Stationary states for the aggregation equation with power law attractive-repulsive potentials

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1 Stationary states for the aggregation equation with power law attractive-repulsive potentials Daniel Balagué Joint work with J.A. Carrillo, T. Laurent and G. Raoul Universitat Autònoma de Barcelona BIRS - Multi-particle systems with non-local interactions January Banff, Canada

2 Introduction We consider the problem given by ρ t = div[ρv] in R N [0, T ] v = W ρ ρ(0) = ρ 0 0. Daniel Balagué Guardia Stationary states for the aggregation equation 2/23

3 Introduction We consider the problem given by ρ t = div[ρv] in R N [0, T ] v = W ρ ρ(0) = ρ 0 0. Here ρ(x, t) is a density of particles located at position x at time t and W is a given interaction potential. Daniel Balagué Guardia Stationary states for the aggregation equation 2/23

4 What do we know about the solutions? Existence and uniqueness of weak solutions for the aggregation equation in P 2 (R N ) L p (R N ), for ρ 0 L p and W W 1,q. Daniel Balagué Guardia Stationary states for the aggregation equation 3/23

5 What do we know about the solutions? Existence and uniqueness of weak solutions for the aggregation equation in P 2 (R N ) L p (R N ), for ρ 0 L p and W W 1,q. Global existence when W is bounded from above. Daniel Balagué Guardia Stationary states for the aggregation equation 3/23

6 What do we know about the solutions? Existence and uniqueness of weak solutions for the aggregation equation in P 2 (R N ) L p (R N ), for ρ 0 L p and W W 1,q. Global existence when W is bounded from above. Weak measure solutions to the the Cauchy problem for the aggregation equation. Daniel Balagué Guardia Stationary states for the aggregation equation 3/23

7 What do we know about the solutions? Existence and uniqueness of weak solutions for the aggregation equation in P 2 (R N ) L p (R N ), for ρ 0 L p and W W 1,q. Global existence when W is bounded from above. Weak measure solutions to the the Cauchy problem for the aggregation equation. The problem is global-in-time well-posed in L p (R N ) under Osgood conditions, and there is blow-up of the L p -norm when this condition is violated. Daniel Balagué Guardia Stationary states for the aggregation equation 3/23

8 Assumptions We will suppose that: the potential W is attractive-repulsive, radially symetric and smooth away from the origin, Daniel Balagué Guardia Stationary states for the aggregation equation 4/23

9 Assumptions We will suppose that: the potential W is attractive-repulsive, radially symetric and smooth away from the origin, if µ is a radially symmetric measure then ˆµ P([0, + )) is defined by r2 d ˆµ(r) = dµ(x). r 1 r 1 < x <r 2 Daniel Balagué Guardia Stationary states for the aggregation equation 4/23

10 Assumptions We will suppose that: the potential W is attractive-repulsive, radially symetric and smooth away from the origin, if µ is a radially symmetric measure then ˆµ P([0, + )) is defined by r2 d ˆµ(r) = dµ(x). r 1 r 1 < x <r 2 Definition (Spherical shell) A delta on a sphere of radius R ( spherical shell, δ B(0,R) ), denoted it by δ R, is a uniform distribution on a sphere B(0, R) = {x R N : x = R}. Daniel Balagué Guardia Stationary states for the aggregation equation 4/23

11 Understanding the velocity The velocity field at a point x generated by a δ R is given by: v = W δ R (x) Daniel Balagué Guardia Stationary states for the aggregation equation 5/23

12 Understanding the velocity The velocity field at a point x generated by a δ R is given by: v = W δ R (x) and, by symmetry, exists a function ω(r 1, r 2 ) such that v = W δ R (x) = ω( x, R) x x. Daniel Balagué Guardia Stationary states for the aggregation equation 5/23

13 Understanding the velocity The velocity field at a point x generated by a δ R is given by: v = W δ R (x) and, by symmetry, exists a function ω(r 1, r 2 ) such that v = W δ R (x) = ω( x, R) x x. Remark Then µ can be written as a sum of δ R, 0 δ r d ˆµ(r). And, v = ( W µ)(x) = 0 ω( x, r)d ˆµ(r) x x. Daniel Balagué Guardia Stationary states for the aggregation equation 5/23

14 The problem in radial coordinates In radially symmetric coordinates, the equation reads: ˆv(r, t) = t ˆµ + r (ˆµˆv) = 0, + 0 ω(r, η)d ˆµ t (η). Daniel Balagué Guardia Stationary states for the aggregation equation 6/23

