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The scalar caseThe system case

Conservation laws on networks

Mauro Garavello

University of Eastern Piedmont

joint works with R.M. Colombo and B. Piccoli

July 30, 2009

Nonlinear conservation laws and applications IMA, July 13-31, 2009


Conservation Laws on Networks

A network is a finite collection of

arcs and vertices




Ii = [ai , bi ]

Ij = [aj , bj ]


arcs and vertices

Each arc is modeled by Ii = [ai , bi ]




(ui )t + f (ui )x = 0

(uj )t + f (uj )x = 0


arcs and vertices

Each arc is modeled by Ii = [ai , bi ]

On each arc we consider the PDE

system (ui )t + f (ui )x = 0



Various applications

Car traffic

LWR model

ρt + (ρv(ρ))x = 0

- Coclite, G., Piccoli, SIAM J. Math. Anal. 36, 2005.- Holden, Risebro, SIAM J. Math. Anal. 26, 1995.- Lighthill, Whitham, Proc. Roy. Soc. London Ser. A 229, 1955.

- Richards, Oper. Res. 4, 1956.




Car traffic

Aw-Rascle-Zhang model

ρt + (ρv)x = 0,(v + p(ρ))t + v(v + p(ρ))x = 0,

- Aw, Rascle, SIAM J. Appl. Math. 60, 2000.- G., Piccoli, Commun. Partial Differential Equations 31, 2006.

- Zhang, Transportation Research B 36, 2002.




Car traffic

Colombo phase transition model

Free flow Congested flow

8

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(ρ, q) ∈ Ωf ,

ρt + [ρ · v ]x = 0,

v =“

1 −ρ

R

”

· V ,

8

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(ρ, q) ∈ Ωc ,

ρt + [ρ · v ]x = 0,

qt + [(q − Q) · v ]x = 0,

v =“

1 −ρ

R

”

·qρ

.

- Colombo, SIAM J. Appl. Math. 63, 2002.- Colombo, Goatin, Piccoli, J. Hyperbolic Differ. Equ., 2009.- Colombo, Goatin, Priuli, Nonlinear Anal. 66, 2007.

- Goatin, Math. Comput. Modelling 44, 2006.




Gas pipelines

p-system

ρt + qx = 0,

qt +(

q2

ρ+ p(ρ)

)

x= 0,

- Banda, Herty, Klar, Netw. Heterog. Media 1, 2006.- Colombo, G., SIAM J. Math. Anal. 39, 2008.- Colombo, Guerra, Herty, Sachers, SIAM J. Contol Optim. 48, 2009.- Colombo, Herty, Sachers, SIAM J. Math. Anal. 40, 2008.- Colombo, Mauri, J. Hyperbolic Diff. Eq. 5, 2008.

- Colombo, Marcellini, J. Matematical Anal. and Appl.




Data networks

ρt + f (ρ)x = 0

- D’Apice, Manzo, Piccoli, SIAM J. Appl. Math. 68, 2008.




Supply chains

ρt + (min µ(t, x), vρ)x = 0

- Armbruster, Degond, Ringhofer, Bull. Inst. Math. Acad. Sin. 2, 2007.- Gottlich, Herty, Klar, Commun. Math. Sci. 4, 2006.

- D’Apice, Gottlich, Herty, Piccoli, SIAM book series, 2009.




Blood circulation

At + mx = 0

mt +(

αm2

A

)

x+ A

ρpx = −K m

A

- Canic, Kim, Math. Meth. Appl. Sci. 26, 2003.

- Fernandez, Milisic, Quarteroni, Multiscale Model. Simul. 4, 2005.




Irrigation channel

De Saint-Venant equation

Ht + (Hv)x = 0vt +

(

12v2 + gH

)

x= gS(H, v)

- Coron, d’Andrea-Novel, Bastin, ECC1999.

- Gugat, Leugering, Ann. Inst. H. Poincare Anal. Non Lineaire 26, 2009.



