Systems of nonlocal interaction PDEs with and without diffusion: - - PowerPoint PPT Presentation

Systems of nonlocal interaction PDEs with and without diffusion: consensus vs. segregation

Marco Di Francesco

University of L’Aquila

Nonlocal Nonlinear Partial Differential Equations and Applications, Anacapri, Sep 14-18 2015

• M. Di Francesco (L’Aquila)

Nonlocal interaction equations Anacapri 2015 1 / 47

Nonlocal interactions

Table of contents

1

Nonlocal interactions

2

Gradient flow structure

3

Systems with many species

4

A predator-prey model

5

The case with quadratic diffusion

6

Conclusions

• M. Di Francesco (L’Aquila)

Nonlocal interaction equations Anacapri 2015 2 / 47

Nonlocal interactions

A discrete particle system

N particles, located at X1(t), . . . , XN(t) ∈ Rd with masses m1, . . . , mN. Subject to binary interaction forces depending on their position. Friction dominated regime: no inertia. Deterministic (no stochastic effects).

Figure: N interacting particles. In green: the forces exerted by X3 on X7

and viceversa.

dXj(t) dt = −

• k=j

mk∇G(Xj(t) − Xk(t)), j = 1, . . . , N. (1)

• M. Di Francesco (L’Aquila)

Nonlocal interaction equations Anacapri 2015 3 / 47

Nonlocal interactions

Interaction potentials

All the interactions are ruled by the interaction potential G. Typical assumptions for the interaction potential G G ∈ C(Rd), with G(0) = 0, Radial symmetry G(x) = g(|x|), Notation: g increasing ⇒ G attractive, g decreasing ⇒ G repulsive. Main motivation: population dynamics Animal swarming: Okubo (1980), Oelschl¨ ager (1989), Morale, Capasso, and Oelschl¨ ager (1998), Mogilner, Edelstein-Keshet (1999), Topaz, Bertozzi, and Lewis (2006).

• M. Di Francesco (L’Aquila)

Nonlocal interaction equations Anacapri 2015 4 / 47

Nonlocal interactions

Typical interaction potentials

Attractive Morse potentials G(x) = −Cae−|x|/La Repulsive Morse potentials G(x) = Cre−|x|/Lr Attractive-repulsive Morse potentials G(x) = −Cae−|x|/La + Cre−|x|/Lr Combination of Gaussian potentials G(x) = −Cae−|x|2/L2

a + Cre−|x|2/L2 r

• M. Di Francesco (L’Aquila)

Nonlocal interaction equations Anacapri 2015 5 / 47

Nonlocal interactions

Hydrodynamic N → +∞ limit

Empirical measure: µN(t) = N

• j=1

mj −1

N

• k=1

mkδXk (t) Formal N → +∞ limit of µN ∂µ ∂t = div(µ∇G ∗ µ) Other related applications Particle physics (vortex dynamics, Vlasov equation, hydrodynamics) Granular media Cell motion and chemotaxis Fractional porous medium equation Opinion formation Crowd movements

• M. Di Francesco (L’Aquila)

Nonlocal interaction equations Anacapri 2015 6 / 47

Gradient flow structure

Table of contents

1

Nonlocal interactions

2

Gradient flow structure

3

Systems with many species

4

A predator-prey model

5

The case with quadratic diffusion

6

Conclusions

• M. Di Francesco (L’Aquila)

Nonlocal interaction equations Anacapri 2015 7 / 47

Gradient flow structure

Finite dimensional gradient flows

Let F ∈ C 1(Rm; R), a curve [0, +∞) ∋ t → X(t) ∈ Rm is a gradient flow of F if ˙ X(t) = −∇F(X(t)). Energy dissipation: d dt F(X(t)) = −|∇F(X(t))|2 Implicit Euler variational derivation: time step τ > 0, Xτ(t) = X n

τ for

t ∈ ((n − 1)τ, nτ], with X n

τ = argmin{ 1

2τ |X − X n

τ |2 + F(X), X ∈ Rm}

D2F ≥ λI implies stability d dt |X1(t) − X2(t)|2 = −2 < X1(t) − X2(t), ∇F(X1(t)) − ∇F(X1(t)) > ≤ −2λ|X1(t) − X2(t)|2.

