Weak Solutions of Kinematic Aggregation Equations

1. Weak Solutions of Kinematic Aggregation Equations Andrea Bertozzi Department of Mathematics UCLA Thanks to Thomas Laurent for contributing to the slides.

2. Finite time singularities- general potentials X X X X X X X X X X X X X X X X Bertozzi, Carrillo, Laurent Nonlinearity, Feature Article, 2009 Last talk addressed finite time blowup from bounded initial conditions. Moreover-finite time blowup for pointy potential can not be described by `first kind’ similarity solution in dimensions N=3,5,7,... Kernel condition: Osgood condition is a necessary and sufficient condition for finite time blowup in any space dimension (under mild monotonicity conditions). Numerics show second similarity solution blowup behavior – leads to a solution that is still in Lp at blowup time for some p.

3. Local existence of Lp solutions ALB, Laurent, Rosado, to appear in CPAM

4. Measure solutions • New preprint: Carrillo, Di Francesco, Figalli, Laurent, Slepcev (5 author paper) • Uses Optimal Transport ideas • Gradient flow in Wasserstein metric • Flow on space of probability measures with bounded second moment • Kernel is lambda-convex:

7. Gradient flow via Wasserstein metric • The notion of gradient flow in the space of probability measures endowed with the Wasserstein metric was introduced in • 1998 Fokker Planck equation, R Jordan D Kinderlehrer, F. Otto • 2001 Porous Media Equation F. Otto

10. A vector field pushes the metric on the manifold according to the continuum equation

12. What are the advantages of this approach?

13. Proof of blowup for general measure solutions

14. Back to the Lp case ALB, Laurent, Rosado, CPAM to appear • 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 Lp bound for Osgood kernel – see next slide

15. Global Existence in Lp: a priori Lp bound Bertozzi, Laurent, Rosado To appear CPAM 2010 Define a length scale using Lp and L1 norms: Again, using potential theory estimates one can prove: For an Osgood kernel one has an a priori pointwise lower bound for the lengthscale R and hence an a priori upper bound for the Lp norm.

16. 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 BR(t)(xm) contains the support, xm is center of mass (conserved), and Thus the Osgood criteria provides a necessaryand sufficient condition on the potential K for finite time blowup from bounded data.

17. Blowup in from bounded dataALB, Laurent, Rosado, CPAM to appear • Local existence using method of characteristics and some analysis (ALB, Laurent) • Uniqueness using optimal transport theory (Rosado) • Finite time singularity forms a power-law singularity – which means it is in some Lp space at the blowup time. Is it good p or bad p?

18. Shape of singularity- pointy potential Huang and Bertozzi Submitted SIAM J. Appl. Math. 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 • 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....

19. Simulations by Y. Huang Second kind Exact self-similar Bad p! Can not continue solution in time as Lp weak solution

20. In one dimension, K(x) = |x|, even initial data, the problem can be transformed exactly to Burgers equation for the integral of u. • Burgers has a well known non-smooth similarity solution –x/t • Discuss • However this is NOT the generic behavior for smooth initial data

21. Second kind similarity solution for Burgers equation • 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 corresponds 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. • Exercise (hard) to give to your students in first year PDE (hint solve along characteristics).

22. Weak Lp solutions and instantaneous mass concentration

23. Instantaneous mass concentration

24. Proof of instantaneous mass concentration

25. Instantaneous Mass ConcentrationALB, Laurent, Rosado, CPAM to appear

26. Critical Lp exponents for local well-posedness vs. ill-posedness Instantaneous mass concentration possible for initial data in Lp for p < d/(d-1)

27. Shape of singularity- pointy potential Huang and Bertozzi preprint 2009 • Aggregation equation in higher dimensions, radially symmetric focusing solution: • Another open problem - what is the shape of the finite-time blowup for different powerlaw potentials, i.e. K(x) = |x|r? For r>=2 blowup is in infinite time - approaches delta-ring solution. 27

28. Lp well-posedness for general potential Bertozzi, Laurent, and Rosado manuscript submitted 2009 • Local existence of solutions in Lp provided that • 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 Lp provided that K satisfies the Osgood condition (derivation of a priori bound for Lp norm - similar to refined potential theory estimates in BCL 2009). • When Osgood condition is violated, solutions blow up in finite time - implies blowup in Lpfor all p>pc. 28

29. Lp well-posedness for general potential Bertozzi, Laurent, and Rosado manuscript in preparation 2009 • Ill-posedness of the problem in Lp for p less than the Holder-critical pc associated with the potential K. • Ill-posedness results because one can construct examples in which mass concentrates instantaneously (for all t>0). • For p> pc , uniqueness in Lp 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. 29

30. Conclusions and Remarks • Attractive potential - regularity of potential determines whether general solutions blow-up in finite time (both for bounded and Lp data, p>pc). Osgood criteria for K. • Radial case - numerics of blowup show second-kind similarity solution - may be related to Burgers shock for odd data in 1D. • 1<p<pcexact solutions prove that Lp problem is ill-posed due to instantaneous mass concentration. • Measure valued data - solutions exist for all time, with possible information loss. • Information loss once a singularity forms, but not before that, for non-Osgood potentials. • Multidimensional `focusing’ analog of shock formation. • No singularities for Osgood potentials.

31. 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 Lp 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 31