The LWR model

ρt + f (ρ)x = 0



The LWR model

ρt + f (ρ)x = 0

• ρ(t, x) denotes the density of cars at time t > 0 and in the position x ∈ [a, b]



The LWR model

ρt + f (ρ)x = 0

• ρ(t, x) denotes the density of cars at time t > 0 and in the position x ∈ [a, b]• f (ρ) is the flux



The LWR model

ρt + f (ρ)x = 0

• ρ(t, x) denotes the density of cars at time t > 0 and in the position x ∈ [a, b]• f (ρ) is the flux and it is given by f (ρ) = ρv , where v is the average velocity



The LWR model

ρt + f (ρ)x = 0

• ρ(t, x) denotes the density of cars at time t > 0 and in the position x ∈ [a, b]• f (ρ) is the flux and it is given by f (ρ) = ρv , where v is the average velocity• v depends only on ρ in a decreasing way



The LWR model

ρt + f (ρ)x = 0

• ρ(t, x) denotes the density of cars at time t > 0 and in the position x ∈ [a, b]• f (ρ) is the flux and it is given by f (ρ) = ρv , where v is the average velocity• v depends only on ρ in a decreasing way• f (0) = f (ρmax ) = 0 is a strictly concave function



The LWR model

ρt + f (ρ)x = 0

• ρ(t, x) denotes the density of cars at time t > 0 and in the position x ∈ [a, b]• f (ρ) is the flux and it is given by f (ρ) = ρv , where v is the average velocity• v depends only on ρ in a decreasing way• f (0) = f (ρmax ) = 0 is a strictly concave function

0 ρρmax

v

vmax

0 ρρmax

f

σ



Solutions at nodes

- J node: n incoming arcs, m outgoing arcs

J

I1

I2

I3

I4

I5

I6

I7

I8

I9



Solutions at nodes

- J node: n incoming arcs, m outgoing arcs- Consider an initial datum ρ0,l in each arc

J

ρ1,0

ρ2,0

ρ3,0

ρ4,0

ρ5,0

ρ6,0

ρ7,0

ρ8,0

ρ9,0



Solutions at nodes

- J node: n incoming arcs, m outgoing arcs- Consider an initial datum ρ0,l in each arc- This corresponds to n + m IBV problems

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(ρl )t + f (ρl )x = 0, l ∈ 1, . . . , n + mρl (0, x) = ρl,0(x), x ∈ Il , l ∈ 1, . . . , n + mρl (t, 0) = ?, t > 0, l ∈ 1, . . . , n + m

J

ρ1,0

ρ2,0

ρ3,0

ρ4,0

ρ5,0

ρ6,0

ρ7,0

ρ8,0

ρ9,0



Solutions at nodes ... the Riemann problem

A Riemann problem at a node is a Cauchy problem with constantinitial conditions on each arc.





(ρl)t + f (ρl)x = 0, l ∈ 1, . . . , n + mρl(0, x) = ρ0,l , x ∈ Il , l ∈ 1, . . . , n + mρ0,l ∈ [0, ρmax ], x ∈ Il , l ∈ 1, . . . , n + m






Giving a solution is equivalent to giving its trace at the node.






Giving a solution is equivalent to giving its trace at the node.

Definition

A Riemann solver at the node J is a function

RS : [0, ρmax ]n+m → [0, ρmax ]

n+m,

which gives the trace at the node of a solution to thecorresponding Riemann problem.



The traces

Given an arc of a node and an initial datum ρl ,0, not all theelements in [0, ρmax ] can be the trace at the node of asolution to a Riemann problem



The traces

Given an arc of a node and an initial datum ρl ,0, not all theelements in [0, ρmax ] can be the trace at the node of asolution to a Riemann problemAn element ρl ∈ [0, ρmax ] can be the trace at a node of asolution to a Riemann problem only if:



The traces


Incoming arcs: the classical Riemann problem (ρl,0, ρl ) issolved with waves with negative speed



The traces


Incoming arcs: the classical Riemann problem (ρl,0, ρl ) issolved with waves with negative speedOutgoing arcs: the classical Riemann problem (ρl , ρl,0) issolved with waves with positive speed



The traces



We prescribe the conservation at J:

n∑

i=1

f (ρi ) =n+m∑

j=n+1

f (ρj)



The traces



We prescribe the conservation at J:

n∑

i=1

f (ρi ) =n+m∑

j=n+1

f (ρj)

There are infinitely many Riemann solvers with theseproperties!