• M. Di Francesco (L’Aquila)

Nonlocal interaction equations Anacapri 2015 8 / 47

Gradient flow structure

Gradient flow structure of the ODE particle system

Consider dXj(t) dt = −

• k=j

mk∇G(Xj(t) − Xk(t)), j = 1, . . . , N. with G(−x) = G(x) and G ∈ C 2(Rd). Weighted metric structure Denote m = (m1, . . . , mN). For X, Y ∈ RdN, let < X, Y >L2

m:=

N

• j=1

mjXjYj, X2

L2

m =< X, X >L2 m .

Frech´ et differential Let F ∈ C 1(RdN). The linear operator gradXF[X] is defined by lim

ǫ→0

F[X + ǫY] − F[X] ǫ =:< gradXF[X], Y >L2

m=

N

• j=1

mj∇Xj F[X] · Yj.

• M. Di Francesco (L’Aquila)

Nonlocal interaction equations Anacapri 2015 9 / 47

Gradient flow structure

Gradient flow structure of the ODE particle system

Energy functional Let X := (X1, . . . , XN)T. G[X] := 1 2

• i,j

mimjG(Xi − Xj) Then ˙ X(t) = −gradXG[X(t)]. (2) Regularity and collisions The above makes sense if G is C 1, G ∈ C 2(Rd): no collisions, The above structure can be extended if G is pointy (e. g. attractive Morse potential) and λ-convex. Collapse in finite time occurs for a class of Non-Osgood potentials including attractive Morse.

• M. Di Francesco (L’Aquila)

Nonlocal interaction equations Anacapri 2015 10 / 47

Gradient flow structure

Finite time collapse for attractive potentials

Assume G satisfies G(x) = g(|x|), g ′(r) > 0 for r > 0, g ′(r) r non-increasing. (3) Proposition (Finite time collapse) Let X1, . . . , XN solve ˙ Xj(t) = −

• Xk (t)=Xj (t)

mk∇G(Xj(t) − Xk(t). Then, Xj(t) = δCm for all t ≥ t∗ for some t∗, iff ε 1 g ′(z)dz < +∞ for some ε > 0. (4) Important remarks t∗ depends on the maximal initial distance of the particles from Cm t∗ does not depend on N.

• M. Di Francesco (L’Aquila)

Nonlocal interaction equations Anacapri 2015 11 / 47

Gradient flow structure

Ingredients for the continuum theory

The measure space µ ∈ P2(Rd) :=

• µ ∈ P(Rd),
• |x|2dµ(x) < +∞
• The Wasserstein distance

d2(µ, ν)2 = inf

• Rd ×Rd |x − y|2dγ(x, y), : γ ∈ Γ(µ, ν)
• Γ(µ, ν) =
• γ ∈ P(Rd × Rd) : µ and µ are the marginals of γ
• The functional

G[µ] = 1 2

• Rd ×Rd G(x − y)dµ(x)dµ(y)

Wasserstein gradient flow ∂µ(t) ∂t + div(µ(t)v(t)) = 0, in D′(Rd × [0, +∞)) v(t) = −∂0G ∗ µ(t) = −

• x=y

∇G(x − y)dµ(y, t).

• M. Di Francesco (L’Aquila)

Nonlocal interaction equations Anacapri 2015 12 / 47

Gradient flow structure

The JKO scheme1

Time discretization: τ > 0 fixed time step. Let µ0 ∈ P(Rd) be fixed. For a given µτ

n ∈ P(Rd), we define the sequence µτ n+1 as

µn+1

τ

∈ argminµ∈P2(Rd ) 1 2τ d2

2 (µn τ, µ) + G [µ]

• .

Define ¯ µτ as the piecewise constant interpolation of µn

τ.

¯ µτ converges to a unique curve of measures µ as τ ց 0, µ is the unique gradient flow solution to the PDE, Stability property d2(µ1(t), µ2(t)) ≤ e|λ|td2(µ0

1, µ0 2),

(5) The property (5) allows to extend the final collapse to all measure solutions by atomization (CDFLS 2011).

