The Constantin-Lax-Majda equation (CLM) was proposed as a one dimensional model for the three dimensional vorticity of an incompressible fluid.
We provide background on the CLM model and discuss some of its drawbacks. Several of its proposed generalizations are also considered. In particular, blowup results and criteria leading to global solutions among these models are compared.
The Constantin-Lax-Majda equation and generalizations thereof
Apr. 08, 2014 1pm (MATH 220)
Grad Algebra/Logic
Jeffrey Shriner (CU Boulder)
X
A chip-firing game on a graph G begins with a configuration (the number of 'chips' on each vertex). When a vertex 'fires', it sends a chip to each adjacent vertex, so the only rule of the game is that a vertex may 'fire' if and only if it has at least deg(v) chips. We discuss a variant of the chip-firing game, called the dollar game, in which one vertex is required to always be in debt. Using the dollar game, we form an abelian group structure K(G), called the critical group of G. We compute the order of K(G), and look at how the dollar game may be used as a tool for analyzing the structure of the group.
The Dollar Game and the Critical Group of a Graph, Part 1
Let be an elliptic curve over with surjective mod-2 representation. Let be the splitting field of the -division polynomial of . We show that for every element of the cohomology group , there exists , a mod-2 companion of and such that maps to under the Kummer map. I will also discuss how a finite subgroup of this cohomology group is related to -extensions of that contain and are unramified outside a finite set of primes. (joint work with Edray Goins and Jamie Weigandt)
The Constantin-Lax-Majda equation and generalizations thereof