Quasi-static evolution and congested transport Inwon Kim Joint with Damon Alexander, Katy Craig and Yao Yao UCLA, UW Madison
Hard congestion in crowd motion The following crowd motion model is proposed by Maury, Roundneff-Chupin and Santambrogio (2010): ρ(x, t): pedestrian population density, which cannot exceed a certain maximal value (which we assume to be 1). Φ(x): The desired velocity field for an individual located at x. It may not be achieved due to the constraint ρ(, t) 1. In the saturated zone {ρ(, t) = 1}, the actual velocity field v(, t) must satisfy v(, t) 0, in order to not increase the density. (If so, we say v is feasible).
Hard congestion in crowd motion In [MRS], they consider the PDE system (P) { ρt + (ρv) = 0 v(, t) = P Cρ Φ, where P Cρ is the projection towards the space of feasible velocity fields in L 2 sense. They link this system with the following gradient flow: For ρ P 2 (R d ), let ˆ ρ(x)φ(x)dx ρ 1 E [ρ] = R d + otherwise.
Gradient flow approach for hard congestion Let ρ (, t) be the gradient flow of E with respect to Wasserstein distance W 2 with initial data ρ 0. Existence and uniqueness of ρ can be shown following the book by Ambrosio, Gigli and Savaré (2005). [MRSV] showed that ρ is a weak solution to (P), however uniqueness of the weak solution to (P) is unknown since the velocity field v = P Cρ Φ is only in L 2. Let Φ 0. Then formally speaking, the velocity field pulls the particles together. In this setting we will show a unique characterization of the velocity field v for ρ.
Modification of the velocity We expect the modified velocity to be of the form v = p + Φ. The continuity equation for ρ then is expected to be of the form ρ t (ρ( p + Φ)) = 0, where p is the pressure generated by the constraint, supported on the congested set {ρ = 1}.
A remark on the assumption Φ 0 When Φ 0, we expect the pressure to be nonzero in all of {ρ = 1}. Moreover the modified velocity should be incompressible in the congested region since the original velocity field only tries to compress the density. Thus p should solve ( p + Φ) = 0 or p = Φ in {ρ = 1}.
Evolution of the congested region Suppose ρ solves the modified discontinuity equation with p discontinuously changing to zero across Ω(t). Then denoting ρ = ρ I χ Ω(t) + ρ O we have 0 = t [ Ω(t) (ρi ) t dx + R n Ω(t) ρo dx] = Ω(t) (ρi ) t + R n Ω(t) (ρo ) t dx + Ω(t) V (ρi ρ O )ds = Ω(t) (ρi ( p + Φ) + Ω(t) (ρ O Φ) + Ω(t) V C (ρi ρ O )d = Ω(t) ρi ν p + (ρ I ρ O )( ν Φ + V )ds, where V = V x,t denotes the (outward) normal velocity of Ω(t).
Evolution of ρ when Φ 0 Above calculation suggests the following evolution for ρ(, t) = χ Ωt + ρ O, where the congested set Ω(t) := {p(, t) > 0} is determined by the following free boundary problem for p 0: p(, t) = Φ in {p(, t) > 0}; V = 1 (1 ρ O ) ( νp) ν Φ on {p(, t) > 0}.
Note that the velocity law V = 1 1 ρ O Dp νφ indicates that there is a generic discontinuity of ρ across Ω(t). The well-posedness of the free boundary problem can be shown by viscosity solutions theory. We are interested in connecting this problem with the gradient flow solution ρ. In addition to the assumption Φ 0, we will also assume that the initial data is patch.
Quasi-static evolution: patch case Suppose ρ 0 = χ Ω0, and consider the following free boundary problem that p solves with the initial data {p(, 0) > 0} = Ω 0 : (FB) { p(, t) = Φ in {p(, t) > 0} =: Ωt ; V = Dp ν Φ on Ω t. Our goal is to prove that the gradient flow solution ρ with initial data ρ 0 satisfies ρ (, t) = χ Ωt.
Approximation by Porous Medium Equation Let ρ m solve the following porous medium equation with drift: ρ t = ρ m + (ρ Φ), with initial data ρ(, 0) = χ Ω0. It is well known (Otto, 2001) that ρ m is the gradient flow for E m [ρ] = 1 ˆ ˆ ρ m dx + ρφdx. m We will show that ρ m converging to ρ as m, with a rate. We will also show that ρ m converges to χ Ωt, yielding the desired statement, ρ = χ Ωt.
Convergence as m Theorem (Alexander-K-Yao., 2013) Let Ω 0 be a compact set in R d with locally Lipschitz boundary, then (a) Assuming Φ 0. Then there is a unique family of compact sets Ω t evolving with (FB). As m, ρ m χ Ωt locally uniformly away from Ω(t). (b) Assume D 2 Φ and inf Φ is finite. Then ρ m (, t) converges to ρ (, t) in W 2 distance uniformly in t [0, T ], with convergence rate sup W 2 (ρ m (t), ρ (t)) 1 t [0,T ] m 1/24.