15 The problem in radial coordinates In radially symmetric coordinates, the equation reads: ˆv(r, t) = The function ω is defined by ω(r, η) = 1 σ N t ˆµ + r (ˆµˆv) = 0, + 0 B(0,1) ω(r, η)d ˆµ t (η). W (re 1 ηy) e 1 dσ(y), where σ N is the surface area of the unit ball in R N and e 1 is the first vector of the canonical basis of R N. Daniel Balagué Guardia Stationary states for the aggregation equation 6/23

16 Stationary states Definition A probability measure µ P(R N ) is a stationary state for the aggregation equation if ( W µ)(x) = 0 for all x supp(µ). Daniel Balagué Guardia Stationary states for the aggregation equation 7/23

17 Stationary states Definition A probability measure µ P(R N ) is a stationary state for the aggregation equation if ( W µ)(x) = 0 for all x supp(µ). According to the given interpretation of the velocity field, a δ R is a stationary state for the aggregation equation if and only if ω(r, R) = 0. Daniel Balagué Guardia Stationary states for the aggregation equation 7/23

18 Energetic point of view The aggregation equation is a gradient flow 1 of the following energy functional E[µ] = 1 W (x y)dµ(x)dµ(y) 2 R N R N w.r.t. the euclidean Wasserstein distance { } d2 2 (ν, ρ) = inf x y 2 dπ(x, y), π Π(ν,ρ) R N R N where Π(ν, ρ) stands for the set of joint distributions with marginals ν i ρ. 1 Ambrosio, L. A.; Gigli, N.; Savaré, G. Gradient flows in metric spaces and in the space of probability measures. Lectures in Mathematics, Birkhäuser, Daniel Balagué Guardia Stationary states for the aggregation equation 8/23

19 Then, the stationary states asymptotically stable are local minimizers of the energy. Let us see the conditions for a δ R Proposition Suppose ω C 1 and let δ R be a stationary state, ω(r, R) = 0. then, (i) If 1 ω(r, R) > 0 exists dr 0 > 0 such that, given 0 < dr < dr 0, for ɛ small enough. E[(1 ɛ)δ R + ɛδ R+dr ] < E[δ R ], (ii) If 1 ω(r, R) + 2 (R, R) > 0 exists dr 0 > 0 such that for all 0 < dr < dr 0. E[δ R+dr0 ] < E[δ R ], Daniel Balagué Guardia Stationary states for the aggregation equation 9/23

20 Instability conditions Theorem Suppose that δ R is a stationary state, ω(r, R) = 0, and that one of the following cases is satisfied: (i) ω C 1 (R 2 +) and 1 ω(r, R) > 0. (ii) ω C(R 2 +) C 1 (R 2 + \ D) and lim 1 ω(r 1, r 2 ) = +. (r 1,r 2 ) D (r 1,r 2 ) (R,R) Conclusion: It is not possible for an L p radially symmetric solution to converge toward a δ R when t. Daniel Balagué Guardia Stationary states for the aggregation equation 10/23

21 Idea for the instability One first observes that ω( x, R) (divv)(x) = W δ R (x) = 1 ω( x, R) + (N 1). r 1 Daniel Balagué Guardia Stationary states for the aggregation equation 11/23

22 Idea for the instability One first observes that ω( x, R) (divv)(x) = W δ R (x) = 1 ω( x, R) + (N 1). r 1 Then we can reformulate the theorem with the condition (divv)(x) = ( W δ R )(x) > 0 for all x B(0, R). Daniel Balagué Guardia Stationary states for the aggregation equation 11/23

23 Idea for the instability One first observes that ω( x, R) (divv)(x) = W δ R (x) = 1 ω( x, R) + (N 1). r 1 Then we can reformulate the theorem with the condition (divv)(x) = ( W δ R )(x) > 0 for all x B(0, R). R δ R R + δ Daniel Balagué Guardia Stationary states for the aggregation equation 11/23

24 Conditions for the stability For the stability we suppose that ω C 1 and that δ R is an stationary state. Moreover, suppose that (i) 1 ω(r, R) < 0, (ii) 1 ω(r, R) + 2 ω(r, R) < 0. Daniel Balagué Guardia Stationary states for the aggregation equation 12/23

