From slow diffusion to a hard height constraint: characterizing congested aggregation

Transcription

1 3 2 1 Time Position From slow iffusion to a har height constraint: characterizing congeste aggregation Katy Craig University of California, Santa Barbara BIRS April 12, 2017

2 2 collective ynamics biological chemotaxis (a colony of slime mol)

3 3 plan collective ynamics optimal transport an Wasserstein graient flow ω-convexity an height constraine aggregation future work

4 4 plan collective ynamics optimal transport an Wasserstein graient flow ω-convexity an height constraine aggregation future work

5 motivation ρ(x,t): Rᵈ R [0, + ) nonnegative ensity mass is conserve ρ(x) x = 1 aggregation equation with egenerate iffusion: = r ((rk ) )+ t m {z } self attraction for K(x) :R! R an m 1 {z} egenerate iffusion interaction kernels: granular meia: K(x) = x 3 K = -1 swarming: K(x) = x ᵃ/a ( - x ᵇ/b, -<b<a 1 chemotaxis: K(x) = 2 log x if =2, C x 2 otherwise. egenerate iffusion: m = r (m m 1 {z } D r ) 5

6 6 collective ynamics: mathematics Aggregation equation with egenerate iffusion: t + r (( rk ) ) = m for K(x) :R! R an m 1 Mathematical interest: Nonlinear Nonlocal Competing effects of attraction/repulsion Rich structure of equilibria [Kolokolnikov, Sun, Uminsky, Bertozzi, 2011]

7 7 collective ynamics: main questions Aggregation equation with egenerate iffusion: t + r (( rk ) ) = m for K(x) :R! R an m 1 Main questions: 1. Do solutions exist? 2. Are they unique? stable? 3. How o they behave in the long time limit? 4. How can we simulate them numerically? Key tool: optimal transport

8 8 plan collective ynamics optimal transport an Wasserstein graient flow ω-convexity an height constraine aggregation future work

9 Wasserstein metric Given two probability measures µ an on R, t : R! R transports µ onto if (B) =µ(t 1 (B)). Write this as t#µ =. The Wasserstein istance between μ an ν P2(Rᵈ) is {z} {z} For simplicity of notation, μ, ν Lᵈ effort to rearrange μ to look like ν, using t(x) t sens μ to ν 9

10 geoesics Not just a metric space a geoesic metric space: there is a constant spee geoesic :[0, 1]! P 2 (R ) connecting any μ an ν. (0) = µ, (1) =, W 2 ( (t), (s)) = t s W 2 (µ, ) Monge Kantorovich µ Wasserstein geoesic (t) µ linear interpolation (1 t)µ + t [Payré, Papaakis, Ouet 2013] 10

11 convexity Since the Wasserstein metric has geoesics, it has a notion of convexity. Recall: in Eucliean space, E: Rᵈ R is convex Eucliean geoesic z } { enpoints D 2 E 0 ()E((1 t)x + ty) apple (1 t)e(x)+te(y) λ-convex D 2 E λ I ()E((1 t)x + ty) apple (1 t)e(x)+te(y) t(1 t) 2 x y 2 Likewise, in the Wasserstein metric, E: P2(Rᵈ) R is λ-convex if E( (t)) apple (1 t)e(µ)+te( ) t(1 t) 2 W 2 2 (µ, ) Wasserstein geoesic enpoints 11

12 graient flow How oes this relate to PDE? Wasserstein graient flow. Informally, a curve x(t):r X is the graient flow of an energy E: X R if t x(t) = r XE(x(t)) x(t) evolves in the irection of steepest escent of E Examples: metric energy functional graient flow Z (L 2 (R ), k k L 2) E(f) =1 rf 2 2 t f = f Z (P 2 (R ),W 2 ) E( ) = log t = E( ) = 1 Z m m 1 t = m 12

13 graient flow ρ(t): R P2(Rᵈ) is the Wasserstein graient flow of energy E: P2(Rᵈ) R if t (t) = r W 2 E( (t)) More precisely, ρ(t) is the graient flow of E if there exists v(t) 2 L 2 loc((0, +1),L 2 ( (t))) so that (x, t)+r (v(x, t) (x, t)) = 0 t for a.e. t>0, If E an ρ are nice, v(t) (t)) ) if as! µ, E( ) = E( flow can be characterize as solutions to a PDE. Z ξ (ν - ρ) z } { h, t iiµ + o(w2(, )), an solutions of the graient 13

14 14 collective ynamics: main questions Aggregation equation with egenerate iffusion: t + r (( rk ) ) = m for K(x) :R! R an m 1 ZZ E(µ) = K(x y)µ(x)µ(y) + 1 Z µ(x) m x m 1 Main questions: 1. Do solutions exist? 2. Are they unique? stable? 3. How o they behave in the long time limit? 4. How can we simulate them numerically?

