Well-posedness of a monotone solver for traffic junctions Carlotta - - PowerPoint PPT Presentation

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Well-posedness of a monotone solver for traffic junctions

Carlotta Donadello1, in collaboration with Boris P . Andreianov2 and Giuseppe M. Coclite3

1Laboratoire de Mathématiques de Besançon

Université de Franche-Comté

2Laboratoire de Mathématiques et Physique Théorique

Université de Tours

3Dipartimento di Matematica, Università di Bari

In the occasion of Alberto Bressan 60th birthday Trieste, June 14, 2016

• C. Donadello (UFC)

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Introduction

Statement of the problem We consider a junction where m incoming and n outgoing roads meet. Incoming roads: x ∈ Ωi = R−, i = 1, . . . , m ; Outgoing roads: x ∈ Ωj = R+, j = m + 1, . . . , m + n ; The junction is located at x = 0.

• C. Donadello (UFC)

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Introduction

Statement of the problem On each road the evolution of traffic is described by ∂tρh + ∂xfh(ρh) = 0, h = 1, . . . , m + n, ρh density of vehicles, fh bell-shaped, non linearly degenerate flux, possibly different. Moreover, we postulate

m

• i=1

fi(ρi(t, 0−)) =

m+n

• j=m+1

fj(ρj(t, 0+)).

• C. Donadello (UFC)

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Introduction

Solutions Fix u0 = (u1

0, . . . , um+n

) s.t. uh

0 ∈ L∞(Ωh, [0, R]), ∀h ∈ {1, . . . , m + n}.

We call solution a (m + n)-uple ρ = (ρ1, . . . , ρm+n) s.t. ρh ∈ L∞(R+ × Ωh, [0, R]) ρh is a Kruzhkov entropy solution in R+ × {Ωh \ ∂Ωh}. Namely ∀k ∈ [0, R] and ∀ϕ ∈ C1

c(R+ × Ωh), ϕ ≥ 0

• R+
• Ωh

|ρh − k|ϕt+sign(ρh − k) (fh(ρh) − fh(k)) ϕx dx dt +

• Ωh

|uh

0(x) − k|ϕ(0, x) dx ≥ 0.

conservation at the junction holds. There is not hope to prove well posedness for solutions.

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Introduction

Example Let m = 1, n = 2, fh(u) = u(1 − u) = f(u) for h = 1, 2, 3. Consider the initial condition (u0

1 = 1/2, u0 2 = 0, u0 3 = 0). Then

ρ1(t, x) = 1/2, f(ρ1) = 1/4, ρ2 = R 1 − √s 2 , 0

• , f(ρ2(t, 0+)) = 1 − s

4 , ρ3 = R 1 − √ 1 − s 2 , 0

• , f(ρ3(t, 0+)) = s

4, where R[ul, ur] is a rarefaction wave from ul to ur, is a solution for any s ∈ [0, 1].

• C. Donadello (UFC)

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Introduction

Many different approaches to single out “suitable” solutions For the Riemann problem (road-wise constant initial conditions) at the junction, for example Holden-Risebro 1995 maximize a concave “entropy” function at the junction ; Coclite-Piccoli 2002, Coclite-Garavello-Piccoli 2005 traffic distribution matrix + optimization ; Lebacque 1996, Lebacque-Khoshyaran 2002 Supply-Demand model. . . . We prove well-posedness for solutions to the general Cauchy problem which are limit of vanishing viscosity approximations.

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Introduction

Vanishing viscosity approximations

[Coclite-Garavello, 2010]

Fix ε > 0 and consider            ρε

h,t + fh(ρε h)x = ερε h,xx,

m

i=1

• fi(ρε

i (t, 0))−ερε i,x(t, 0)

• = m+n

j=m+1

• fj(ρε

j (t, 0))−ερε j,x(t, 0)

• ,

ρε

h(t, 0) = ρε h′(t, 0),

ρε

h(0, x) = u0 h,ε(x),

where the approximated initial conditions u0,ε satisfy u0

h,ε ∈ W 2,1 ∩ C∞(Ωh; [0, R]),

u0

h,ε −

→ u0

h,

a.e. and in Lp(Ωh), 1 ≤ p < ∞, as ε → 0,

• u0

h,ε

• L1(Ωh) ≤
• u0

h

• L1(Ωh),
• (u0

h,ε)x

• L1(Ωh) ≤ TV(u0

h),

ε

• (u0

h,ε)xx

• L1(Ωh) ≤ C0,

with C0 > 0 independent from ε, h.