Convergence as m Corollary Since ρ m (, t) converges to both χ Ωt and ρ (, t) as m, χ Ωt and ρ (, t) must be equal almost everywhere. While the gradient flow approach cannot directly deal with general nonconvex bounded domains (with e.g. Neumann boundary conditions), the viscosity solution approach still applies and we have ρ m χ Ωt. On the other hand, without the condition Φ 0, the gradient flow approach still works and we still have ρ m (, t) ρ (, t) but the characterization of the modified velocity remains open.
ρ m χ Ωt : Heuristics First let us discuss the convergence of ρ m to χ Ωt. The following heuristics suggest that ρ m (, t) should converge to χ Ωt as m. The main trick is to consider the limit of the corresponding pressure variable p m = m m 1 ρm 1 m rather than ρ m, which remains continuous as m. The equation for p m is (ρ m ) t (ρ m ( p m + Φ) = 0 One then considers the corresponding PDE for p m, and show the convergence of p m to p by barrier (viscosity solutions) argument.
Convergence of ρ m, p m as m : Relevant work When Φ = 0 but with source term, in the patch case: Gil and Quirós (1998), K(2003). Weak solutions theory is developed for general initial data in Perthame-Quiros-Vazquez (2013), where they study the m limit of ρ t (ρ m ) = ρg(p).
ρ m χ Ωt We first define viscosity solution for (FB), and prove the comparison principle. Then we construct the lower limit of {p m } m as m : u 2 (x, t) := lim inf n m n (x,t) (y,s) <1/n p m (y, s), and use comparison arguments to show that u 2 is a supersolution of the Hele-Shaw problem (P) with the initial pressure satisfying {p 0 > 0} = Ω 0.
ρ m χ Ωt Roughly speaking, the strategy is then to show that the corresponding upper limit u 1 of {p m } m is a subsolution of (P) with the initial data p 0. Then due to the comparison principle we can conclude that u 1 (u 2 ). Since u 1 u 2 by definition, it follows that u 1 = (u 2 ) and (u 1 ) = u 2. Let Ω t := {u 1 (, t) > 0}. Then (Ω t ) 0 = (Ω t ) 0 and ρ m uniformly converges to χ Ωt away from Ω t with initial data u 0 = χ Ω0.
ρ m ρ : based on the JKO scheme Next we proceed to show ρ m ρ. The goal is to show W 2 (ρ m, ρ ) Cm 1/24, where W 2 is the 2-Wasserstein distance. To this end, let us first compare discrete-time solutions over one time step. Let µ m and µ be the respective minimizer of the following JKO scheme for one time step: µ m = argmin E m [ρ] + 1 ρ P 2 (R d ) 2 t W 2 2 (ρ, ρ 0 ) µ = argmin E [ρ] + 1 ρ P 2 (R d ) 2 t W 2 2 (ρ, ρ 0 ) We want to estimate W 2 (µ m, µ ): the main difficulty is that µ m may not be in L.
ρ m ρ Towards a contradiction, suppose W 2 (µ m, µ ) is large; in this case we want to find a better competitor µ, such that E m [ µ] + 1 2 t W 2 2 ( µ, ρ 0 ) + E [ µ] + 1 2 t W 2 2 ( µ, ρ 0 ) <E m [µ m ] + 1 2 t W 2 2 (µ m, ρ 0 ) + E [µ ] + 1 2 t W 2 2 (µ, ρ 0 ) This means µ would at least beat one of µ m and µ! How do we find such µ? First guess: Choose µ as the midpoint (along the generalized geodesics) of µ m and µ. This choice saves the distance. But E [ µ] may be infinite.
ρ m ρ A suitable competitor µ can be found as follows: 1 Even though the maximum density of µ m may exceed 1, (µm 1) + dx m 1/2 for m > 2. 2 Thus we can find some η m 1, such that W 2 (η m, µ m ) m 1/4, and E m [η m ] E m [µ m ]. 3 One can then choose µ as the midpoint (along the generalized geodesics) of η m and µ. (Note that µ 1.) 4 µ would be better than either µ m or µ if W 2 (µ m, µ ) m 1/8.
ρ m ρ :Controlling the distance for multiple time steps m m ρ 3 m m ρ 2 m ρ 0 m ρ1 m ρ 1 d m η 2 1 δ d 2 d 3 δ ρ 2 m η 3 δ ρ 3 m
Confinement and long time behavior For 1 < m, if Φ(x) as x, using the comparison principle, we know that if the initial data is compactly supported, the discrete solution to JKO scheme will be uniformly confined for all time steps. If Φ is strictly convex, ρ converges to the global minimizer (which is χ {Φ(x) C} for some C) exponentially fast in W 2 distance. χ Ωt t equilibrium profile χ {Φ(x) C}
Many open questions remain. The ultimate goal would be to try to generalize the continuity equation with L constraint, possibly characterized as the singular limit of the porous medium equation with drift ρ t (ρ m ) + ( vρ) = 0. as m.
