − sign(ρn − k)f(k)φ′ϕ

• dxdt ≥ 0,

Macroscopic model Microscopic models Convergence Convergence to the entropy solution

Proof (Main result) Step 1: {ρn}n converges (up to a subsequence) almost everywhere and in L1

• n R × [0, T] to a certain function ρ (Generalised Aubin-Lions lemma (R. Rossi

and G. Savar´ e, 2003)). Step 2: {ρn}n satisfies, for every k ≥ 0 and ϕ ∈ C∞

c (R × (0, T)) with ϕ ≥ 0,

lim inf

n→+∞

T

• R
• |ρn − k|ϕt + sign(ρn − k)

f(ρn) − f(k) φϕx − sign(ρn − k)f(k)φ′ϕ

• dxdt ≥ 0,

Step 3: ρ satisfies the entropy condition.

Macroscopic model Microscopic models Convergence Convergence to the entropy solution

Proof (Main result) Step 1: {ρn}n converges (up to a subsequence) almost everywhere and in L1

• n R × [0, T] to a certain function ρ (Generalised Aubin-Lions lemma (R. Rossi

and G. Savar´ e, 2003)). Step 2: {ρn}n satisfies, for every k ≥ 0 and ϕ ∈ C∞

c (R × (0, T)) with ϕ ≥ 0,

lim inf

n→+∞

T

• R
• |ρn − k|ϕt + sign(ρn − k)

f(ρn) − f(k) φϕx − sign(ρn − k)f(k)φ′ϕ

• dxdt ≥ 0,

Step 3: ρ satisfies the entropy condition. Step 4: ρ(·, t) → ρ strongly in L1(R) as t ց 0.

Macroscopic model Microscopic models Convergence Convergence to the entropy solution

Proof (Main result) Step 1: {ρn}n converges (up to a subsequence) almost everywhere and in L1

• n R × [0, T] to a certain function ρ (Generalised Aubin-Lions lemma (R. Rossi

and G. Savar´ e, 2003)). Step 2: {ρn}n satisfies, for every k ≥ 0 and ϕ ∈ C∞

c (R × (0, T)) with ϕ ≥ 0,

lim inf

n→+∞

T

• R
• |ρn − k|ϕt + sign(ρn − k)

f(ρn) − f(k) φϕx − sign(ρn − k)f(k)φ′ϕ

• dxdt ≥ 0,

Step 3: ρ satisfies the entropy condition. Step 4: ρ(·, t) → ρ strongly in L1(R) as t ց 0. Step 5: ρ is the unique entropy solution (L1 contraction property (K.H. Karlsen and N.H. Risebro, 2003)).

Macroscopic model Microscopic models Convergence

Present and future prospects Extension of the FTL approach for Hughes model (in preparation with M. Di Francesco and M.D. Rosini).

Macroscopic model Microscopic models Convergence

Present and future prospects Extension of the FTL approach for Hughes model (in preparation with M. Di Francesco and M.D. Rosini). Extension to external potentials depending also on time.

Macroscopic model Microscopic models Convergence

Present and future prospects Extension of the FTL approach for Hughes model (in preparation with M. Di Francesco and M.D. Rosini). Extension to external potentials depending also on time. Extension to the multi-dimensional case.

Macroscopic model Microscopic models Convergence

Present and future prospects Extension of the FTL approach for Hughes model (in preparation with M. Di Francesco and M.D. Rosini). Extension to external potentials depending also on time. Extension to the multi-dimensional case.

Thank you for the attention!