e Well-posedness of a monotone solver for traffic junctions Carlotta Donadello 1 , . Andreianov 2 and Giuseppe M. Coclite 3 in collaboration with Boris P 1 Laboratoire de Mathématiques de Besançon Université de Franche-Comté 2 Laboratoire de Mathématiques et Physique Théorique Université de Tours 3 Dipartimento di Matematica, Università di Bari In the occasion of Alberto Bressan 60 th birthday Trieste, June 14, 2016 C. Donadello (UFC) 1 / 21
Introduction Statement of the problem We consider a junction where m incoming and n outgoing roads meet. Incoming roads: x ∈ Ω i = R − , i = 1 , . . . , m ; Outgoing roads: x ∈ Ω j = R + , j = m + 1 , . . . , m + n ; The junction is located at x = 0. C. Donadello (UFC) 2 / 21
Introduction Statement of the problem On each road the evolution of traffic is described by ∂ t ρ h + ∂ x f h ( ρ h ) = 0 , h = 1 , . . . , m + n , ρ h density of vehicles, f h bell-shaped, non linearly degenerate flux, possibly different. Moreover, we postulate m m + n � � f j ( ρ j ( t , 0 + )) . f i ( ρ i ( t , 0 − )) = i = 1 j = m + 1 C. Donadello (UFC) 3 / 21
Introduction Solutions u 0 = ( u 1 0 , . . . , u m + n ) s.t. u h Fix � 0 ∈ L ∞ (Ω h , [ 0 , R ]) , ∀ h ∈ { 1 , . . . , m + n } . 0 We call solution a ( m + n ) -uple � ρ = ( ρ 1 , . . . , ρ m + n ) s.t. ρ h ∈ L ∞ ( R + × Ω h , [ 0 , R ]) ρ h is a Kruzhkov entropy solution in R + × { Ω h \ ∂ Ω h } . Namely ∀ k ∈ [ 0 , R ] and ∀ ϕ ∈ C 1 c ( R + × Ω h ) , ϕ ≥ 0 � � | ρ h − k | ϕ t + sign ( ρ h − k ) ( f h ( ρ h ) − f h ( k )) ϕ x dx dt R + Ω h � | u h + 0 ( x ) − k | ϕ ( 0 , x ) dx ≥ 0 . Ω h conservation at the junction holds. There is not hope to prove well posedness for solutions . C. Donadello (UFC) 4 / 21
Introduction Example Let m = 1, n = 2, f h ( u ) = u ( 1 − u ) = f ( u ) for h = 1 , 2 , 3. Consider the initial condition ( u 0 1 = 1 / 2 , u 0 2 = 0 , u 0 3 = 0 ) . Then ρ 1 ( t , x ) = 1 / 2, f ( ρ 1 ) = 1 / 4, � 1 − √ s � , f ( ρ 2 ( t , 0 + )) = 1 − s ρ 2 = R , 0 , 2 4 √ � 1 − � 1 − s , f ( ρ 3 ( t , 0 + )) = s ρ 3 = R , 0 4, 2 where R [ u l , u r ] is a rarefaction wave from u l to u r , is a solution for any s ∈ [ 0 , 1 ] . C. Donadello (UFC) 5 / 21
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. C. Donadello (UFC) 6 / 21
Introduction Vanishing viscosity approximations [Coclite-Garavello, 2010] Fix ε > 0 and consider ρ ε h , t + f h ( ρ ε h ) x = ερ ε h , xx , � � � � � m = � m + n f i ( ρ ε i ( t , 0 )) − ερ ε f j ( ρ ε j ( t , 0 )) − ερ ε i , x ( t , 0 ) j , x ( t , 0 ) , i = 1 j = m + 1 ρ ε h ( t , 0 ) = ρ ε h ′ ( t , 0 ) , ρ ε h ( 0 , x ) = u 0 h ,ε ( x ) , where the approximated initial conditions � u 0 ,ε satisfy h ,ε ∈ W 2 , 1 ∩ C ∞ (Ω h ; [ 0 , R ]) , u 0 u 0 → u 0 a.e. and in L p (Ω h ) , 1 ≤ p < ∞ , h ,ε − h , as ε → 0 , � u 0 � u 0 � ( u 0 L 1 (Ω h ) ≤ TV ( u 0 � ( u 0 � � � � � � � � L 1 (Ω h ) ≤ L 1 (Ω h ) , h ,ε ) x h ) , ε h ,ε ) xx L 1 (Ω h ) ≤ C 0 , h ,ε � h � � � with C 0 > 0 independent from ε, h . C. Donadello (UFC) 7 / 21
Introduction [Coclite-Garavello, 2010] ρ ε s.t. Theory of semigroups ⇒ for any fixed ε > 0 there exists a unique � ρ ε h ∈ C ([ 0 , ∞ ); L 2 (Ω h )) ∩ L 1 loc (( 0 , ∞ ); W 2 , 1 (Ω h )) , h ∈ { 1 , . . . , m + n } , m + n m + n 0 ≤ ρ ε � � ρ ε � � u 0 � � h ≤ R , h ( t , · ) � L 1 (Ω h ) ≤ L 1 (Ω h ) , ∀ t ≥ 0 , h � h = 1 h = 1 + additional a priori estimates. Compensated compactness ⇒ existence of a sequence { ε ℓ } ℓ ∈ N , ε ℓ → 0 and a solution � ρ of the inviscid Cauchy problem at the junction s.t. ρ ε ℓ a.e. and in L p h − → ρ h , loc ( R + × Ω h ) , 1 ≤ p < ∞ , (1) for every h ∈ { 1 , . . . , m + n } . Uniqueness of solutions for the inviscid problem is only proved in the case m = n . [Coclite-Garavello-Piccoli, 2005] C. Donadello (UFC) 8 / 21