A particular Riemann solver

1 Fix a node J: n incoming and m outgoing arcs





2 Fix a distribution matrix A ∈ M(m × n)






3 Impose the constraints

A · (f (ρ1), . . . , f (ρn))T = (f (ρn+1), . . . , f (ρn+m))T

on the fluxes at J








on the fluxes at J

4 Choose the only solution, which maximizes∑n

i=1 f (ρi )








on the fluxes at J

4 Choose the only solution, which maximizes∑n

i=1 f (ρi )

Remark

If we “invert” the order of the last two rules, i.e. first we maximizethe functional and then we impose some constraints, then weobtain a different Riemann solver.



Properties (P1), (P2) and (P3)

Property (P1)

RS has the property (P1) if, given (ρ1,0, . . . , ρn+m,0) and(ρ′1,0, . . . , ρ

′n+m,0) such that ρl ,0 = ρ′l ,0 whenever ρl ,0 or ρ′l ,0 is a

bad datum, then

RS(ρ1,0, . . . , ρn+m,0) = RS(ρ′1,0, . . . , ρ′n+m,0).




Property (P1)



bad datum, then

RS(ρ1,0, . . . , ρn+m,0) = RS(ρ′1,0, . . . , ρ′n+m,0).

The datum ρl ,0 is called bad if

ρl ,0 < σ, l ≤ n or ρl ,0 > σ, l ≥ n + 1




Property (P1)



bad datum, then

RS(ρ1,0, . . . , ρn+m,0) = RS(ρ′1,0, . . . , ρ′n+m,0).

The datum ρl ,0 is called bad if

ρl ,0 < σ, l ≤ n or ρl ,0 > σ, l ≥ n + 1

i.e. ρl ,0 gives a non trivial constraint for the flux solution at J




RS has the property (P2) if there exists C ≥ 1 such that, for everyequilibrium (ρ1,0, . . . , ρn+m,0) for RS and for every wave (ρl,0, ρl ) (l ∈1, . . . , n + m) interacting with J at time t, it holds

Tot.Var.f (t+) − Tot.Var.f (t−) ≤ C min˘

˛

˛f (ρl,0) − f (ρl )˛

˛ , |Γ(t+) − Γ(t−)|¯

.

- An equilibrium is a fixed point of RS

- The functionals Γ(t) and Tot.Var.f (t) are defined by

Γ(t) :=∑n

i=1 f (ρi (t, 0−))

Tot.Var.f (t) :=∑n+m

l=1 Tot.Var.f (ρl(t, ·))






˛

˛f (ρl,0) − f (ρl )˛

˛ , |Γ(t+) − Γ(t−)|¯

.

I1

I2

I3

I4

ρ1

t

ρ1,0

ρ2,0

ρ3,0

ρ4,0

I1

I2

I3

I4

J

ρ1

ρ1,0

ρ2,0

ρ3,0

ρ4,0






˛

˛f (ρl,0) − f (ρl )˛

˛ , |Γ(t+) − Γ(t−)|¯

.

I1

I2

I3

I4

ρ1

t

ρ1,0

ρ2,0

ρ3,0

ρ4,0

ρ1

ρ2

ρ3

ρ4

I1

I2

I3

I4

J

ρ1

ρ2,0

ρ3,0

ρ4,0

ρ1

ρ2

ρ3

ρ4




RS satisfies the property (P3) if, for every equilibrium(ρ1,0, . . . , ρn+m,0) of RS and for every wave (ρl ,0, ρl) (l ∈1, . . . , n + m) with f (ρl) < f (ρl ,0), interacting with J at timet > 0 it holds

Γ(t+) ≤ Γ(t−).



