Aggregation Equations – well posedness, self-similarity, and collapse

Andrea Bertozzi

Riviere-Fabes Symposium 2016

The Problem

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A density advected by a field that is the the gradient of K convolved with itself.

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Active scalar problem in gradient flow format.

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Analysis follows ideas from both fluid mechanics (active scalars – divergence free flow) and optimal transport (gradient flow).

• Previous Results
• For smooth K the solution blows up in infinite time
• For `pointy’ K (biological kernel such as K=e^-|x| ) blows up in finite time for special radial data in any space dimension.

Bertozzi Carrillo, Laurent

Nonlinearity 2009

New result:

Osgood condition

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is a necessary and sufficient condition for finite time blowup in any space dimension

(under mild monotonicity conditions).

Moreover-finite time blowup for

pointy potential can not be described by `first kind’ similarity solution in dimensions N=3,5,7,...

Osgood uniqueness criteria for ODEs

• dX/dt = F(X), system of first order ODEs
• Picard theorem – F is Lipschitz continuous
• Generalization of Picard is Osgood criteria(sharp) : w(x) is modulus of continuity of F, i.e.

|F(X)-F(Y)|< w(|X-Y|)

• 1/w(z) is not integrable at the origin for unique solutions.
• Example: dx/dt = x |ln x| has unique solutions.
• Example: dx/dt = sqrt(x) does not have unique solutions.

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Bertozzi, Carrillo,

Laurent

Nonlinearity 2009

COMPARISON PRINCIPLE: Proof of finite time collapse for non-Osgood potentials – let R(t) denote the particle farthest away from the center of mass (conserved), then

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When the Osgood criteria is violated particles collapse together in finite time. When the Osgood criteria is satisfied we have global existence and uniqueness of a solution of the particle equations.

particles

Bertozzi, Carrillo, Laurent

Nonlinearity 2009

COMPARISON PRINCIPLE: The proof for a finite number of particles extends naturally to the continuum limit. Proof of finite time blowup for non-Osgood potentials - assumes compact support of solution. One can prove that there exists an R(t) such that B_R(t)(x_m) contains the support, x_m is center of mass (conserved), and

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Thus the Osgood criteria provides a sufficient condition on the potential K for finite time blowup from bounded data. To prove the condition is also necessary we must do further potential theory estimates.

First an easier result - C² kernels

• A priori bound
• One can easily prove a Gronwall estimate
• L-infty of density controlled by L-infty of div v.
• But, div v = Laplacian of K convolved with the density, which we assume to be in L¹.
• If the kernel is C² we have an a priori bound.
• We need more refined potential theory estimates for general Osgood condition.

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Bertozzi, Carrillo, Laurent

Nonlinearity 2009

Vorticity Stream form of 3D Euler Equations

omega is vector vorticity and K₃ is Biot-Savart Kernel in 3D

Bertozzi, Carrillo, Laurent

Nonlinearity 2009

l₂

l₁

Bertozzi, Carrillo, Laurent

Nonlinearity 2009

Bertozzi, Carrillo, Laurent

Nonlinearity 2009

• Similarity solution of form

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• The equation implies
• Conservation of mass would imply - NO
• Second kind similarity solution - no mass conservation – power < 2.
• Experimentally, the exponents vary smoothly with dimension of space, and there is no mass concentration in the blowup....

Huang and Bertozzi

SIAP 2010-radially symmetric numerics

DCDS 2012 – general power law kernel

``Finite time blowup for `pointy’ potential, K=|x|, can not be described by `first kind’ similarity solution in dimensions N=3,5,7,...’’ - proof Bertozzi, Carrillo, Laurent, Dai preprint – general N. What happens when the solution blows up? Let’s compute it.

Simulations by Y. Huang

Second

kind

Exact

self-similar

• CONNECTION TO BURGERS SHOCKS
• In one dimension, K(x) = |x|, even initial data, the problem can be transformed exactly to Burgers equation for the integral of u.

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• Burgers equation for odd initial data has an exact similarity solution for the blowup - it is an initial shock formation, with a 1/3 power singularity at x=0.

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• There is no jump discontinuity at the initial shock time, which correponds to a zero-mass blowup for the aggregation problem. However immediately after the initial shock formation a jump discontinuity opens up - corresponds to mass concentration in the aggregation problem instantaneously after the initial blowup.
• This Burgers solution is (a) self-similar, (b) of `second kind’, and (c) generic for odd initial data. There is a one parameter family of such solutions (also true in higher D).
• For the original u equation, this corresponds to beta = 3/2.

