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Weak Solutions of Kinematic Aggregation Equations Andrea Bertozzi Department of Mathematics UCLA Thanks to Thomas Laurent for contributing to the slides. Slide 2 Finite time singularities- general potentials 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,... X X X X X X XX X X X X X X X X Bertozzi, Carrillo, Laurent Nonlinearity, Feature Article, 2009 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 L p at blowup time for some p. Slide 3 Local existence of L p solutions 3 ALB, Laurent, Rosado, to appear in CPAM Slide 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: 4 Slide 5 5 Slide 6 6 Slide 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 7 Slide 8 8 Slide 9 9 Slide 10 10 A vector field pushes the metric on the manifold according to the continuum equation Slide 11 11 Slide 12 What are the advantages of this approach? 12 Slide 13 Proof of blowup for general measure solutions 13 Slide 14 Back to the L p 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 L p bound for Osgood kernel see next slide 14 Slide 15 Global Existence in L p : a priori L p bound Bertozzi, Laurent, Rosado To appear CPAM 2010 Define a length scale using L p and L 1 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 L p norm. Slide 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 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. Slide 17 Blowup in from bounded data ALB, 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 L p space at the blowup time. Is it good p or bad p? 17 Slide 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 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 Slide 19 Simulations by Y. Huang Second kind Exact self-similar Bad p! Can not continue solution in time as Lp weak solution Slide 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 Slide 21 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). Second kind similarity solution for Burgers equation Slide 22 Weak L p solutions and instantaneous mass concentration 22 Slide 23 Instantaneous mass concentration 23 Slide 24 Proof of instantaneous mass concentration 24 Slide 25 Instantaneous Mass Concentration ALB, Laurent, Rosado, CPAM to appear 25 Slide 26 26 Critical L p exponents for local well-posedness vs. ill-posedness Instantaneous mass concentration possible for initial data in L p for p < d/(d-1) Slide 27 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 Shape of singularity- pointy potential Huang and Bertozzi preprint 2009 Slide 28 Local existence of solutions in L p 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 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. 28 L p well-posedness for general potential Bertozzi, Laurent, and Rosado manuscript submitted 2009 Slide 29 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. 29 L p well-posedness for general potential Bertozzi, Laurent, and Rosado manuscript in preparation 2009 Slide 30 Attractive potential - regularity of potential determines whether general solutions blow-up in finite time (both for bounded and L p data, p>p c ). 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