1Jordan, Kinderlehrer, Otto - 1998

• M. Di Francesco (L’Aquila)

Nonlocal interaction equations Anacapri 2015 13 / 47

Systems with many species

Table of contents

1

Nonlocal interactions

2

Gradient flow structure

3

Systems with many species

4

A predator-prey model

5

The case with quadratic diffusion

6

Conclusions

• M. Di Francesco (L’Aquila)

Nonlocal interaction equations Anacapri 2015 14 / 47

Systems with many species

A two species model2

X1, . . . , XN particles of the first species with masses n1, . . . , nN, Y1, . . . , YM are particles of the second species with masses m1, . . . , mM. Particle system: ˙ Xi(t) = −

Xi =Xk nk∇K11(Xi(t) − Xk(t)) − Xi =Yk mk∇K12(Xi(t) − Yk(t))

˙ Yj(t) = −

Yj =Yk mk∇K22(Yj(t) − Yk(t)) − Yj =Xk nk∇K21(Yj(t) − Xk(t))

. Continuum version: ∂tµ1 = div (µ1∇K11 ∗ µ1 + µ1∇K12 ∗ µ2) ∂tµ2 = div (µ2∇K22 ∗ µ2 + µ2∇K21 ∗ µ1) . Motivation Pedestrian movements, lane formation [Degond et al. 2011, Colombo et al. 2012] Opinion formation [Josek - 2009], [D¨ uring et al. 2009] Two species chemotaxis [Espejo et al. 2009] Predator–Prey type interaction

2[DF, Fagioli - Nonlinearity 2013]

• M. Di Francesco (L’Aquila)

Nonlocal interaction equations Anacapri 2015 15 / 47

Systems with many species

Symmetrizable case

• ∂tµ1 = div (µ1∇K11 ∗ µ1 + µ1∇K12 ∗ µ2)

∂tµ2 = div (µ2∇K22 ∗ µ2 + µ2∇K21 ∗ µ1) , with K21 = αK12, α > 0. (6) System (6) has a gradient flow structure, with functional F(µ1, µ2) = 1 2

• Rd K11 ∗ µ1dµ1 + 1

2α

• Rd K22 ∗ µ2dµ2 +
• Rd K12 ∗ µ2dµ1.

As a byproduct, due to the property K21 = αK12, one can prove the following energy identity d dt F(µ1(t), µ2(t)) = −

• |∇K11 ∗ µ1 + ∇K12 ∗ µ2|2 dµ1(x)

− α

• 1

α∇K22 ∗ µ2 + ∇K12 ∗ µ1

• 2

dµ2(x) ≤ 0

• M. Di Francesco (L’Aquila)

Nonlocal interaction equations Anacapri 2015 16 / 47

Systems with many species

Results in the symmetrizable case

Main assumption All the kernels Kij are mildly singular and λij–convex. Preserved quantity: cM,α := α

• xdµ1(x) +
• xdµ2(x).

Metric product structure µ = (µ1, µ2) ∈ P2(Rd) × P2(Rd), W2

2,α(µ, ν) = W 2 2 (µ1, ν1) + 1

αW 2

2 (µ2, ν2).

Results λ convexity of F(µ1, µ2) on a suitable sub-differential structure. Existence, uniqueness, and stability of gradient flow solutions, by suitably generalizing the one-species theory. Finite time collapse if all the kernels are of Non–Osgood type. Intermediate clustering of each species if the cross interaction kernel decays at infinity.

• M. Di Francesco (L’Aquila)

Nonlocal interaction equations Anacapri 2015 17 / 47

Systems with many species

Intermediate clustering

Figure: Evolution of two initially separated species, intermediate clustering at time t1, total collapse at time t2.

• M. Di Francesco (L’Aquila)

Nonlocal interaction equations Anacapri 2015 18 / 47

Systems with many species

General case: a semi-implicit JKO scheme

Assume now no relation at all between K12 and K21. Typical example: opinion dynamics with opinion leaders. Two species of individuals: µ1 opinion leaders, and µ2 followers. Both species have internal dynamics through their self interaction kernels K11 and K22. The followers are affected by the opinion of the opinion leaders via the kernel K21. The opinion leaders are affected by the opinion of the followers via K12, which has much smaller amplitude of K21, and is not necessarily related with K12. Difficulty: No gradient flow structure, no variational formulation, no functional. How to deal with singular potentials? Main idea: semi-implicit version of the JKO scheme.