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Huang and Bertozzi

SIAP 2010

Local existence of L^p solutions

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ALB, Laurent, Rosado, CPAM 2011

More on L^p �ALB, Laurent, Rosado, CPAM 2011

• Local existence using method of characteristics and some analysis (ALB, Laurent)
• Uniqueness using optimal transport theory (Rosado)
• Global existence vs local existence – the Osgood criteria comes back again. Why?
• Mass concentration eventually happens in finite time for non-Osgood kernel
• A priori L^p bound for Osgood kernel – see next slide

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• Local existence of solutions in L^p provided that

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• where q is the Holder conjugate of p (characteristics).
• Existence proof constructs solutions using characteristics, in a similar fashion to weak solutions (B. and Brandman Comm. Math. Sci. - to appear).
• Global existence of the same solutions in L^p provided that K satisfies the Osgood condition (derivation of a priori bound for L^p norm - similar to refined potential theory estimates in BCL 2009).
• When Osgood condition is violated, solutions blow up in finite time - implies blowup in L^p for all p>p_c.

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Bertozzi, Laurent, and Rosado

CPAM 2011.

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• Ill-posedness of the problem in L^p for p less than the Holder-critical p_c associated with the potential K.
• Ill-posedness results because one can construct examples in which mass concentrates instantaneously (for all t>0).
• For p> p_{c ,} uniqueness in L^p can be proved for initial data also having bounded second moment, the proof uses ideas from optimal transport.
• The problem is globally well-posed with measure-valued data (preprint of Carrillo, DiFrancesco, Figalli, Laurent, and Slepcev - using optimal transport ideas).
• Even so, for non-Osgood potentials K, there is loss of information as time increases.
• Analogous to information loss in the case of compressive shocks for scalar conservation laws.

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Bertozzi, Laurent, and Rosado

CPAM 2011

Followup on Lp

• Result by Hongjie Dong proving that the pc is sharp for all powerlaw kernels
• Another paper SIMA 2012 by ALB, Garnett, Laurent studying monotonicity of radially symmetric solutions with mass concentration –delta
• Existence of solutions for all powers down to Newtonial potential – requires Lagrangian form of the equation.
• Newtonial potential is easy because radial symmetry reduces the PDE to Burgers equation in 1D and you can prove everything.
• Uniqueness is open for powerlaw kernels between |x| and Newtonian case.

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Mixed Potentials – the World Cup Example�joint work with T. Kolokolnikov, H. Sun, D. Uminsky�Phys. Rev. E 2011

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• K’(r ) =

Tanh((1-r)a)

+b

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Patterns as

Complex as

The surface of a

A soccer ball.

Predicting pattern formation in particle interactions (3D linear theory)�M3AS 2012, von Brecht, Uminsky, Kolokolnikov, ALB

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Linear stability of spherical shells

• Funk-Hecke theorem for spherical harmonics
• Analytical computation predicts shapes of patterns

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• Pattern

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• Linearly

Unstable mode

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Fully nonlinear theory for multidimensional sheet solutions

• Joint work with James von Brecht, in Comm Math Phys, 2013.
• Existence/uniqueness of solutions
• Local well-posedness depends on the kernel – sometimes not locally well-posed even for `reasonable’ kernels (fission to non-sheet behavior)
• Collapse in finite time – Osgood condition comes back again
• Also can expand to infinity in finite time

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Aggregation Patches�Joint work with Flavien Leger and Thomas Laurent, M3AS 2012

• These are like vortex patches….only different
• V = grad N*u where N is Newtonian potential
• Flow is orthogonal to the case of the vortex patch
• Solution will either contract or expand depending on the sign of the kernel

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Key aspects of the problem

• General equations

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• Newtonian case

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• Density is specified along particle paths
• It means there are exact solutions that are patches – like the vortex patch only they blow up in finite time, and the measure of the support shrinks to zero.

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• These solutions exist in any dimension.