15 collective ynamics: main questions Aggregation equation with egenerate iffusion: If K(x) is λ-convex, λ 0, so is E(μ) [CDFLS, 2011]. But what about when K(x) isn t λ-convex? t + r (( rk ) ) = m for K(x) :R! R an m 1 ZZ E(µ) = K(x y)µ(x)µ(y) + 1 Z µ(x) m x m 1 Theorem (Ambrosio, Gigli, Savaré 2005): If the energy is λ-convex, 1. Do solutions exist? Yes (JKO) 2. Are they unique? Yes stable? contract (λ>0)/expan (λ 0) exponentially 3. How o they behave in the long time limit? For λ>0, there is a unique steay state, which solutions approach exponentially quickly. 4. How can we simulate them numerically? 15

16 16 collective ynamics: applications Aggregation equation with egenerate iffusion: t + r (( rk ) ) = m for K(x) :R! R an m 1 Applie interest: Slime mol (chemotaxis): K(x) = ( 1 2 log x if =2, C x 2 otherwise. Swarming: K(x) = x a /a x b /b, <b<a not λ-convex Granular meia: K(x) = x 3 merely 0-convex

17 17 plan collective ynamics optimal transport an Wasserstein graient flow ω-convexity an height constraine aggregation future work

18 height constraine aggregation a new moel (C., Kim, Yao 2016): inspire by the aggregation equation with egenerate iffusion, we consier a height constraine aggregation equation, for K = Δ ¹ = r ((rk ) )+ t m ( t Both moels have self-attraction from K ρ. = r (r(k ) ) if < 1 apple 1 always The role of repulsion is playe by har height constraint instea of egenerate iffusion. Heuristically, har height constraint is singular limit of egenerate iffusion: ( +1 if > 1 Iea: m = r (m m 1 {z } D r ), so as m +, D! 0 if < 1 18

19 19 height constraine aggregation ( t = r (r(k ) ) if < 1 apple 1 always Har height constraint appeare in previous work by [Maury, Rouneff- Chupin, Santambrogio 2010] instea of K*ρ(x) ha V(x). Has a (formal) Wasserstein graient flow structure: equation energy = r ((rk ) )+ t m E(µ) = ZZ K(x y)µ(x)µ(y) + 1 m 1 Z µ(x) m x ( t = r ((rk ) ) if < 1 apple 1 always E 1 (µ) = (RR K(x y)µ(x)µ(y) if kµkl 1 apple 1 +1 otherwise Since K(x) is not λ-convex, E falls outsie the scope of the existing theory.

20 ω-convexity Even though we on t have E 1 ( (t)) apple (1 t)e 1 (µ)+te 1 ( ) 2 t(1 t)w 2 2 (µ, ) λ-convexity E oes satisfy a similar inequality for a moulus of convexity ω(x) = x log(x). E 1 ( (t)) apple (1 t)e 1 (µ)+te 1 ( ) 2 (1 t)! t 2 W 2 2 (µ, ) + t! (1 t) 2 W 2 2 (µ, ) [Carrillo, McCann, Villani, 2006] [Ambrosio, Serfaty, 2008] [Carrillo, Lisini, Mainini, 2014] ω-convexity Inequalities coincie for ω(x) = x; ω-convexity generalizes λ-convexity. Sufficient conition: above the tangent line inequality E(µ 1 ) E(µ 0 ) E(µ ) =0 2!(W 2 2 (µ 0,µ 1 )) 20

21 21 collective ynamics: main questions Aggregation equation with egenerate iffusion: t + r (( rk ) ) = m for K(x) :R! R an m 1 ZZ E(µ) = K(x y)µ(x)µ(y) + 1 Z µ(x) m x m 1 Main questions: 1. Do solutions exist? 2. Are they unique? stable? 3. How o they behave in the long time limit? 4. How can we simulate them numerically?