• M. Di Francesco (L’Aquila)

Nonlocal interaction equations Anacapri 2015 19 / 47

Systems with many species

Semi implicit JKO scheme: a toy model

Consider the ODE (not a gradient flow) ˙ X(t) = −∇F(X(t)) + W (X(t)). We consider a time discretization in which the non gradient term W is explicit: X n+1 − X n τ + ∇XF(X n+1) − W (X n) = 0, which can be set in variational form X n+1 = argminX 1 2τ |X − X n|2 + F(X) + X · W (X n)

• .

Hence, at each step n we are performing a steepest descent for a relative functional ˜ F(X|Y ) := F(X) + X · W (Y ) X n+1 = argminX 1 2τ |X − X n|2 + ˜ F(X|X n)

• .
• M. Di Francesco (L’Aquila)

Nonlocal interaction equations Anacapri 2015 20 / 47

Systems with many species

Semi implicit JKO scheme: relative energy

We see our system as follows

• ∂tµ1 = div (µ1∇K11 ∗ µ1 + µ1∇K12 ∗ ν2)

∂tµ2 = div (µ2∇K22 ∗ µ2 + µ2∇K21 ∗ ν1) . Relative interaction energy For all pairs µ = (µ1, µ2), ν = (ν1, ν2) ∈ P(Rd)2 we set F[µ|ν] = 1 2

• Rd K11 ∗ µ1dµ1 +
• Rd K12 ∗ ν2dµ1 + 1

2

• Rd K22 ∗ µ2dµ2 +
• Rd K21 ∗ ν1dµ2.

Let τ > 0 be a fixed time step, and let µ0 = (µ0,1, µ0,2) ∈ P(Rd)2 be a fixed initial pair

• f probability measures. For a given µτ

n ∈ P(Rd)2, we define the sequence µτ n+1 as

µn+1

τ

∈ argminµ∈P2(Rd )×P2(Rd ) 1 2τ W2

2(µn τ, µ) + F [µ|µn τ]

• .
• M. Di Francesco (L’Aquila)

Nonlocal interaction equations Anacapri 2015 21 / 47

Systems with many species

General case: the results

Existence of weak measure solutions d dt

• φ(x)dµ1(x, t) = −1

2

• ∇H1(x − y) · (∇φ(x) − ∇φ(y))dµ1(x)dµ1(y)

−

• ∇K1(x − y) · ∇φ(x)dµ1(x)dµ2(y)

d dt

• ψ(x)dµ2(x, t) = −1

2

• ∇H2(x − y) · (∇ψ(x) − ∇ψ(y))dµ2(x)dµ2(y)

−

• ∇K2(x − y) · ψ(x)dµ2(x)dµ1(y).

as limit of the semi-implicit JKO scheme. Uniqueness in case Hj and Kj are W 2,∞, via a variant of the characteristics method.

• M. Di Francesco (L’Aquila)

Nonlocal interaction equations Anacapri 2015 22 / 47

Systems with many species

Interpretation: a dynamic game

Assume ν = (ν1, ν2) ∈ P(Rd)2 is given (a given strategy for both species). At the next (discrete) step the two species chose a new strategy µ as a best reply

• w. r. t. ν.

The payoff functional is P[µ|ν] = 1 2τ W2

2(µ, ν) + F [µ|ν]

F [µ|ν] = 1 2

• Rd K11 ∗ µ1dµ1 +
• Rd K12 ∗ ν2dµ1

+ 1 2

• Rd K22 ∗ µ2dµ2 +
• Rd K21 ∗ ν1dµ2.

Open problems: Existence of ESS? Large time behavior? Difficulty: no preserved quantity.