Main result

Theorem [AIP 2009]

Let RS be a Riemann solver satisfying properties (P1)–(P3). Fix initialconditions ρl,0 ∈ BV .For every T > 0, there exists a solution (ρ1, . . . , ρn+m) to the Cauchyproblem

∂∂t

ρl + ∂∂x

f (ρl) = 0

ρl(0, x) = ρl,0(x)l = 1, . . . , n + m

such that

RS(ρ1(t, 0), . . . , ρn+m(t, 0)) = (ρ1(t, 0), . . . , ρn+m(t, 0))

for a.e. t ∈ [0,T ].



Main result

Theorem [AIP 2009]

Let RS be a Riemann solver satisfying properties (P1)–(P3). Fix initialconditions ρl,0 ∈ BV .For every T > 0, there exists a solution (ρ1, . . . , ρn+m) to the Cauchyproblem

∂∂t

ρl + ∂∂x

f (ρl) = 0

ρl(0, x) = ρl,0(x)l = 1, . . . , n + m

such that

RS(ρ1(t, 0), . . . , ρn+m(t, 0)) = (ρ1(t, 0), . . . , ρn+m(t, 0))

for a.e. t ∈ [0,T ].

Remark [JDE 2009]

The previous result holds also in the case of Riemann solvers RSdepending on time



proof

- The proof is based on the wave-front tracking method

I1

I2

I3

I4



proof

- The proof is based on the wave-front tracking method- We approximate the initial datum by p.c. functions

I1

I2

I3

I4



proof

- The proof is based on the wave-front tracking method- We approximate the initial datum by p.c. functions- We solve each Riemann problem until the first interaction happens

I1

I2

I3

I4



proof

- The proof is based on the wave-front tracking method- We approximate the initial datum by p.c. functions- We solve each Riemann problem until the first interaction happens- We repeat the previous steps

I1

I2

I3

I4



Proof

1 Estimate on the number of waves



Proof

1 Estimate on the number of waves2 Estimate on the total variation of the flux



Proof


1 Properties (P1)–(P3) imply

Tot.Var.+Γ(·) ≤ CTot.Var.f (0+)



Proof




2 The total variation of Γ is bounded, since Γ is bounded



Proof




2 The total variation of Γ is bounded, since Γ is bounded3 The total variation of the flux is bounded

Tot.Var.f (·) ≤ C1Tot.Var.f (0+) + C1nf (σi )



Proof




2 The total variation of Γ is bounded, since Γ is bounded3 The total variation of the flux is bounded

Tot.Var.f (·) ≤ C1Tot.Var.f (0+) + C1nf (σi )

3 In general, there is not a bound for the total variation of thedensity



Continuous dependence

Theorem [AIP 2009]

Fix a node J and a Riemann solver RS satisfying (P1), (P2) and(P3) and such that

Tot.Var.f (t+) ≤ Tot.Var.f (t−)

for every time t at which an interaction of a wave with J happens.Then there exists a unique solution to the Cauchy problem and thesolution depends in a Lipschitz continuous way on the initialdatum with respect to the L1-topology.



Continuous dependence (idea of the proof)

1 Introduce a differential structure on L1.[ Bressan, Colombo, 1995 ] [ Bressan, Crasta, Piccoli, 2000 ]

γ : [0, 1] → L1 a curve s.t. γ(θ) is a piecewise constant functionswith N discontinuities: x1(θ) < x2(θ) < · · · < xN(θ).Tangent vector γ(θ) = (v , ξ)(θ) ∈ L1 × R

N if

L1 ∋ v(θ, x)= limh→0

γ(θ + h, x) − γ(θ, x)

h, for a.e. x ,

ξi (θ)= limh→0

xi (θ + h) − xi (θ)

h, i = 1, ...,N.

The norm of (v , ξ)(θ) is defined by:

‖(v , ξ)(θ)‖ = ‖v(θ)‖L1 +

N∑

i=1

|ξi (θ)||γ(θ, xi+) − γ(θ, xi−)|.




