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 X 1 ( t ) , . . . , X N ( t ) ∈ R d with masses m 1 , . . . , m N . 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 X 3 on X 7 and viceversa. dX j ( t ) � = − m k ∇ G ( X j ( t ) − X k ( t )) , j = 1 , . . . , N . (1) dt k � = j 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 ( R d ), 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 ) = − C a e −| x | / L a Repulsive Morse potentials G ( x ) = C r e −| x | / L r Attractive-repulsive Morse potentials G ( x ) = − C a e −| x | / L a + C r e −| x | / L r Combination of Gaussian potentials G ( x ) = − C a e −| x | 2 / L 2 a + C r e −| x | 2 / L 2 r M. Di Francesco (L’Aquila) Nonlocal interaction equations Anacapri 2015 5 / 47

Nonlocal interactions Hydrodynamic N → + ∞ limit Empirical measure: � N � − 1 N � � µ N ( t ) = m j m k δ X k ( t ) j =1 k =1 Formal N → + ∞ limit of µ N ∂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 ( R m ; R ), a curve [0 , + ∞ ) ∋ t �→ X ( t ) ∈ R m 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, X τ ( t ) = X n τ for t ∈ (( n − 1) τ, n τ ], with τ = argmin { 1 τ | 2 + F ( X ) , X ∈ R m } X n 2 τ | X − X n D 2 F ≥ λ I implies stability d dt | X 1 ( t ) − X 2 ( t ) | 2 = − 2 < X 1 ( t ) − X 2 ( t ) , ∇ F ( X 1 ( t )) − ∇ F ( X 1 ( t )) > ≤ − 2 λ | X 1 ( t ) − X 2 ( 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 dX j ( t ) � = − m k ∇ G ( X j ( t ) − X k ( t )) , j = 1 , . . . , N . dt k � = j with G ( − x ) = G ( x ) and G ∈ C 2 ( R d ). Weighted metric structure Denote m = ( m 1 , . . . , m N ). For X , Y ∈ R dN , let N � � X � 2 < X , Y > L 2 m := m j X j Y j , m = < X , X > L 2 m . L 2 j =1 Frech´ et differential Let F ∈ C 1 ( R dN ). The linear operator grad X F [ X ] is defined by N F [ X + ǫ Y ] − F [ X ] � lim =: < grad X F [ X ] , Y > L 2 m = m j ∇ X j F [ X ] · Y j . ǫ ǫ → 0 j =1 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 := ( X 1 , . . . , X N ) T . G [ X ] := 1 � m i m j G ( X i − X j ) 2 i , j Then ˙ X ( t ) = − grad X G [ X ( t )] . (2) Regularity and collisions The above makes sense if G is C 1 , G ∈ C 2 ( R d ): 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 ′ ( r ) g ′ ( r ) > 0 for r > 0 , G ( x ) = g ( | x | ) , non-increasing . (3) r Proposition (Finite time collapse) Let X 1 , . . . , X N solve ˙ � X j ( t ) = − m k ∇ G ( X j ( t ) − X k ( t ) . X k ( t ) � = X j ( t ) Then, X j ( t ) = δ C m for all t ≥ t ∗ for some t ∗ , iff � ε 1 g ′ ( z ) dz < + ∞ for some ε > 0 . (4) 0 Important remarks t ∗ depends on the maximal initial distance of the particles from C m 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 ∈ P 2 ( R d ) := µ ∈ P ( R d ) , | x | 2 d µ ( x ) < + ∞ The Wasserstein distance ��� � d 2 ( µ, ν ) 2 = inf R d × R d | x − y | 2 d γ ( x , y ) , : γ ∈ Γ( µ, ν ) γ ∈ P ( R d × R d ) : µ and µ are the marginals of γ � � Γ( µ, ν ) = The functional G [ µ ] = 1 �� R d × R d G ( x − y ) d µ ( x ) d µ ( y ) 2 Wasserstein gradient flow ∂µ ( t ) in D ′ ( R d × [0 , + ∞ )) + div ( µ ( t ) v ( t )) = 0 , ∂ t � v ( t ) = − ∂ 0 G ∗ µ ( t ) = − ∇ G ( x − y ) d µ ( y , t ) . x � = y M. Di Francesco (L’Aquila) Nonlocal interaction equations Anacapri 2015 12 / 47

Gradient flow structure The JKO scheme 1 Time discretization: τ > 0 fixed time step. Let µ 0 ∈ P ( R d ) be fixed. For a given µ τ n ∈ P ( R d ), we define the sequence µ τ n +1 as � 1 � µ n +1 2 τ d 2 2 ( µ n ∈ argmin µ ∈ P 2 ( R d ) τ , µ ) + G [ µ ] . τ µ τ as the piecewise constant interpolation of µ n Define ¯ τ . µ τ converges to a unique curve of measures µ as τ ց 0, ¯ µ is the unique gradient flow solution to the PDE, Stability property d 2 ( µ 1 ( t ) , µ 2 ( t )) ≤ e | λ | t d 2 ( µ 0 1 , µ 0 2 ) , (5) The property (5) allows to extend the final collapse to all measure solutions by atomization (CDFLS 2011). 1 Jordan, 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 model 2 X 1 , . . . , X N particles of the first species with masses n 1 , . . . , n N , Y 1 , . . . , Y M are particles of the second species with masses m 1 , . . . , m M . Particle system: � ˙ X i ( t ) = − � X i � = X k n k ∇ K 11 ( X i ( t ) − X k ( t )) − � X i � = Y k m k ∇ K 12 ( X i ( t ) − Y k ( t )) . ˙ Y j ( t ) = − � Y j � = Y k m k ∇ K 22 ( Y j ( t ) − Y k ( t )) − � Y j � = X k n k ∇ K 21 ( Y j ( t ) − X k ( t )) Continuum version: � ∂ t µ 1 = div ( µ 1 ∇ K 11 ∗ µ 1 + µ 1 ∇ K 12 ∗ µ 2 ) ∂ t µ 2 = div ( µ 2 ∇ K 22 ∗ µ 2 + µ 2 ∇ K 21 ∗ µ 1 ) . 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 � ∂ t µ 1 = div ( µ 1 ∇ K 11 ∗ µ 1 + µ 1 ∇ K 12 ∗ µ 2 ) with K 21 = α K 12 , α > 0 . (6) ∂ t µ 2 = div ( µ 2 ∇ K 22 ∗ µ 2 + µ 2 ∇ K 21 ∗ µ 1 ) , System (6) has a gradient flow structure, with functional F ( µ 1 , µ 2 ) = 1 � R d K 11 ∗ µ 1 d µ 1 + 1 � � R d K 22 ∗ µ 2 d µ 2 + R d K 12 ∗ µ 2 d µ 1 . 2 2 α As a byproduct, due to the property K 21 = α K 12 , one can prove the following energy identity d dt F ( µ 1 ( t ) , µ 2 ( t )) � |∇ K 11 ∗ µ 1 + ∇ K 12 ∗ µ 2 | 2 d µ 1 ( x ) = − 2 � � � 1 � � − α α ∇ K 22 ∗ µ 2 + ∇ K 12 ∗ µ 1 d µ 2 ( x ) ≤ 0 � � � � M. Di Francesco (L’Aquila) Nonlocal interaction equations Anacapri 2015 16 / 47