• M. Di Francesco (L’Aquila)

Nonlocal interaction equations Anacapri 2015 23 / 47

A predator-prey model

Table of contents

1

Nonlocal interactions

2

Gradient flow structure

3

Systems with many species

4

A predator-prey model

5

The case with quadratic diffusion

6

Conclusions

• M. Di Francesco (L’Aquila)

Nonlocal interaction equations Anacapri 2015 24 / 47

A predator-prey model

A swarming approach to predator-prey dynamics3

N-predator, M-prey: ˙ Xi(t) = − N

k=1 mk X∇S1(Xi(t) − Xk(t)) − M h=1 mh Y ∇K(Xi(t) − Yh(t)),

˙ Yj(t) = − M

h=1 mh Y ∇S2(Yj(t) − Yh(t)) + α M k=1 mk X∇K(Yj(t) − Xk(t)),

K is a given attractive potential. PDE version:

• ∂tµ1 = div(µ1(∇S1 ∗ µ1 + ∇K ∗ µ2))

∂tµ2 = div(µ2(∇S2 ∗ µ2 − α∇K ∗ µ1)), Results: Stability conditions on the kernels for particle steady states in the ODE system. Local nonlinear stability of steady states in the PDE model.

3DF - Fagioli, to appear

• M. Di Francesco (L’Aquila)

Nonlocal interaction equations Anacapri 2015 25 / 47

A predator-prey model

Numerical simulations in dimension two

Figure: Normalized Gaussian self-attractive potentials for both predators and prey.

• M. Di Francesco (L’Aquila)

Nonlocal interaction equations Anacapri 2015 26 / 47

A predator-prey model

Numerical simulations in dimension two

Figure: Normalized Gaussian self-attractive potentials predators and repulsive for prey.

• M. Di Francesco (L’Aquila)

Nonlocal interaction equations Anacapri 2015 27 / 47

A predator-prey model

Numerical simulations in dimension two

Figure: A realistic catching with cross-interaction given by K(x) = 1 − (|x| + 1)e−|x|.

• M. Di Francesco (L’Aquila)

Nonlocal interaction equations Anacapri 2015 28 / 47

The case with quadratic diffusion

Table of contents

1

Nonlocal interactions

2

Gradient flow structure

3

Systems with many species

4

A predator-prey model

5

The case with quadratic diffusion

6

Conclusions

• M. Di Francesco (L’Aquila)

Nonlocal interaction equations Anacapri 2015 29 / 47

The case with quadratic diffusion

The one species case

∂ρ ∂t = div(ρ∇(ǫρ − G ∗ ρ)). ǫ > 0 is a diffusion constant. G = G(x) is radially symmetric. The G-term is attractive. G ∈ C2 ∩ L1, G ≥ 0. The solution ρ(·, t) is absolutely continuous w.r.t. Lebesgue measure (due to the porous medium term). Wasserstein gradient flow of the energy functional F[ρ] = ǫ 2

• ρ2dx − 1

2

• ρG ∗ ρdx.

Competition between Diffusive effect, driving the solution toward a slow decay to zero. Nonlocal aggregation effect, leading to concentration towards the center of mass.

• M. Di Francesco (L’Aquila)

Nonlocal interaction equations Anacapri 2015 30 / 47

The case with quadratic diffusion

Why quadratic diffusion?

Consider the energy ǫ 2

• ρR ∗ ρdx − 1

2

• ρG ∗ ρdx,

with R a repulsive kernel, −G an attractive kernel, both R and G are ≥ 0, in L1, with unit mass. Scale R as Rγ(x) = γ−dR(xγ−1) , Rγ → δ0 as γ ց 0. In the γ ց 0 limit we obtain the energy F[ρ] = ǫ 2

• ρ2dx − 1

2

• ρG ∗ ρdx.

The energy F is the outcome of long range attraction and short range repulsion. Applications in population biology (see Okubo, Capasso et al.).

• M. Di Francesco (L’Aquila)

Nonlocal interaction equations Anacapri 2015 31 / 47

The case with quadratic diffusion

Stationary states

The existence of global minimisers for F and of non trivial (integrable) steady state for the equation depend on the parameters in the problem, see Burger, DF, Franek (2013) and Bedrossian (2012). If ǫ ≥ GL1, then the only integrable steady state (and the global minimiser for F) is ρ = 0. If ǫ < GL1, then there exists a non trivial L1 global minimiser for F. In one space dimension, provided supp(G) = R and G is strictly decreasing on [0, +∞), a unique steady state ρ∞ exists for fixed mass and center of mass. ρ∞ satisfies (see Burger, DF, Frnek 2013) ρ∞ is compactly supported and symmetric, ρ∞ is C 1 on its support, ρ∞ has a unique stationary point and global maximum. Proof: relies on Krein-Rutman theorem.