1 Introduce a differential structure on L1.

2 Define a distance between piecewise constant functions

d(u, u′) = inf

∫ 1

0‖γ(t)‖

d(u, u′) ∼∥

∥u − u′∥

∥

L1




1 Introduce a differential structure on L1.

2 Define a distance between piecewise constant functions

3 Prove that the norm of tangent vectors are not increasing intime along wave front tracking solutions




u0

u′0




u0

u′0

γ0(θ)




u0

u′0

u(t)

u′(t)

γ0(θ)




u0

u′0

u(t)

u′(t)

γ0(θ)

γt (θ)




u0

u′0

u(t)

u′(t)

γ0(θ)

γt (θ)

For a.e. θ, the norm ‖γs(θ)‖ is not increasing with respect to s.



The p-system in a tube

∂tρ + ∂xq = 0

∂tq + ∂x

(

q2

ρ+ p(ρ)

)

= 0

ρ(t, x) density of the fluid




∂tρ + ∂xq = 0

∂tq + ∂x

(

q2

ρ+ p(ρ)

)

= 0


q(t, x) = ρ(t, x)v(t, x) linear momentum density




∂tρ + ∂xq = 0

∂tq + ∂x

(

q2

ρ+ p(ρ)

)

= 0



v(t, x) fluid speed




∂tρ + ∂xq = 0

∂tq + ∂x

(

q2

ρ+ p(ρ)

)

= 0



v(t, x) fluid speed

p(ρ) pressure law



The pressure law p

p ∈ C 2 (]0, +∞[ ; ]0, +∞[) p′ > 0 p′′ ≥ 0

Typical case: p(ρ) = kργ γ ≥ 1

ρ

p

p(ρ) = ρ

p(ρ) = ρ2



Lax curves and other quantities

“dynamic pressure”: P(ρ, q) = q2

ρ+ p(ρ)

ρ

q





ρ+ p(ρ)

eigenvalues: λ1(ρ, q) = qρ−

p

p′(ρ), λ2(ρ, q) = qρ

+p

p′(ρ)

ρ

q λ1 = 0

λ2 = 0





ρ+ p(ρ)


p

p′(ρ), λ2(ρ, q) = qρ

+p

p′(ρ)

zones: A+, A+0 , A−

0 , A−

ρ

q λ1 = 0

λ2 = 0

A+

A−

A+0

A−

0





ρ+ p(ρ)


p

p′(ρ), λ2(ρ, q) = qρ

+p

p′(ρ)

zones: A+, A+0 , A−

0 , A−

Lax curves:

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:

S1(ρ) =ρ

ρo

qo −

s

ρ

ρo

(ρ − ρo ) (p(ρ) − p(ρo )) ρ ≥ ρo

S2(ρ) =ρ

ρo

qo −

s

ρ

ρo

(ρ − ρo ) (p(ρ) − p(ρo )) ρ ≤ ρo

R1(ρ) =ρ

ρo

qo − ρ

Z

ρ

ρo

p

p′(r)

rdr ρ ≤ ρo

R2(ρ) =ρ

ρo

qo + ρ

Z

ρ

ρo

p

p′(r)

rdr ρ ≥ ρo .

R1(ρ)

R2(ρ)

S1(ρ)S2(ρ)

ρρ0

q0

q



The Riemann problem at the junction

(M):∑

‖νl‖ql(t, 0) = 0 (conservation of mass)




(M):∑


(P): P (ρl(t, 0), ql(t, 0)) = P∗ (condition on dynamic pressure)




(M):∑


(P): P (ρl(t, 0), ql(t, 0)) = P∗ (condition on dynamic pressure)

(E):∑

‖νl‖F (ρl(t, 0), ql(t, 0)) ≤ 0 (entropy inequality)

E(ρ, q) =q2

2ρ+ ρ

Z ρ

ρ∗

p(r)

r2dr

F (ρ, q) =q

ρ· (E(ρ, q) + p(ρ))

‖νl‖ = section of the l-th tube



Stability of solutions for Riemann problems

Theorem [NHM 2006]

Fix n initial states (bρl , bql ) ∈ A0 satisfying

8

<

:

P

‖νl‖ bql = 0,

P(bρl , bql ) = P∗ l = 1, . . . , nP

‖νl‖F (bρl , bql ) < 0 .