25 Conditions for the stability For the stability we suppose that ω C 1 and that δ R is an stationary state. Moreover, suppose that (i) 1 ω(r, R) < 0, (ii) 1 ω(r, R) + 2 ω(r, R) < 0. The interpretation is similar... R δ R R + δ Daniel Balagué Guardia Stationary states for the aggregation equation 12/23

26 Instability: non-radial case Consider an steady state µ of the form f (x) dµ(x) = f (x)φ(x) dσ(x), f C(R N ), R N M where M is a C 2 hypersurface and dσ is the volume element on M. Proposition Suppose that one of the two conditions holds: (i) W is locally integrable on hypersurfaces, φ L (M) and (div v(x) := ( W µ)(x) > 0 x supp(µ), (ii) W is not locally integrable on hypersurfaces and φ(x) > φ 0 > 0 for all x M. Conclusion: it is not possible for an L p solution to converge to µ w.r.t. the d -topology. Daniel Balagué Guardia Stationary states for the aggregation equation 13/23

27 An example with powers Consider now power law potentials: W (x) = x a a x b b 2 N < b < a Daniel Balagué Guardia Stationary states for the aggregation equation 14/23

28 An example with powers Consider now power law potentials: Some properties: W (x) = x a a x b b 2 N < b < a The condition b < a ensures that the potential is repulsive in the short range and attractive in the long range. Daniel Balagué Guardia Stationary states for the aggregation equation 14/23

29 An example with powers Consider now power law potentials: Some properties: W (x) = x a a x b b 2 N < b < a The condition b < a ensures that the potential is repulsive in the short range and attractive in the long range. The condition 2 N < b ensures that the potential is in W 1,q loc (RN ) for some 1 < q <. Daniel Balagué Guardia Stationary states for the aggregation equation 14/23

30 δ R in the power law case For the case of powers one has a formula for the function ω(r, η): ω(r, η) = r b 1 ψ b (η/r) r a 1 ψ a (η/r), Daniel Balagué Guardia Stationary states for the aggregation equation 15/23

31 δ R in the power law case For the case of powers one has a formula for the function ω(r, η): where ω(r, η) = r b 1 ψ b (η/r) r a 1 ψ a (η/r), ψ a (s) = σ N 1 σ N π 0 (1 s cos θ)(sin θ) N 2 dθ. (1 + s 2 2s cos θ) 2 a 2 Daniel Balagué Guardia Stationary states for the aggregation equation 15/23

32 δ R in the power law case For the case of powers one has a formula for the function ω(r, η): where ω(r, η) = r b 1 ψ b (η/r) r a 1 ψ a (η/r), ψ a (s) = σ N 1 σ N π 0 (1 s cos θ)(sin θ) N 2 dθ. (1 + s 2 2s cos θ) 2 a 2 Then, a δ R is a stationary state for the aggregation equation if and only if ( ) 1 ψb (1) a b ω(r, R) = 0, R = R ab =. ψ a (1) Daniel Balagué Guardia Stationary states for the aggregation equation 15/23

33 Value of R? In the power law case, the radius R can be computed explicitly: R = R ab = 1 2 ( β( b+n 1 2, N 1 2 ) β( a+n 1 2, N 1 2 ) ) 1 a b Daniel Balagué Guardia Stationary states for the aggregation equation 16/23

34 In/stability for powers Suppose that W (x) = x a a state. x b b and consider a δ Rab an stationary Daniel Balagué Guardia Stationary states for the aggregation equation 17/23

35 In/stability for powers Suppose that W (x) = x a a state. x b b and consider a δ Rab an stationary (i) If 2 N < b 3 N then ω C(R 2 +) C 1 (R 2 + \ D) and for all (R, R) D lim 1 ω(r 1, r 2 ) = +. (r 1,r 2 ) D (r 1,r 2 ) (R,R) Daniel Balagué Guardia Stationary states for the aggregation equation 17/23

36 In/stability for powers Suppose that W (x) = x a a state. x b b and consider a δ Rab an stationary (i) If 2 N < b 3 N then ω C(R 2 +) C 1 (R 2 + \ D) and for all (R, R) D (ii) If b lim 1 ω(r 1, r 2 ) = +. (r 1,r 2 ) D (r 1,r 2 ) (R,R) ) (3 N, (3 N)a 10+7N N2 a+n 3 then ω C 1 and 1 ω(r ab, R ab ) > 0. Daniel Balagué Guardia Stationary states for the aggregation equation 17/23