Part II: Aggregation with Height constraint Next we discuss a different but relevant problem, which can be formally viewed as the singular limit of Patlak-Keller-Segel (PKS) equation: ρ t (ρ m ) (ρ (ρ N )) = 0. This is joint work with Katy Craig (UCLA) and Yao Yao (UW Madison).
Interaction energy Let ρ (, t) be the gradient flow of { Ẽ [ρ] = R ρ(ρ N )dx for ρ 2 1 + otherwise where N is the Newtonian potential. Uniqueness of the minimizer follows from certain convexity properties of Ẽ given by Carrillo, Lisini and Mainini (2012). The stability of the JKO scheme is much weaker due to the weak convexity properties of the energy Ẽ.
Open questions: Does the discrete-time solutions converge to a gradient flow solution in a stable way? Can one characterize the corresponding gradient flow solution? As t, does ρ eventually converge to the characteristic function of a ball? yes if n=2 Does the solutions of (PKS) converge to ρ as m? Open
Convexity property of E To show the convergence of discrete-time scheme with E, we recall the notion of ω-convexity: Definition (Carrillo-Lisini-Mainini) E is called ω- convex if for ρ 1, ρ 2 P 2,ac (R d ) we have E(ρ t ) (1 t)e(ρ 0 ) + te(ρ 1 ) +C[(1 t)ω(t 2 W 2 2 (ρ 0, ρ 1 )) + tω((1 t) 2 W 2 2 (ρ 0, ρ 1 ))], where ω(x) = x ln x for small x.
Contraction Inequality for ω-convex energy We then have the following contraction inequality. Theorem ( K. Craig) Ẽ is ω-convex. Suppose E is ω-convex, then the corresponding solutions of the JKO scheme satisfies f τ (W 2 (µ τ, ν τ )) W 2 2 (µ 0, ν 0 ) + Cτ 2 ln τ, where f τ (x) = x cτω(x). Based on above inequality, one can follow the argument of Crandall-Liggett to obtain a recursive inequality to estimate W 2 (µ τ, ν h ), and thus to show that JKO scheme converges to a unique limit ρ in Wasserstein distance.
Characterization of the gradient flow solution Next we consider characterizing ρ with a free boundary problem. The corresponding approximating energy is ˆ 1 Ẽ m (ρ) := m ρm + ρn ρ, But then we realize that the corresponding gradient flow of above energy is hard to analyze due to the lack of convexity properties of Ẽ m. The main difficulty lies in the lack of L -bound for the discrete-time gradient flow solutions. In fact it is open whether the discrete-time solutions converges to the continuum solutions of (PKS) in spite of their formal connection.
Characterization of the gradient flow solution Hence we will use instead the following energy ˆ 1 E m (ρ) := m ρm + ρn ρ, which is ω-convex, and show that the corresponding gradient flow solution ρ m converges.
aggregation with density constraint: preliminary results As before, we only consider the patch case, i.e. when ρ 0 = χ Ω0. Here the corresponding free boundary problem is: (FB) p = Φ = 1 in {p(, t) > 0} =: Ω(t); V = Dp ν Φ on Ω(t), where Φ = χ Ω(t) N. Theorem (Craig-K-Yao) ρ m converges to ρ = χ Ω(t) where Ω(t) solves (FB).
Long time behavior of ρ Using the free boundary formulation for ρ, we can show the following: Theorem (Craig-K-Yao) When n = 2, Ω t converges to a ball as t. We prove this by showing that the second moment ρ (x, t) x 2 dx decreases in time unless ρ = χ B.
Computing the second moment The evolution of M 2 [ρ(t)] is given by ˆ ˆ d dt M 2[ρ(t)] = ρ (ρ N ) xdx ρ p xdx R 2 R 2 = 1 ˆ ˆ ˆ ( x y) x ρ(x)ρ(y) 2π R 2 R 2 x y 2 dydx = 1 ˆ ˆ ˆ ρ(x)ρ(y)dydx + 2 p(x)dx 4π R 2 R 2 Ω(t) = 1 ˆ 4π Ω(t) 2 + 2 p(x)dx. Ω(t) The quantity above is negative unless Ω(t) is a ball due to [Talenti, 1976]. Ω(t) p xdx
References D. Alexander, I. Kim and Y. Yao, Quasi-static evolution and crowded transport, Nonlinearity (2014), Vol. 27 no. 4, 823-858. J. A.Carrillo, S. Lisini, and E. Mainini. Uniqueness for Keller-Segel-type chemotaxis models. Discrete Contin. Dyn. Syst., 34(4):13191338, 2014. B. Maury, A. Roudneff-Chupin, F. Santambrogio and J. Venel, Handling Congestion in Crowd Motion Modeling. Networks and Heterogeneous Media (2011), Vol 6, no. 3, pp. 485 519. B. Maury, A. Roudneff-Chupin and F. Santambrogio, A macroscopic Crowd Motion Model of the gradient-flow type, M3AS (2010), Vol 20 no. 10, pp. 1787 1821. Thank you for your attention!