22 22 In general, for ω(x) satisfying Osgoo s conition, i.e. collective ynamics: main x questions Aggregation equation with we obtain egenerate the stability iffusion: estimate t + r (( rk ) ) F = 2t (W m 2 2 ( 1 (t), 2 (t))) apple W for K(x) :R 2 2 ( 1 (0), 2 (0))! R an m 1 ZZ E(µ) = K(x y)µ(x)µ(y) + 1 Z t F t(x) =!(F t (x)) from which we recover µ(x) m x m[ags, ] & [CMV, 2006]. Theorem (C. 2016): If the energy is ω-convex, ω(x) = x log(x), 1. Do solutions exist? Yes (JKO) 2. Are they unique? Yes stable? expan at most ouble-exponentially Z 1 0!(x) =+1 W 2 2 ( 1 (t), 2 (t)) apple W 2 2 ( 1 (0), 2 (0)) e2 t

23 ω-convexity: applications ( t = r (r(k ) ) if < 1 apple 1 always Slime mol singular limit: K(x) = Slime mol (chemotaxis): = r ((rk ) )+ t m K(x) = ( 1 2 ( 1 2 log x if =2 C x 2 otherwise log x if =2 C x 2 otherwise ω-convex ω-convex on boune measures Swarming: K(x) = x a /a x b /b, <b<a ω-convex on L p measures for 2- b<a Granular meia: K(x) = x 3 ω-convex on measures with fixe center of mass an ω(x) = x 3/2 23

24 24 collective ynamics: main questions Aggregation equation with egenerate iffusion: t + r (( rk ) ) = m for K(x) :R! R an m 1 Main questions: 1. Do solutions exist? 2. Are they unique? stable? 3. How o they behave in the long time limit? epens on choice of K(x) 4. How can we simulate them numerically?

25 25 long time behavior: K = Δ - ¹ For K = Δ - ¹ an 1 m +, long time behavior of Keller-Segel equation has been the subject of recent interest. Supercritical power (m 2-2/): Profiles of steay states known for certain of m; solutions can blow up to a Dirac mass in finite time or remain boune. [Sugiyama 2006, 2007], [Luckhaus an Sugiyama 2006, 2007], [Blanchet, Carlen, Carrillo 2012], [Chen, Liu, Wang 2012] Subcritical power (m > 2-2/): All steay states are raially symmetric an ecreasing; still, convergence to equilibrium is only known in =1, 2 an for raial solutions in higher imensions. [Carrillo, Hitter, Volzone, Yao 2016], [Kim, Yao 2012]

26 long time behavior: K = Δ - ¹, m =+ In the case of the height constraine aggregation equation, we obtaine quantitative rates of convergence to equilibrium for patch solutions: Theorem (C., Kim, Yao 2016): Suppose ρ(x,t) solves congeste aggregation eqn with ρ(x,0) = 1Ω(0)(x). Then, in two imensions, an (x, t) Lp! 1 B (x) for all 1 apple p<+1 E 1 ( (,t)) E 1 (1 B ) apple C (0) t 1/6 In any imension, the Riesz Rearrangement Inequality guarantees that the unique minimizer of E is 1B(x). The tricky part is showing mass of ρ(x,t) oesn t escape to +. To o this, we characterize the ynamics of patch solutions in terms of a free bounary problem an control M₂(ρ(t)) by Talenti inequality (=2). 26

27 27 collective ynamics: main questions Aggregation equation with egenerate iffusion: t + r (( rk ) ) = m for K(x) :R! R an m 1 Main questions: 1. Do solutions exist? 2. Are they unique? stable? 3. How o they behave in the long time limit? 4. How can we simulate them numerically?

28 numerics For nice velocity fiels an = 1 P N, N i=1 x i (t) (x, t)+r (v(x, t) (x, t)) = 0 t t x i(t) =v(x i (t),t), 8i =1,...,N For any ρ(x), there exist x1,, xn so that W 2, 1 N P N i=1 x i N!+1! 0 General Numerical Strategy: to approximate a solution ρ(x,t) of a PDE P 1 N N i=1 x i 1) Approximate ρ(x,0) by ρn(x,0)=. 2) Compute the solution with initial ata ρn by numerically solving the corresponing system of ODEs. 3) Use stability of PDE to conclue that the numerical solution ρn(x,t) must be close to ρ(x,t) on boune time intervals. What about when v(x,t) is not nice? 28

29 29 numerics None of the v(x,t) mentione so far are nice! We nee to make them nice. Aggregation equation without iffusion: Regularize K by convolution with a mollifier ( blob ) = r ((rk ) ) t Theorem [C., Bertozzi 2014]: If you remove the mollification as you a particles, the particle blob metho converges.