Tools for our approach Godunov’s flux Given the Riemann problem u t + f ( u ) x = 0 , ( t , x ) ∈ R + × R � a , if x < 0 , u 0 ( x ) = b , if x > 0 . The Godunov flux is the function ( a , b ) �→ f ( u ( t , 0 − )) = f ( u ( t , 0 + )) . Analytically � min s ∈ [ a , b ] f ( s ) if a ≤ b , G ( a , b ) = max s ∈ [ 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 ≤ ∂ a G ( a , b ) ≤ L , − L ≤ ∂ b G ( a , b ) ≤ 0 . C. Donadello (UFC) 9 / 21
Tools for our approach Bardos-LeRoux-Nédélec boundary conditions Consider the initial and boundary value problem u t + f ( u ) x = 0 , for ( t , x ) in R + × R − u ( t , 0 − ) = u b ( t ) , u ( 0 , x ) = u 0 ( 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 − ) , u b ( t )) , sign ( u ( t , 0 − ) − u b ( t )) � f ( u ( t , 0 − )) − f ( k ) � ≥ 0 . (2) Remark u satisfies (2) ⇐ ⇒ f ( u ( t , 0 − )) = G ( u ( t , 0 − ) , u b ( t )) . C. Donadello (UFC) 10 / 21
Tools for our approach The junction as a family of IBVPs u 0 = ( u 1 0 , . . . , u m + n ) s.t. u h 0 ∈ L ∞ (Ω h , [ 0 , R ]) , ∀ h ∈ { 1 , . . . , m + n } . Fix � 0 We look for � ρ = ( ρ 1 , . . . , ρ m + n ) s.t. ∀ h , ρ h ∈ L ∞ ( R + × Ω h , [ 0 , R ]) is a weak entropy solution of ρ h , t + f h ( ρ h ) x = 0 , on ] 0 , T [ × Ω h , ρ h ( t , 0 ) = v h ( t ) , on ] 0 , T [ , ρ h ( 0 , x ) = u h 0 ( x ) , on Ω h , v : R + → [ 0 , R ] m + n is to be fixed where � depending on the model, in order to ensure conservation at the junction. C. Donadello (UFC) 11 / 21
Tools for our approach The junction as a family of IBVPs u 0 = ( u 1 0 , . . . , u m + n ) s.t. u h 0 ∈ L ∞ (Ω h , [ 0 , R ]) , ∀ h ∈ { 1 , . . . , m + n } . Fix � 0 We look for � ρ = ( ρ 1 , . . . , ρ m + n ) s.t. ∀ h , ρ h ∈ L ∞ ( R + × Ω h , [ 0 , R ]) is a weak entropy solution of ρ h , t + f h ( ρ h ) x = 0 , on ] 0 , T [ × Ω h , ρ h ( t , 0 ) = v h ( t ) , on ] 0 , T [ , ρ h ( 0 , x ) = u h 0 ( x ) , on Ω h , v : R + → [ 0 , R ] m + n is to be fixed where � depending on the model, in order to ensure conservation at the junction. We look for limits of vanishing viscosity approximations ⇒ we postulate ∀ h , h ′ ∈ { 1 , . . . , m + n } . v h ( t ) = v h ′ ( t ) , 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) 11 / 21
Admissibility condition and vanishing viscosity germ Admissible solutions at the junction Given � u 0 i.c., we say that � ρ = ( ρ 1 , . . . , ρ m + 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 ρ h is weak entropy solution for the IBVP ρ h , t + f h ( ρ h ) x = 0 , on ] 0 , T [ × Ω h , ρ h ( t , 0 ) = p ( t ) , on ] 0 , T [ , ρ h ( 0 , x ) = u h 0 ( x ) , on Ω h , and m m + n � � G ( p ( t ) , ρ j ( t , 0 + )) , G ( ρ i ( t , 0 − ) , p ( t )) = for a.e. t ∈ R + . i = 1 j = m + 1 Of course, any admissible solution is a solution . C. Donadello (UFC) 12 / 21
Admissibility condition and vanishing viscosity germ The vanishing viscosity germ G VV is the set of all stationary road-wise constant admissible solutions � u = ( u 1 , . . . , u m + n ) : ∃ p ∈ [ 0 , R ] s.t. : m m + n � � G i ( u i , p ) = G j ( p , u j ) G VV = . i = 1 j = m + 1 G i ( u i , p ) = f i ( u i ) , G j ( p , u j ) = f j ( u j ) , ∀ i , j u = ( u 1 , . . . , u m + n ) ∈ [ 0 , R ] m + n there exists p ∈ [ 0 , R ] such that Given any � conservation at the junction holds. The value of the flux at the junction, � m i = 1 G i ( u i , p ) , is unique while p is not. C. Donadello (UFC) 13 / 21
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