## Applied Math Talk: Patterns deformed by spatial inhomogeneity give by Prof. Jasper Weinburd (HMC)

### November 25, 2019 @ 4:15 pm - 5:15 pm

At the turn of the twentieth century, physicist Henri Bénard heated a shallow plate of fluid from below. For temperatures above a critical value, the fluid’s evenly heated state became unstable as thermal convection took hold; heated fluid rose in localized areas while cooler fluid fell nearby. The rising and falling fluid created hexagonal convection cells, squares, and stripes.

Suppose that we modify Bénard’s experiment by heating only the left half plate. We expect the fluid on the right to remain stationary and only the the fluid on the left to form patterns. We confirm this intuition mathematically and, more surprisingly, find that the step-type inhomogeneity restricts the spatial period of the resulting patterns on the left. We examine this phenomenon using a universal partial differential equation model. The main difficulty arrises at the location of the discontinuous inhomogeneity because results on either side cannot be directly compared. We construct a transformation of variables that bridges this jump and allows a heteroclinic glueing argument from left to right. The explicit form of this transformation determines the widths of patterns that may occur in the inhomogeneous environment.