37 In/stability for powers Suppose that W (x) = x a a state. x b b and consider a δ Rab an stationary (i) If 2 N < b 3 N then ω C(R 2 +) C 1 (R 2 + \ D) and for all (R, R) D (ii) If b (iii) If b lim 1 ω(r 1, r 2 ) = +. (r 1,r 2 ) D (r 1,r 2 ) (R,R) ) (3 N, (3 N)a 10+7N N2 a+n 3 then ω C 1 and ( (3 N)a 10+7N N 2 a+n 3 1 ω(r ab, R ab ) > 0. ), a then ω C 1 and 1 ω(r ab, R ab ) < 0 i 1 ω(r ab, R ab ) + 2 ω(r ab, R ab ) < 0. Daniel Balagué Guardia Stationary states for the aggregation equation 17/23

38 Idea of the proof (i) (i) Case 2 N < b 3 N. One has ω C 1 ((R + R + )\D) and ω r (r, η) = r b 2[ ] (b 1)ψ b (η/r) (η/r)ψ b (η/r) r a 2[ ] (a 1)ψ a (η/r) (η/r)ψ a(η/r) in (R +, R + )\D. Daniel Balagué Guardia Stationary states for the aggregation equation 18/23

39 Idea of the proof (i) (i) Case 2 N < b 3 N. One has ω C 1 ((R + R + )\D) and ω r (r, η) = r b 2[ ] (b 1)ψ b (η/r) (η/r)ψ b (η/r) r a 2[ ] (a 1)ψ a (η/r) (η/r)ψ a(η/r) in (R +, R + )\D. Calculating the limit, lim (r,η) (R,R) (r,η)/ D ω (r, η) = +. r Daniel Balagué Guardia Stationary states for the aggregation equation 18/23

40 Idea of the proof (ii) (ii) Case 3 N < b < (3 N)a 10+7N N2 a+n 3. In this case, ω C 1 (R + R + ). Daniel Balagué Guardia Stationary states for the aggregation equation 19/23

41 Idea of the proof (ii) (ii) Case 3 N < b < (3 N)a 10+7N N2 a+n 3. In this case, ω C 1 (R + R + ). Condition ω r (R ab, R ab ) > 0 a ψ a(1) ψ a (1) < b ψ b (1) ψ b (1) Daniel Balagué Guardia Stationary states for the aggregation equation 19/23

42 Idea of the proof (iii) A change of variable in the expression of ψ a (1) shows that ψ a (1) = ω ( N 1 a + N 1 2 a+n 3 β, N 1 ), ω N 2 2 Daniel Balagué Guardia Stationary states for the aggregation equation 20/23

43 Idea of the proof (iii) A change of variable in the expression of ψ a (1) shows that ψ a (1) = ω ( N 1 a + N 1 2 a+n 3 β, N 1 ), ω N 2 2 and for ψ a(1) ψ a(1) = ω ( N 1 (a 2)(a + N 2) a + N 3 2 N+a 4 β, N + 1 ). ω N N Daniel Balagué Guardia Stationary states for the aggregation equation 20/23

44 Idea of the proof (iii) A change of variable in the expression of ψ a (1) shows that ψ a (1) = ω ( N 1 a + N 1 2 a+n 3 β, N 1 ), ω N 2 2 and for ψ a(1) ψ a(1) = ω ( N 1 (a 2)(a + N 2) a + N 3 2 N+a 4 β, N + 1 ). ω N N Then the quotient ψ a (1) ψ a(1) becomes ψ a(1) ψ a (1) = 1 2 (a 2)(a + N 2). a + N 3 Daniel Balagué Guardia Stationary states for the aggregation equation 20/23

45 Some pictures for the velocity field ω(r, R ab ), N = 2 a=4, b=2, Rab= a=2, b=1, Rab= Daniel Balagué Guardia Stationary states for the aggregation equation 21/23

46 Summary b = (3 N)a 10+7N N 2 a+n b = a N 3 N Daniel Balagué Guardia Stationary states for the aggregation equation 22/23

47 Thank you for your attention. Daniel Balagué Guardia Stationary states for the aggregation equation 23/23

arxiv: v1 [math.ap] 25 Oct 2012

arxiv: v1 [math.ap] 25 Oct 2012 DIMENSIONALITY OF LOCAL MINIMIZERS OF THE INTERACTION ENERGY D. BALAGUÉ 1, J. A. CARRILLO 2, T. LAURENT 3, AND G. RAOUL 4 arxiv:1210.6795v1 [math.ap] 25 Oct 2012 Abstract. In this work we consider local