Introduction The problem Results Traveling-wave solutions for models of collective movements with degenerate diffusivities Andrea Corli Department of Mathematics and Computer Science University of Ferrara - Italy June 13-17th 2016 11th Meeting on Nonlinear Hyperbolic PDEs and Applications On the occasion of the 60th birthday of Alberto Bressan S.I.S.S.A. Trieste Joint work with Luisa Malaguti and Lorenzo di Ruvo (University of Modena - Reggio Emilia, Italy) 1 / 31
Introduction The problem Results A tribute to Alberto u t + A ( u ) u x = ǫu xx 2 / 31
Introduction The problem Results A tribute to Alberto u t + A ( u ) u x = ǫu xx S. Bianchini - A. Bressan: Vanishing viscosity solutions of nonlinear hyperbolic systems, Ann. Math. 161 (2005), 223–342, dedicated to Prof. Constantine Dafermos on the occasion of his 60th birthday. 2 / 31
Introduction The problem Results A tribute to Alberto u t + A ( u ) u x = ǫu xx S. Bianchini - A. Bressan: Vanishing viscosity solutions of nonlinear hyperbolic systems, Ann. Math. 161 (2005), 223–342, dedicated to Prof. Constantine Dafermos on the occasion of his 60th birthday. . . . and if ǫ = ǫ ( u ) ? In the case n = 1 , for special solutions. . . 2 / 31
Introduction The problem Results A simple motivation The simplest continuum (macroscopic) model for collective movements (traffic flow) is Lighthill-Whitham-Richards (1955) equation (conservation of mass) � � ρ t + ρv ( ρ ) x = 0 or ρ t + q ( ρ ) x = 0 (LWR) where t, x ∈ R ρ ∈ [0 , ρ ] is the density (of cars, pedestrians), ρ the maximal density v = v ( ρ ) the (assigned) speed, usually v ′ ( ρ ) < 0 and v ( ρ ) = 0 q = q ( ρ ) = ρv ( ρ ) the flux: usually q ′′ ( ρ ) < 0 (ok if v ′′ ( ρ ) < 0 ). ✻ ✻ q v ✲ ✲ ρ ρ ρ ρ Huge literature, in particular in the last twenty years: systems (different populations), networks, control. 3 / 31
Introduction The problem Results Diffusion in collective movements Lighthill and Whitham proposed to include a linear diffusion term in (LWR) to avoid the appearance of shock waves: � � ρ t + ρv ( ρ ) x = Dρ xx . The density ρ is still conserved. What is the “correct” diffusion? Several approaches: Payne (1971), Nelson (2000), Helbing (2001). 4 / 31
Introduction The problem Results Diffusion in collective movements Lighthill and Whitham proposed to include a linear diffusion term in (LWR) to avoid the appearance of shock waves: � � ρ t + ρv ( ρ ) x = Dρ xx . The density ρ is still conserved. What is the “correct” diffusion? Several approaches: Payne (1971), Nelson (2000), Helbing (2001). The simplest idea for generating diffusion (vehicular flow): drivers adjust their speed to the local density; however, there is a reaction time τ in the response to events; drivers compensate for this delay by adjusting to the density seen at some anticipation distance δ ahead of their current position. 4 / 31
Introduction The problem Results Diffusion in collective movements Lighthill and Whitham proposed to include a linear diffusion term in (LWR) to avoid the appearance of shock waves: � � ρ t + ρv ( ρ ) x = Dρ xx . The density ρ is still conserved. What is the “correct” diffusion? Several approaches: Payne (1971), Nelson (2000), Helbing (2001). The simplest idea for generating diffusion (vehicular flow): drivers adjust their speed to the local density; however, there is a reaction time τ in the response to events; drivers compensate for this delay by adjusting to the density seen at some anticipation distance δ ahead of their current position. The actual mean speed is then � � V ( x, t )= v ρ ( x + δ − vτ, t − τ ) ∼ v + δv ′ ρ x − τv ′ ρ x − τv ′ ρ t = v + δv ′ ρ x + τ ( v ′ ) 2 ρρ x expanding in δ and τ at first order. Then � � 2 � � δv ′ ( ρ ) + τρ v ′ ( ρ ) q = q + ρ ˜ ρ x and we end up with � � � � ρ t + ρv ( ρ ) x = D ( ρ ) ρ x x for � 2 . D ( ρ ) = − δρv ′ ( ρ ) − τρ 2 � v ′ ( ρ ) 4 / 31
Introduction The problem Results Properties of the diffusivity Since v ′ ( ρ ) < 0 : � � � � 2 δv ′ ( ρ ) v ′ ( ρ ) D ( ρ ) = − ρ + τρ = D δ ( ρ ) + D τ ( ρ ) . � �� � � �� � � �� � � �� � ≤ 0 ≥ 0 ≤ 0 ≥ 0 Usually drivers anticipate so as to more than compensate for the delay: D ( ρ ) ≥ 0 . There are situations where the reaction time dominates anticipation: D ( ρ ) < 0 for large ρ . E.g.: roads providing limited sight distance ahead (fog). 5 / 31