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Expanding case – aggregation patch

• In the expanding case the patch grows at a known rate
• In the long time limit the expanding patch converges in L1 to an exact similarity solution
• The similarity solution is an expanding ball:

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• Proof of convergence to the similarity solution has a power-law rate – proved to be sharp in 2D

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Aggregation Patch “Kirchoff Ellipse”

• Exact analytic solution – aggregation analogue of the Kirchoff ellipse – collapse onto a line segment – weighted measure

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Aggregation Patches – attractive case 2D – collapse onto skeletons

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Movie Aggregation Patch 3D Cube Attractive Case

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Movie Aggregation Patch – Teapot – Attractive Case

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3D knot collapse

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2D repulsive patch – rescaled variables

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Repulsive case – 2D particles rescaled variables

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Local and Global Regularity of Aggregation Patch Boundary�preprint 2015 ALB, J. Garnett, T. Laurent, J. Verdera

• Inspired by work from 1990s by Chemin and ALB/Constantin on global regularity of the vortex patch boundary
• Uses level set formulation and proves a priori bound on Holder norm of the gradient of the level set function
• Harmonic analysis lemma involving cancellation on half-spheres – generalized from 2D to higher Dimensions

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Swarming on Random Graphs I & II

• J. Stat. Phys. 151(1-2), pp. 150-173, 2013; 158(3), pp. 699-734, 2014
• Joint work with T. Kolokolnikov, J. von Brecht, B. Sudakov, and H. Sun
• Pairwise particle interactions with attractive-repulsive potential
• Connections between states occur with probability p – Erdos-Renyi graph
• Ground state with p=1 – compromise model with two states

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Swarming on Random Graphs

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Results-stability of compromise state

• Stability occurs for probability p> pc.
• Pc scales like C Ln N/N for some constant C
• Constant C is larger than the critical constant for a fully connected graph – due to compromise state
• Rigorous theory for multiple dimensions

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Inviscid Aggregation Equations Analysis

• ALB, J. B. Garnett, T. Laurent, J. Verdera, regularity boundary aggregation patch, 2015
• ALB, J. B. Garnett, T. Laurent, - existence of solutions with measure and singular potentials, 1D radial, SIMA 2012
• von Brecht and ALB – general analysis of “sheet solutions” diffeomorphic to spherical shells – Comm. Math. Phys., 319(2), pp. 451-477, 2013
• ALB, F. Leger, T. Laurent, aggregation patches and the Newtonial Potential, M3AS 2012.
• ALB, Jose A. Carrillo, and Thomas Laurent, Nonlinearity, 2009. - Osgood criteria for finite time blowup, similarity solutions in odd dimension
• ALB, Thomas Laurent, Jesus Rosado, CPAM 2011.
• Full L^p theory
• ALB and Jeremy Brandman, Comm. Math. Sci, 2010.
• L infinity weak solutions of the aggregation problem
• ALB and T. Laurent, Comm. Math. Phys., 274, p. 717-735, 2007.
• Finite time blowup in all space dimensions for pointy kernels

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Numerics and Asymptotics

• Kolokolnikov, Sun, Uminsky, Bertozzi, `world cup’ Phys. Rev. E 2011
• von Brecht, Uminsky, Kolokolnikov, ALB, M3AS 2012
• Von Brecht and Uminsky, 2012
• Yanghong Huang, ALB, SIAP 2010.
• Simulation of finite time blowup
• DCDS 2012 results for all powerlaw kernels
• Huang, Witelski, and ALB. Asymptotic theory to explain selection of anomalous exponents for blowup solution – in dimensions 3 and 5 for K=|x|. Applied Mathematics Letters 2012
• Sun, Uminsky, ALB – extension of 2D vortex sheets to problems with aggregating kernels –SIAP 2012.
• Von Brecht, Uminsky ALB- 3D pattern formation in particle aggregation systems,– M3AS 2012

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Papers and preprints on viscous aggregation equations

Topaz, Bertozzi, and Lewis, Bull. Math. Bio. 2006.

Bertozzi and Slepcev, CPAA, 2010

Rodriguez and Bertozzi – M3AS 2010 crime models.

Bedrossian, Rodriguez, and Bertozzi, Nonlinearity 2011

Bedrossian – CMS 2011, AML 2011.

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Students and Postdocs and Collaborators

• Masters: Flavien Leger
• PhD students: Jeremy Brandman, Yanghong Huang, Hui Sun, Nancy Rodriguez, Jacob Bedrossian, Jesus Rosado, James von Brecht
• Postdocs: Chad Topaz, Thomas Laurent, Dejan Slepcev, David Uminsky
• Collaborators: Jose Antonio Carrillo, John Garnett, Theo Kolokolnikov, Tom Witelski

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