Conference abstracts

Session B3 - Symbolic Analysis - Semi-plenary talk

July 15, 14:30 ~ 15:20 - Room B2

The Good, the Bad, and the Ugly: the Cartan algorithm for overdetermined PDE systems

Jeanne Clelland

University of Colorado, Boulder, USA   -   Jeanne.Clelland@colorado.edu

Cartan's theory of exterior differential systems provides tools for analyzing spaces of local solutions to systems of PDEs that don't fit nicely into any standard classification. In particular, the Cartan-Kahler Theorem -- which generalizes the Cauchy-Kowalewski Theorem for determined systems -- can often be used to compute the size of the local solution space, as long as the system itself and all initial data are assumed to be real analytic. This can be a powerful framework for analyzing the solution spaces to overdetermined PDE systems, including many that arise naturally in geometric contexts.

In this talk, we will illustrate the application of Cartan's algorithm to the problem of finding strong Beltrami fields with nonconstant proportionality factor, i.e., vector fields $\mathbf{u}$ on an open set $U \subset \mathbb{R}^3$ with the property that \[ \nabla \times \mathbf{u} = f \mathbf{u}, \qquad \nabla \cdot \mathbf{u} = 0 \] for some nonconstant function $f:U \to \mathbb{R}$. This example illustrates both the power of the method and some significant challenges that may arise during its application.

Joint work with Taylor Klotz (University of Colorado, Boulder).

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