30 numerics Aggregation equation w/ eg. iffusion: Regularize both K an v by convolution m = r (m m 1 r ) =r ((m m 2 r ) {z } v = r ((rk ) )+ t m Theorem [Carrillo, C., Patacchini (in progress)]: If you remove the mollification as you a particles, the particle blob metho Γ-converges. ) Newtonian attraction (K = Δ - ¹) an m=2 an m=100 iffusion 30

31 31 future work: Does Keller-Segel converge to congeste aggregation? = r ((rk ) )+ t m m + - For V(x) convex, [Alexaner, Kim, Yao 2014] showe t = r ((rv ) )+ m m + - Connecting Keller-Segel an the congeste aggregation eqn woul lea to greater insight in long-time behavior of supercritical (m>2-2/) Keller- Segel. Further examples of ω-convex energies? More applications with a height constraint? ( t = r (r(k ) ) if < 1 apple 1 always ( t = r ((rv ) ) if < 1 apple 1 always

32 Thank you!

33 33 motivation for free bounary problem How oes congeste aggregation equation relate to free bounary problem? ( t = r (r(k ) ) if < 1 apple 1 always Consier patch solutions. For a omain Ω, suppose that ρ(x,t) is a solution with initial ata (x, 0) = ( 1 if x 2, 0 otherwise. Since K= Δ - ¹, K ρ causes self-attraction. Thus, we expect ρ(x,t) to remain a characteristic function. Let Ω(t)={ρ=1} be congeste region, so ρ(x,t)=1ω(t)(x). What free bounary problem escribes evolution of Ω(t)?

34 34 formal erivation Here is a formal erivation of the relate free bounary problem. Suppose ρ(x,t) solves ( t = r (r(k ) ) if < 1 apple 1 always Since mass is conserve, we expect ρ(x,t) satisfies a continuity equation t = r ((rk + rp) {z } ) v where p(x,t) is the pressure arising from the height constraint. Height constraint is active on the congeste region {p>0} = Ω(t). Height constraint is inactive outsie the congeste region {p=0}= Ω(t) c.

35 35 formal erivation Given = r ((rk + rp) t {z } ) v what happens on congeste region? Because of har height constraint, on the congeste region Ω(t)={ρ=1}, the velocity fiel is incompressible, v=0. Since K= Δ - ¹, r v = K + p = + p, so incompressibility means p = on (t) ={ =1} Using that the height constraint is active on the congeste region, Ω(t)={p>0}, we obtain the following equation for the pressure: p = 1 on {p > 0}

36 36 formal erivation Given By conservation of mass, what about by of congeste region? Using that ρ(x,t) solves the above continuity equation, this equals = Z = r ((rk + rp) t {z } ) v (t) 0= t Z (t) = r ((rk + rp) )+ Z Z (t) outwar normal velocity of Ω(t) t + V = Using that ρ(x,t)=1ω(t)(x), for Ω(t)={p>0}, we again obtain an equation for p, Z (t) V (@ K p + V K 1 {p>0} p + V = 0 > 0}

37 free bounary problem Combining the observations that on the congeste region, p = 1 on {p > 0} an on the bounary of the congeste region, outwar normal velocity of K 1 {p>0} p + V = 0 > 0} Theorem (C., Kim, Yao 2016): Suppose ρ(x,t) solves congeste aggregation eqn with ρ(x,0) = 1Ω(0)(x). Then ρ(x,t)=1ω(t)(x), for Ω(t) = {p(x,t)>0}, where p a viscosity solution of ( p = 1 on {p > 0} V K 1 p > 0}. 37

38 38 collective ynamics (x, t) :R R! [0, +1) nonnegative ensity Mass is conserve (assume ρ(x) x = 1), an ρ(x,t) evolves accoring to a continuity equation: (x, t)+r (v(x, t) (x, t)) = 0 t Particle approximation: Suppose = 1 N P N i=1 x i (t) For nice velocity fiels, ρ(x,t) solves the continuity equation iff t x i(t) =v(x i (t),t), 8i =1,...,N

39 collective ynamics: slime mol In the case of the slime mol, we have 1) self-attraction an 2) iffusion. 1) Self-Attraction At the particle level, we may formulate self-attraction as t x i(t) = 1 N NX j=1 rk(x i (t) x j (t)) K(x) = ( 1 2 log x if =2, C x 2 otherwise. Since = 1 P N N i=1 x i (t), we write the resulting velocity fiel as 1 NX Z rk(x x j (t)) = rk(x y) (y) = rk (x) N j=1 39

40 collective ynamics: slime mol In the case of the slime mol, we have 1) self-attraction an 2) iffusion. 2) Diffusion Combining self-attraction with iffusion gives the Keller-Segel equation + r (( rk ) ) = t More generally, we can consier egenerate iffusion for m 1 + r (( rk ) ) = m t m = r (m m 1 {z } D r ) 40