• M. Di Francesco (L’Aquila)

Nonlocal interaction equations Anacapri 2015 32 / 47

The case with quadratic diffusion

The two species case

Consider two species ρ1 and ρ2, and the functional F[ρ1, ρ2] = ǫ 2

• (ρ1 + ρ2)2dx − 1

2

• ρ1S1 ∗ ρ1dx

− 1 2

• ρ2S2 ∗ ρ2dx −
• ρ1K ∗ ρ2dx.

The two species are subject to short range repulsion and long range attraction. The two species are indistinguishable in the way they diffuse (every individual is repelled by any other one in the same way). Nonlocal interactions characterise each species. We consider symmetric cross-interactions for simplicity. The kernels S1, S2, and K have the same assumptions of the one-species case.

• M. Di Francesco (L’Aquila)

Nonlocal interaction equations Anacapri 2015 33 / 47

The case with quadratic diffusion

The PDE system

The (formal) gradient flow of F is ∂tρ1 = div (ρ1∇(ǫ(ρ1 + ρ2) − S1 ∗ ρ1 − K ∗ ρ2)) ∂tρ2 = div (ρ2∇(ǫ(ρ1 + ρ2) − S2 ∗ ρ2 − K ∗ ρ1)) . System of nonlocal interaction equations with self-diffusion and cross-diffusion. The diffusion term is degenerate, i. e. it does not provide a smoothing effect for both species. In the context of reaction-diffusion systems with cross diffusion (see J¨ ungel 2014), the diffusion matrix is not semi-definite. Hence, no satisfactory theory for the time-dependent problem so far! The JKO scheme converges to a limit, but no characterisation can be done in terms of the PDE system (the diffusion term is not in Laplacian form, lack of compactness).

• M. Di Francesco (L’Aquila)

Nonlocal interaction equations Anacapri 2015 34 / 47

The case with quadratic diffusion

Asymptotic segregation

Numerical simulations suggest that this system produces segregated states for large times. The profile resemble Barenblatt profiles. ρ1 and ρ2 are discontinuous, hence no parabolic regularity can be expected.

• M. Di Francesco (L’Aquila)

Nonlocal interaction equations Anacapri 2015 35 / 47

The case with quadratic diffusion

Necessary conditions for steady states

The functional in Fourier variables becomes F[ρ1, ρ2] =

• (ˆ

ρ1(ξ), ˆ ρ2(ξ)) · A(ξ) · (ˆ ρ1(ξ), ˆ ρ2(ξ))Tdξ, with A(ξ) = 1

2(ǫ − ˆ

S1(ξ))

1 2(ǫ − ˆ

K(ξ))

1 2(ǫ − ˆ

K(ξ))

1 2(ǫ − ˆ

S2(ξ))

• .

The conditions ǫ > sup

ξ

min{ˆ S1(ξ), ˆ S2(ξ)} , (ǫ − ˆ S1(ξ))(ǫ − ˆ S2(ξ)) > (ǫ − ˆ K(ξ))2 imply that the global minimiser is zero. First condition: analogous to one species, diffusion dominates attraction. Second condition: less trivial. In case of Gaussian kernels, it is met if

KL1 > max{S1L1, S2L1}, the variance of K is much smaller than those of S1 and S2, cross-interaction dominating self-interaction at small distances.

• M. Di Francesco (L’Aquila)

Nonlocal interaction equations Anacapri 2015 36 / 47

The case with quadratic diffusion

Solution via Krein-Rutman theorem

Assuming supp(ρ1) = [−L1, L1] =: I1 supp(ρ2) = [−L2, −L1] ∪ [L1, L2] =: I2, the stationary system can be written as        x ∈ I1 ǫρ1(x) =

