Kofroth, Collin. Local Energy Decay for Damped Waves on Stationary, Asymptotically Flat Space-Times. PhD Dissertation, 2022. 114 pages. ProQuest Dissertations and Theses. (2665564597)
Kofroth, Collin. Asymptotic Stability of Biharmonic Shallow Water Equations. Honors Thesis, 2017. 45 pages. ASU KEEP.
More writings (notes and course projects) can be found here.
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 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. 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)
Undergraduate Honors Thesis
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
Undergraduate Research Experiences
In the summer of 2016, I took part in the MTCP/CSUMS REU, and I performed two projects simultaneously. I was funded to present my research at the 2017 Joint Mathematics meeting. My project topics:
My first project, performed under the guidance of Dr. Mohamed Moustaoui, concerned the study of momentum transport through stably-stratified flows. It involved studying mountain wave phenomena, wave-wave interactions, and momentum transport through the tropopause. We derived analytical solutions and performed numerical analysis of the two-layer and three-layer models, including altitude-dependent profiles such as those involving wind shear. We were able to compare to real data obtained using the WRF model to validate our models.
My second project, performed under the guidance of Dr. Don Jones, studied how the choice of numerical discretization affected the dynamics of nonlinear partial differential equations. Using the shallow water equations as a test case, we performed numerical experiments to observe the dynamical effects from the choices of mesh, time scheme, time step, and filter. We utilized stability analysis, dispersive relations of (linearized, inviscid) SWE, kinetic energy analysis, and spectral analysis to determine which choices were superior. We came to the conclusion that the Arakawa B-grid was the most stable spatial discretization when varying other parameters (along with displaying better convergence properties and behavior at lower resolutions).
Velocity contours of gravity waves with wind shear profile