1. Bounded Solutions of Kinematic Aggregation Equations Andrea Bertozzi Department of Mathematics, UCLA Thanks to support from ARO, NSF, and ONR and to contributions from numerous collaborators. X X X X X X XX X X X X X X X X

2. Propagation of constant density groups in 2D Conserved population Velocity depends nonlocally, linearly on density Topaz and Bertozzi (SIAM J. Appl. Math., 2004) Assumptions: incompressibility Potential flow

3. Mathematical model Sense averaged nearby pop. Climb gradients K spatially decaying, isotropic Weight 1, length scale 1 Social attraction: X X X X X X XX X X X X X X X X Topaz, Bertozzi, and Lewis Bull. Math. Bio. 2006

4. Mathematical model Sense averaged nearby pop. Climb gradients K spatially decaying, isotropic Weight 1, length scale 1 X X X X X X XX X X X X X X X X Descend pop. gradients Short length scale (local) Strength ~ density Characteristic speed r Social attraction: Dispersal (overcrowding): X X X X X X XX X X X X X X X X

5. Finite time singularities- pointy potentials Previous Results For smooth K the solution blows up in infinite time For n=1, and `pointy’ K (biological kernel: K=e -|x| ) blows up in finite time due to delta in K xx X X X X X X XX X X X X X X X X Bertozzi and Laurent Comm. Math. Phys. 2007 2007 result: For `pointy’ kernel one can have smooth initial data that blows up in finite time in any space dimension. Proof: uses Lyapunov function and some potential theory estimates. Well posedness proved via classical Sobolev space estimates.

6. Finite time singularities- pointy potentials Local well-posedness of the problem with initial data in Existence theory uses method of characteristics - anologous to Yudovich theory for 2D Euler. X X X X X X XX X X X X X X X X Bertozzi and Brandman Comm. Math. Sci. 2009 - Majda issue Harder than Euler case- we have to pass to the limit in the amplification term. Easier than Euler case- the potential K is less singular, moreover bounded data implies more regularity for characteristic velocity. Uniqueness - proved using primitive equation as in Yudovich. Finite time blowup as in BL 2007 paper. For the aggregation problem, result in all space dimensions, N>1.

7. Finite time singularities- general potentials 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. X X X X X X XX X X X X X X X X Bertozzi Carrillo, Laurent Nonlinearity `featured article’ 2009 New result: Osgood condition 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,...

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9. 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. 9

10. Collapse of particles in the ODE 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 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

11. 11 Proof (non-Osgood case):

12. What about continuum PDE? Finite time blowup – proof follows the argument for collapse of ODEs, only for continuum density (some analysis). Global existence – requires an a priori UPPER bound for the density Known for C 2 kernels, what about the more general Osgood case? First prove for C 2 kernels…. 12

13. Proof of global existence: Connection to 3D Euler Bertozzi, Carrillo, Laurent Nonlinearity 2009 Vorticity Stream form of 3D Euler Equations omega is vector vorticity and K 3 is Biot-Savart Kernel in 3D

14. Finite time singularities- general potentials Bertozzi, Carrillo, Laurent Nonlinearity 2009 l2l2 l1l1

15. Finite time singularities- general potentials Bertozzi, Carrillo, Laurent Nonlinearity 2009

16. Finite time singularities- general potentials Bertozzi, Carrillo, Laurent Nonlinearity 2009

17. Finite time singularities- general potentials Bertozzi, Carrillo, Laurent Nonlinearity 2009 COMPARISON PRINCIPLE: 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 Thus the Osgood criteria provides a necessary and sufficient condition on the potential K for finite time blowup from bounded data.

18. Similarity solution of form The equation implies Conservation of mass would imply - NO Second kind similarity solution - no mass conservation Experimentally, the exponents vary smoothly with dimension of space, and there is no mass concentration in the blowup.... Shape of singularity- pointy potential Huang and Bertozzi preprint 2009-radially symmetric numerics ``Finite time blowup for `pointy’ potential, K=|x|, can not be described by `first kind’ similarity solution in dimensions N=3,5,7,...’’ - Bertozzi, Carrillo, Laurent

19. Simulations by Y. Huang Second kind Exact self-similar

20. 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. 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. 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. Shape of singularity- pointy potential Huang and Bertozzi preprint 2009

21. 21 Papers - Inviscid Aggregation Equations Andrea L. Bertozzi, Jose A. Carrillo, and Thomas Laurent, Nonlinearity, 2009. - Osgood criteria for finite time blowup, similarity solutions in odd dimension Andrea L. Bertozzi, Thomas Laurent, Jesus Rosado, CPAM to appear 2010. –Full L p theory Yanghong Huang, Andrea L. Bertozzi, submitted to SIAP, 2009. –Simulation of finite time blowup Andrea L. Bertozzi and Jeremy Brandman, Comm. Math. Sci, 2010. –L infinity weak solutions of the aggregation problem A. L. Bertozzi and T. Laurent, Comm. Math. Phys., 274, p. 717-735, 2007. –Finite time blowup in all space dimensions for pointy kernels 21