• I1

S1(x − y)ρ1(y)dy +

• I2

K(x − y)ρ2(y)dy + C1 x ∈ I2 ǫρ2(x) =

• I2

S2(x − y)ρ2(y)dy +

• I1

K(x − y)ρ1(y)dy + C2 Set ¯ w = ρ1(L1) = ρ2(L1), p = −ρ′

1, q = −ρ′ 2, we get

       x ∈ I1 ǫp(x) =

• I1

S1(x − y)p(y)dy +

• I2

K(x − y)q(y)dy + ¯ wA1(x) x ∈ I2 ǫq(x) =

• I2

S2(x − y)q(y)dy +

• I1

K(x − y)p(y)dy + ¯ wA2(x) , A1(x) = S1(x − L1) − S1(x + L1) + K(x + L1) − K(x − L1) A2(x) = K(x − L1) − K(x + L1) + S2(x + L1) − S2(x − L1)

• M. Di Francesco (L’Aquila)

Nonlocal interaction equations Anacapri 2015 37 / 47

The case with quadratic diffusion

Solution via Krein-Rutman theorem

Assuming All kernels are as in the one species case, S1(x − L1) − S1(x + L1) ≥ K(x − L1) − K(x + L1), S2(x − L1) − S2(x + L1) ≤ K(x − L1) − K(x + L1), S′

1(L1) < K ′(L1) and S′ 2(L1) < K ′(L1),

then The functional equation can be solved via Krein-Rutman theorem, The corresponding eigenvalue ǫ(L1, L2) is the diffusion constant, The result is partially unsatisfactory because there is no information about ǫ(L1, L2).

• M. Di Francesco (L’Aquila)

Nonlocal interaction equations Anacapri 2015 38 / 47

The case with quadratic diffusion

A result for small diffusion

Formulation in the pseudo-inverse variable:                        ǫ 2∂z

• (∂zu1(z))−2

=

• J1

S′

1 (u1(z) − u1(ζ)) dζ

+

• J2

K ′ (u1(z) − u2(ζ)) dζ, z ∈ [(1 − z1)/2, (1 + z1)/2] ǫ 2∂z

• (∂zu2(z))−2

=

• J2

S′

2 (u2(z) − u2(ζ)) dζ

+

• J1

K ′ (u2(z) − u1(ζ)) dζ, z ∈ [0, 1/2 − z1] ∪ [1/2 − z1, 1]. Perturbations of the δ state: u1(z) = δv1(z) , u2(z) = δv2(z) , δ = ǫ1/3.

• M. Di Francesco (L’Aquila)

Nonlocal interaction equations Anacapri 2015 39 / 47

The case with quadratic diffusion

Some manipulations

Multiply the first equation by δ∂Zv1 and the second one by δ∂zv2. Integrate over z. δ2 ∂zv1 =

• J1

S1 (δ(v1(z) − v1(ζ))) dζ +

• J2

K (δ(v1(z) − v2(ζ))) dζ + α1 δ2 ∂zv2 =

• J2

S2 (δ(v2(z) − v2(ζ))) dζ +

• J1

K (δ(v2(z) − v1(ζ))) dζ + α2. Impose ∂zv2(1) = +∞ and ∂zv1((1 + z1)/2) = ∂v2((1 + z1)/2). Multiply the first equation by δ∂Zv1 and the second one by δ∂zv2. Set v1(1/2) = 0, v2(1) = λ, v1((1 + z1)/2) = v2((1 + z1)/2). Set ˜ z = (1 + z1)/2, G ′

1 = S1, G ′ 2 = S2, H′ = K.

We obtain:

• M. Di Francesco (L’Aquila)

Nonlocal interaction equations Anacapri 2015 40 / 47

The case with quadratic diffusion

A functional system

z − 1 2 = δ−3

• J1

(G1(δ(v1(z) − v1(ζ))) − δv1(z)S1(δ(µ − v1(ζ)))) dζ +

• J2

(H(δ(v1(z) − v2(ζ))) − δv1(z)K(δ(µ − v2(ζ)))) dζ

• + δ−2v1(z)
• J2

(S2(δ(µ − v2(ζ))) − S2(δ(λ − v2(ζ)))) dζ +

• J1

(K(δ(µ − v1(ζ))) − K(δ(λ − v1(ζ)))) dζ

• ,

z ∈ [1/2, ˜ z] , z − ˜ z = δ−3

• J2

(G2(δ(v2(z) − v2(ζ))) − δv2(z)S2(δ(λ − v2(ζ)))) dζ +

• J1

(H(δ(v2(z) − v1(ζ))) − δv2(z)K(δ(λ − v1(ζ)))) dζ

• − δ−3
• J2

(G2(δ(µ − v2(ζ))) − δµS2(δ(λ − v2(ζ)))) dζ +

• J1

(H(δ(µ − v1(ζ))) − δµK(δ(λ − v1(ζ)))) dζ

• ,

z ∈ [˜ z, 1/2] .