• C. Donadello (UFC)

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Introduction

[Coclite-Garavello, 2010] Theory of semigroups ⇒ for any fixed ε > 0 there exists a unique ρε s.t. ρε

h ∈ C([0, ∞); L2(Ωh)) ∩ L1 loc((0, ∞); W 2,1(Ωh)),

h ∈ {1, . . . , m + n}, 0 ≤ ρε

h ≤ R, m+n

• h=1

ρε

h(t, ·)L1(Ωh) ≤ m+n

• h=1
• u0

h

• L1(Ωh),

∀t ≥ 0, + additional a priori estimates. Compensated compactness ⇒ existence of a sequence {εℓ}ℓ∈N, εℓ → 0 and a solution ρ of the inviscid Cauchy problem at the junction s.t. ρεℓ

h −

→ ρh, a.e. and in Lp

loc(R+ × Ωh), 1 ≤ p < ∞,

(1) for every h ∈ {1, . . . , m + n}. Uniqueness of solutions for the inviscid problem is

• nly proved in the case m = n.

[Coclite-Garavello-Piccoli, 2005]

• C. Donadello (UFC)

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Tools for our approach

Godunov’s flux Given the Riemann problem      ut + f(u)x = 0, (t, x) ∈ R+ × R u0(x) =

• a,

if x < 0, b, if x > 0. The Godunov flux is the function (a, b) → f(u(t, 0−)) = f(u(t, 0+)). Analytically G(a, b) =

• mins∈[a,b] f(s)

if a ≤ b, maxs∈[b,a] f(s) if a ≥ b. Basic properties: Consistency: for all a ∈ [0, R], G(a, a) = f(a); Monotonicity and Lipschitz continuity: There exists L > 0 such that for all (a, b) ∈ [0, R]2 we have 0 ≤ ∂aG(a, b) ≤ L, −L ≤ ∂bG(a, b) ≤ 0.

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Tools for our approach

Bardos-LeRoux-Nédélec boundary conditions Consider the initial and boundary value problem      ut + f(u)x = 0, for (t, x) in R+ × R− u(t, 0−) = ub(t), u(0, x) = u0(x), u is a weak entropy solution for the IBVP if u is a Kruzhkov entropy solution in the interior of R+ × R−, u satisfies the boundary condition in the (BLN) sense ∀k ∈ I(u(t, 0−), ub(t)), sign(u(t, 0−)−ub(t))

• f(u(t, 0−)) − f(k)
• ≥ 0. (2)

Remark u satisfies (2) ⇐ ⇒ f(u(t, 0−)) = G(u(t, 0−), ub(t)).

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Tools for our approach

The junction as a family of IBVPs

Fix u0 = (u1

0, . . . , um+n

) s.t. uh

0 ∈ L∞(Ωh, [0, R]), ∀h ∈ {1, . . . , m + n}.

We look for ρ = (ρ1, . . . , ρm+n) s.t. ∀h, ρh ∈ L∞(R+ × Ωh, [0, R]) is a weak entropy solution of      ρh,t + fh(ρh)x = 0,

• n ]0, T[×Ωh,

ρh(t, 0) = vh(t),

• n ]0, T[,

ρh(0, x) = uh

0(x),

• n Ωh,

where v : R+ → [0, R]m+n is to be fixed depending on the model, in order to ensure conservation at the junction.

• C. Donadello (UFC)

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Tools for our approach

The junction as a family of IBVPs

Fix u0 = (u1

0, . . . , um+n

) s.t. uh

0 ∈ L∞(Ωh, [0, R]), ∀h ∈ {1, . . . , m + n}.

We look for ρ = (ρ1, . . . , ρm+n) s.t. ∀h, ρh ∈ L∞(R+ × Ωh, [0, R]) is a weak entropy solution of      ρh,t + fh(ρh)x = 0,

• n ]0, T[×Ωh,

ρh(t, 0) = vh(t),

• n ]0, T[,

ρh(0, x) = uh

0(x),

• n Ωh,

where v : R+ → [0, R]m+n is to be fixed depending on the model, in order to ensure conservation at the junction. We look for limits of vanishing viscosity approximations ⇒ we postulate vh(t) = vh′(t), ∀h, h′ ∈ {1, . . . , m + n}. See [Diehl, 2009], [Andreianov-Mitrovi´ c, 2015], for the m = n = 1 case. See [Andreianov-Cancès, 2013 and 2015] for different coupling conditions.

