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## Research Papers

The hyperlinked article titles will bring you to the most up-to-date (and, hopefully, correct) versions of each paper.

Kofroth, Collin. Integrated Local Energy Decay for Damped Magnetic Wave Equations on Stationary Space-Times. Submitted for publication, 2023.

Kofroth, Collin.Accepted for publication in the SIAM Journal on Mathematical Analysis, 2022.

Kofroth, CollinPhD Dissertation, 2022. 114 pages. ProQuest Dissertations and Theses. (2665564597)

Kofroth, CollinAsymptotic Stability of Biharmonic Shallow Water Equations. Honors Thesis, 2017. 45 pages. ASU KEEP.

More writings (notes and course projects) can be found here.

## Dissertation Research

During my time at UNC, I completed a Ph.D. dissertation under Dr. Jason Metcalfe. In my dissertation research, I studied local energy estimates for damped wave equations on asymptotically flat space-times, with the goal of establishing local energy decay. Local energy decay is a particularly powerful measure of dispersion which implies that the energy of a wave must decay over time within any bounded region in space. It is provably unavailable for non-damped waves on space-times with trapping, which contain places where waves (and their energy) must live for an infinite amount of time. Examples of space-times with trapping are black holes. Black holes possess a region called the photon sphere where light is trapped forever, stuck sitting between the region where it can escape the enormous gravitational effect of the black hole and the region where it succumbs to this pull and is torn into the black hole (the latter region is called the event horizon). Such a phenomenon makes genuine energy decay for non-damped waves impossible within this region.

To that end, we consider the related problem of analyzing local energy decay of damped waves on backgrounds that allow for trapping. Damping creates energy dissipation, and we enforce a condition called geometric control which stipulates that all trapped trajectories must encounter the region where the damping takes effect. Since local energy decay is known for waves on non-trapping space-times, the hope is that the aforementioned modifications allow us to enforce decay in the trapped region – the only place where issues occur (and, indeed, this is the content of my dissertation). This involves utilizing tools from spectral theory, microlocal analysis, and classical PDE theory (e.g. elliptic theory).

An example of geometric control on a trapped set (blue-shaded region represents the support of the damping in phase space)

During my undergraduate studies, I completed an honors thesis under Dr. Don Jones. In my thesis, I proved a global existence result on the biharmonic shallow water equations, a set of three nonlinear partial differential equations modeling the behavior of a hydrostatically-balanced fluid (velocity varies negligibly in the vertical direction). The results were established using a priori energy estimates and bootstrapping arguments.

Height field contours of oceanic basin modeled with biharmonic SWE