• M. Di Francesco (L’Aquila)

Nonlocal interaction equations Anacapri 2015 41 / 47

The case with quadratic diffusion

Solution for δ = 0

Extension of the functional system by Taylor expansions and symmetries: F1[(v1, µ, v2λ); 0](z) = 1 2 − z − C1 6 v1(z)3 + 1 2[C1µ2 + C2(λ2 − µ2)]v1(z) , F2[(v1, µ, v2λ); 0](z) = ˜ z − z − C2 6 v2(z)3 + C2 2 λ2v2(z) + C2 6 µ3 − C2 2 λ2µ , C1 = −S′′

1 (0)|J1| − K ′′(0)|J2| ,

C2 = −S′′

2 (0)|J2| − K ′′(0)|J1| .

System for µ and λ ˜ z − 1 2 = C1 3 − C2 2

• µ3 + C2

2 µλ2 , 1 − ˜ z = C2 6 µ3 + C2 3 λ3 − C2 2 µλ2 , admits a unique solution. There exists a unique solution for δ =, having the shape of a pair of segregated Barenblatt profiles.

• M. Di Francesco (L’Aquila)

Nonlocal interaction equations Anacapri 2015 42 / 47

The case with quadratic diffusion

Solution via implicit function theorem

For small enough δ > 0, The above functional operator is continuously differentiable around δ = 0, on a suitably defined Banach space B (incorporating H¨

• lder seminorms at z = 1 (hard

computation!). The Fr´ echet derivative at δ = 0 is a linear isomorphism on B. Hence, a unique solution to the functional system exists for small δ, i. e. for small ǫ. Going back to the density formulation, this is a unique segregated stationary solution (ρ1, ρ2), with a support of order ǫ1/3, with

• ρ1dx = z1 ,
• ρ2dx = 1 − z1.
• M. Di Francesco (L’Aquila)

Nonlocal interaction equations Anacapri 2015 43 / 47

Conclusions

Table of contents

1

Nonlocal interactions

2

Gradient flow structure

3

Systems with many species

4

A predator-prey model

5

The case with quadratic diffusion

6

Conclusions

• M. Di Francesco (L’Aquila)

Nonlocal interaction equations Anacapri 2015 44 / 47

Conclusions

Concluding remarks

Many species subject to nonlocal aggregation are subject to asymptotic consensus when the structure of the system features some symmetry properties in the interaction laws. Such consensus can be reached after intermediate states of same-species clustering. Without symmetry, in general no information about the asymptotic state. Predator prey: repulsive forces enter the game, non trivial delta states. Chasing patterns arising in nature are produced. Introducing local quadratic diffusion turns the consensus (singular) states into segregated (L1) steady states. Future work: Generalisation of the necessary conditions and existence of minimisers for more general diffusions. Case without diffusion: more singular kernels without symmetric cross-interaction. Time dependent problem, existence of solutions with degenerate diffusion (hard). Predator-prey models with gain-loss terms. Derivation from microscopic models.

• M. Di Francesco (L’Aquila)

Nonlocal interaction equations Anacapri 2015 45 / 47

Conclusions

References

• M. Di Francesco, and S. Fagioli, Measure solutions for nonlocal interaction PDEs

with two species - Nonlinearity 26 (2013), 2777-2808.

• M. Di Francesco and S. Fagioli, Steady states for a two species system of nonlocal

interaction PDEs of predator-prey type, to appear on M3AS. Simone’s PhD thesis.

• M. Burger, M. Di Francesco, S. Fagioli, A. Stevens, in preparation.
• M. Di Francesco (L’Aquila)

Nonlocal interaction equations Anacapri 2015 46 / 47

Conclusions

End of the talk

Thank you for your attention!

• M. Di Francesco (L’Aquila)

Nonlocal interaction equations Anacapri 2015 47 / 47