Mathematics Colloquium, Spring 2004
Special Lecture presented by

Dynamic Scaling in Models of Coalescence

The ideas of scaling and self-similarity are central principles in physics. Scaling laws have a natural appeal: they are experimentally robust, and often rely only on back of the envelope calculations. Yet they are hard to pin down with complete rigor. An important class of "dynamic" scaling laws describe "coarsening". This is the process of mass transfer from small to large scales. For example, the formation of raindrops in clouds, the separation of oil and water, and the gravitational accretion of dust into planets. I will survey rigorous results for coarsening in: (1) A free surface instability in fluid mechanics (the Saffman-Taylor instability). (2) A mean-field model of coalescence (Smoluchowski's coagulation equation).

A common theme is coarsening by coalescence, but the methods are distinct. In problem (1), we use PDE methods (scale-invariant estimates, hyperbolic conservation laws) to describe the evolution in regimes of physical interest. In problem (2), there is a striking duality with classical probabilistic limit theorems (domains of attraction in the normal law, Levy-Khintchine characterizations, Doeblin's universal laws). Here I will describe both our work, and some deeper probabilistic insights of Jean Bertoin.

Problem (1) is joint work with Felix Otto; problem (2) with Bob Pego.

Thursday, January 15, 2004, 245 Altgeld Hall, 4:00 p.m.