Then, for every initial datum closed to (bρl , bql ), there exists (locally) a unique solution

(ρl , ql ) to the Riemann problem.

The proof is based on the Implicit Function Theorem



The Cauchy problem

Theorem [SIMA 2008]

Fix n subsonic states (ρ, q) ∈ (A0) such that

8

<

:

P

‖νl‖ ql = 0,

P(ρl , ql ) = P∗ l = 1, . . . , nP

‖νl‖F (ρl , ql ) < 0 .

Then, there exists a map S : [0, +∞[×D 7→ D, with the properties:

D ⊇˘

(ρ, q) ∈ L1(R+; R+ × R)n : Tot.Var.(ρ, q) ≤ δ0

¯

;



The Cauchy problem

Theorem [SIMA 2008]


8

<

:

P

‖νl‖ ql = 0,

P(ρl , ql ) = P∗ l = 1, . . . , nP

‖νl‖F (ρl , ql ) < 0 .


D ⊇˘


¯

;

for (ρ, q) ∈ D, S0(ρ, q) = (ρ, q) and for s, t ≥ 0, SsSt(ρ, q) = Ss+t(ρ, q);



The Cauchy problem

Theorem [SIMA 2008]


8

<

:

P

‖νl‖ ql = 0,

P(ρl , ql ) = P∗ l = 1, . . . , nP

‖νl‖F (ρl , ql ) < 0 .


D ⊇˘


¯

;


for (ρ, q), (ρ′, q′) ∈ D and s, t ≥ 0,

‚

‚St(ρ, q) − Ss(ρ′, q′)

‚

‚

L1 ≤ L ·`

‚

‚(ρ, q) − (ρ′, q′)‚

‚

L1 + |t − s|´

.



The Cauchy problem

Theorem [SIMA 2008]


8

<

:

P

‖νl‖ ql = 0,

P(ρl , ql ) = P∗ l = 1, . . . , nP

‖νl‖F (ρl , ql ) < 0 .


D ⊇˘


¯

;


for (ρ, q), (ρ′, q′) ∈ D and s, t ≥ 0,

‚

‚St(ρ, q) − Ss(ρ′, q′)

‚

‚

L1 ≤ L ·`

‚

‚(ρ, q) − (ρ′, q′)‚

‚

L1 + |t − s|´

.

the map t 7→ St(ρ, q) is a weak entropy solution to the Cauchy Problem.



The Cauchy problem

Interaction estimate at the junction

For any 1-waves σ−l

hitting the junction and producing the 2-waves σ+l

, it holds

nX

l=1

˛

˛σ+l

˛

˛ ≤ KJ ·n

X

l=1

˛

˛

˛

σ−l

˛

˛

˛

.



The Cauchy problem

Glimm functional

V (t) =n

X

l=1

X

α∈Jl

ˆ

2 KJ ·˛

˛σl,1,α

˛

˛ +˛

˛σl,2,α

˛

˛ +˛

˛σl,3,α

˛

˛

˜

Q(t) =

nX

l=1

X

˘

˛

˛σl,i,α σl,j,β

˛

˛ : (σl,i,α, σl,j,β) ∈ Al

¯

Υ(t) = V (t) + K1 · Q(t) ,



The Cauchy problem

Liu-Yang functional

Φ(u1, u2) =n

X

l=1

2X

i=1

Z +∞

0

˛

˛sl,i (x)˛

˛ Wl,i (x) dx ,

Wl,i (x) = 1 + κ1 Al,i (x) + κ1 κ2 [Υ (u1) + Υ (u2)] .

- Liu-Yang, Comm. Pure Appl. Math. 52, 1999.

- Bressan-Liu-Yang, Ration. Mech. Anal. 149, 1999.


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