Introduction The problem Results Properties of the diffusivity Since v ′ ( ρ ) < 0 : � � � � 2 δv ′ ( ρ ) v ′ ( ρ ) D ( ρ ) = − ρ + τρ = D δ ( ρ ) + D τ ( ρ ) . � �� � � �� � � �� � � �� � ≤ 0 ≥ 0 ≤ 0 ≥ 0 Usually drivers anticipate so as to more than compensate for the delay: D ( ρ ) ≥ 0 . There are situations where the reaction time dominates anticipation: D ( ρ ) < 0 for large ρ . E.g.: roads providing limited sight distance ahead (fog). D (0) = 0 : if no density, then no diffusivity! 5 / 31
Introduction The problem Results Properties of the diffusivity Since v ′ ( ρ ) < 0 : � � � � 2 δv ′ ( ρ ) v ′ ( ρ ) D ( ρ ) = − ρ + τρ = D δ ( ρ ) + D τ ( ρ ) . � �� � � �� � � �� � � �� � ≤ 0 ≥ 0 ≤ 0 ≥ 0 Usually drivers anticipate so as to more than compensate for the delay: D ( ρ ) ≥ 0 . There are situations where the reaction time dominates anticipation: D ( ρ ) < 0 for large ρ . E.g.: roads providing limited sight distance ahead (fog). D (0) = 0 : if no density, then no diffusivity! Analogously one could assume: D ( ρ ) = 0 : if no movement, then no diffusivity! 5 / 31
Introduction The problem Results Properties of the diffusivity Since v ′ ( ρ ) < 0 : � � � � 2 δv ′ ( ρ ) v ′ ( ρ ) D ( ρ ) = − ρ + τρ = D δ ( ρ ) + D τ ( ρ ) . � �� � � �� � � �� � � �� � ≤ 0 ≥ 0 ≤ 0 ≥ 0 Usually drivers anticipate so as to more than compensate for the delay: D ( ρ ) ≥ 0 . There are situations where the reaction time dominates anticipation: D ( ρ ) < 0 for large ρ . E.g.: roads providing limited sight distance ahead (fog). D (0) = 0 : if no density, then no diffusivity! Analogously one could assume: D ( ρ ) = 0 : if no movement, then no diffusivity! τ = 0 for pedestrians (Bruno-Tricerri-Tosin-Venuti (2011)). There: � � � � ρ − 1 1 1 − e − γ D ( ρ ) = − δρv ′ ( ρ ) ≥ 0 , v ( ρ ) = v , ρ ∈ [0 , ρ ] ρ where γ > 0 is obtained through experimental data. v and D are exponentially flat at 0 (L=leisure, C=commuters, R=rush hours). Velocities [BTTV] Fluxes [BTTV] Diffusion coefficients [BTTV] 2 2 2 L L L C C C 1.8 1.8 1.8 R R R 1.6 1.6 1.6 1.4 1.4 1.4 1.2 1.2 1.2 ρ · v( ρ ) 1 1 D( ρ ) 1 v( ρ ) 0.8 0.8 0.8 0.6 0.6 0.6 0.4 0.4 0.4 0.2 0.2 0.2 0 0 0 0 1 2 3 4 5 6 0 1 2 3 4 5 6 0 1 2 3 4 5 6 ρ ρ ρ 5 / 31
Introduction The problem Results Source terms Now � � � � ρ t + ρv ( ρ ) x = D ( ρ ) ρ x x + g ( ρ ) . Source terms in modeling collective movements are often localized in space (Bagnerini-Colombo-C. (2006)): g = g ( x, ρ ) = χ [ a,b ] ( x ) g 1 ( ρ ) � entry : g ′ g 1 ( ρ ) ≥ 0 , g ( ρ ) = 0 , 1 ( ρ ) ≤ 0; with g ′ exit : g 1 ( ρ ) ≤ 0 , g (0) = 0 , 1 ( ρ ) ≥ 0 . Entry: g 1 ( ρ ) = 0 : there is no room for further entries if the maximal density is reached; g ′ 1 ( ρ ) ≤ 0 : the less the cars on the road, the more the cars enter; for example: g ( ρ ) = L · ( ρ − ρ ) α , L > 0 , α > 0 . 6 / 31
Introduction The problem Results Source terms Now � � � � ρ t + ρv ( ρ ) x = D ( ρ ) ρ x x + g ( ρ ) . Source terms in modeling collective movements are often localized in space (Bagnerini-Colombo-C. (2006)): g = g ( x, ρ ) = χ [ a,b ] ( x ) g 1 ( ρ ) � entry : g ′ g 1 ( ρ ) ≥ 0 , g ( ρ ) = 0 , 1 ( ρ ) ≤ 0; with g ′ exit : g 1 ( ρ ) ≤ 0 , g (0) = 0 , 1 ( ρ ) ≥ 0 . Entry: g 1 ( ρ ) = 0 : there is no room for further entries if the maximal density is reached; g ′ 1 ( ρ ) ≤ 0 : the less the cars on the road, the more the cars enter; for example: g ( ρ ) = L · ( ρ − ρ ) α , L > 0 , α > 0 . However, source terms can be “diffused”: g = g ( ρ ) = g 1 ( ρ ) . For instance, pedestrians moving along a long corridor (street): if the number of side entries (cross streets) is large, one drops a model with many localized entries for a model with a diffuse source term. g ✻ ✻ ρ ✲ g ρ ρ ✲ ρ exits entries 6 / 31
Introduction The problem Results Some references Degenerate diffusivities are common in physics (porous media, heat conduction), neurophysics and biophysics, chemical physics, biology (chemotaxis), population genetics, tumor growth, and mathematical ecology, jointly with source terms vanishing at some point (Kalashnikov (1987)). Diffusivities changing sign: in traffic, Kerner-Osipov (1994); in biology, Maini-Malaguti (2006) (diffusion-aggregation phenomena, also with source terms). For traveling waves (degenerate diffusivities): Gilding-Kersner (2004). Engler (1985) gave a general and interesting transformation that reduces the degenerate diffusivity to a constant one. Not used in the following, but there are some common features. 7 / 31
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