• C. Donadello (UFC)

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Admissibility condition and vanishing viscosity germ

Admissible solutions at the junction Given u0 i.c., we say that ρ = (ρ1, . . . , ρm+n) is an admissible solution for the Cauchy problem at the junction, if there exists p in L∞(R+, [0, R]) such that each component ρh is weak entropy solution for the IBVP      ρh,t + fh(ρh)x = 0,

• n ]0, T[×Ωh,

ρh(t, 0) = p(t),

• n ]0, T[,

ρh(0, x) = uh

0(x),

• n Ωh,

and

m

• i=1

G(ρi(t, 0−), p(t)) =

m+n

• j=m+1

G(p(t), ρj(t, 0+)), for a.e. t ∈ R+. Of course, any admissible solution is a solution.

• C. Donadello (UFC)

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Admissibility condition and vanishing viscosity germ

The vanishing viscosity germ GVV is the set of all stationary road-wise constant admissible solutions GVV =           

• u = (u1, . . . , um+n) : ∃p ∈ [0, R] s.t. :

m

• i=1

Gi(ui, p) =

m+n

• j=m+1

Gj(p, uj) Gi(ui, p) = fi(ui), Gj(p, uj) = fj(uj), ∀i, j            . Given any u = (u1, . . . , um+n) ∈ [0, R]m+n there exists p ∈ [0, R] such that conservation at the junction holds. The value of the flux at the junction, m

i=1 Gi(ui, p), is unique while p is not.

• C. Donadello (UFC)

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Admissibility condition and vanishing viscosity germ

The vanishing viscosity germ GVV is the set of all stationary road-wise constant admissible solutions GVV =           

• u = (u1, . . . , um+n) : ∃p ∈ [0, R] s.t. :

m

• i=1

Gi(ui, p) =

m+n

• j=m+1

Gj(p, uj) Gi(ui, p) = fi(ui), Gj(p, uj) = fj(uj), ∀i, j            . Lemma Given any u0 ∈ [0, R]m+n road-wise constant initial condition, there exists a self-similar admissible solution ρ = (ρ1, . . . , ρm+n) for the Riemann problem at the junction. The vector of traces γρ = (ρ1(0−), . . . , ρm+n(0+)) is in GVV.

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Admissibility condition and vanishing viscosity germ

Example

Consider a junction consisting of two incoming and one outgoing roads. f1(x) = −x2 + 1, f2(x) = −2x2 + 2, f3(x) = −3x2 + 3. Given the initial condition (u1

0 = −

• 1/2, u2

0 = 1/4, u3 0 =

• 1/6) one can trace

p → Gi(ui

0, p), i = 1, 2, p → G3(p, u3 0)

For all p ∈ [−

• 1/6, 0],

2

• i=1

Gi(ui

0, p) = G3(p, u3 0) = 2.5.

This is the maximum value for the flux at the junction, given the initial conditions.

• C. Donadello (UFC)

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Admissibility condition and vanishing viscosity germ

Equivalent definitions of admissible solution

• ρ = (ρ1, . . . , ρm+n) in L∞

R+ × Rm

− × Rn +; [0, R]m+n

is an admissible solution associated with the i.c. u0 ∈ L∞ Rm

− × Rn +; [0, R]m+n

if and only if Definition (used to prove uniqueness ) ∀h ∈ {1, . . . , m + n}, uh is a Kruzhkov entropy solution in the interior of R+ × Ωh; for a.e. t in R+, the vector of traces γρ(t) = (ρ1(t, 0−), . . . , ρm+n(t, 0+)) is in GVV. See [Garavello-Natalini-Piccoli-Terracina, 2007]. Definition (used to prove existence) ∀h ∈ {1, . . . , m + n}, uh is a Kruzhkov entropy solution in the interior of R+ × Ωh; ∀ k ∈ GVV, ρ satisfies adapted entropy inequality on the network, namely ∀ξ ∈ D(R+ × R), ξ ≥ 0,

m+n

• h=1
• R+
• Ωh

{|ρh − kh|ξt + sign(ρh − kh)(fh(ρh) − fh(kh))ξx} dx dt

• ≥ 0.

See [Baiti-Jenssen, 1997] and [Audusse-Perthame, 2005].

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Our results

Theorem: Well-posedness for admissible solutions

For any u0 = (u1

0, . . . , um+n

) ∈ L∞(Rm

− × Rn +; Rm+n) there exists at most one

admissible solution ρ in L∞(R+ × (Rm

− × Rn +); [0, R]m+n).

If ρ and ρ∗ are admissible solutions corresponding to u0 and v0, then

m+n

• h=1

ρh(t) − ρ∗

h(t)L1(Ωh;R) ≤ m+n

• h=1
• u0

h − v 0 h

• L1(Ωh;R)

The proof relies on three fundamental properties of GVV, completeness : given any Riemann datum, the corresponding admissible solution is in GVV. dissipativity : for any k1, k2 in GVV with kℓ = (k ℓ

1, . . . , k ℓ m+n), ℓ = 1, 2, m

• i=1

sign(k 1

i −k 2 i )

• fi(k 1

i ) − fi(k 2 i )

• −

m+n

• j=m+1

sign(k 1

j −k 2 j )

• fj(k 1

j ) − fj(k 2 j )

• ≥ 0. (3)

maximality : if k1 satisfies (3) for all k2 in GVV, then k1 ∈ GVV.

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Our results

Existence of admissible solutions

Can be proved by a convergent finite volumes numerical scheme based on Godunov fluxes; It is an immediate corollary of Theorem Vanishing viscosity limits are admissible solutions

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Our results

Existence of admissible solutions

Can be proved by a convergent finite volumes numerical scheme based on Godunov fluxes; It is an immediate corollary of Theorem Vanishing viscosity limits are admissible solutions Step 0 Standard theory shows that each component ρh of ρ, s.t. ρε

h → ρh, is a

Kurzhkov entropy solution in the interior of R+ × Ωh.

• C. Donadello (UFC)

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Our results

Existence of admissible solutions

Can be proved by a convergent finite volumes numerical scheme based on Godunov fluxes; It is an immediate corollary of Theorem Vanishing viscosity limits are admissible solutions Step 0 Standard theory shows that each component ρh of ρ, s.t. ρε

h → ρh, is a

Kurzhkov entropy solution in the interior of R+ × Ωh. Step 1 Almost all k ∈ GVV are vanishing viscosity limits. For any k in Go

VV ⊂ GVV, there exists p ∈ [0, R] and

k ε = (k ε

1 , . . . , k ε m+n) in

L∞(Rm

− × Rn +; [0, R]m+n) s.t. each component k ε h solves

     fh(ρε

h)x = ε(ρε h)xx,

• n Ωh

ρε

h(0) = p,

limx∈Ωh, |x|→+∞ ρε

h(x) = kh.

Moreover, k ε → k as ε → 0.

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Our results

Theorem Vanishing viscosity limits are admissible solutions Step 0 Let ρε → ρ, then each component ρh of ρ, is a Kurzhkov entropy solution in the interior of R+ × Ωh. Step 1 Almost all k ∈ GVV are vanishing viscosity limits. Step 2 Coclite and Garavello proved that vanishing viscosity approximations satisfy Kato’s type inequalities. In particular,

m+n

• h=1
• R+
• Ωh

{|ρε

h − k ε h |ξt + qh(ρε h, k ε h )ξx + ε|ρε h − k ε h |xξx} dx dt

• ≥ 0.

where qh(u, v) = sign(u − v)(fh(u) − fh(v)), ∀ξ ∈ D(R+ × R), ξ ≥ 0.

Step 3 The function ρ, s.t. ρε → ρ, satisfies the adapted entropy inequalities,

m+n

• h=1
• R+
• Ωh

{|ρh − kh|ξt + sign(ρh − kh)(fh(ρh) − fh(kh))ξx} dx dt

• ≥ 0.

∀ξ ∈ D(R+ × R), ξ ≥ 0.

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Thank you for your attention!!!!

and happy birthday Alberto!!!!!!

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