# PDE Seminar Fall 2015

### Fridays, 4:10pm, Stevenson Center 1307

Date: ** Thursday, September 24, 2015 **(notice the date; this is a colloquium on a PDE topic)

- Speaker:
**Marcelo Disconzi, Vanderbilt University** - Title: From mathematics to cosmology.

- Abstract: In this talk, I will present the results of a recent paper that has received widespread media coverage. The paper in question was written in collaboration with Robert Scherrer and Thomas Kephart from the Physics department, and grew out of my earlier results on the existence, uniqueness, and causality of Einstein’s equations coupled to the Navier-Stokes equations, known as the Einstein-Navier-Stokes system (ENSS). The first part of the talk will be a fairly mathematical discussion of the ENNS. The second part will describe the work with Scherrer and Kephart, where we applied the ENSS to cosmology. The main implication is that the resulting model favors a scenario known as “big rip,” where the universe comes to an end in finite time (approximately 22 billion years from now). If time allows, I will say a few words about the press coverage.

Date: **Friday, Sept 25, 2015**

- Speaker:
**Mihai Tohaneanu, University of Kentucky** - Title: Global existence for quasilinear wave equations close to Schwarzschild.

- Abstract: We study the quasilinear wave equation $\Box_{g} u = 0$, where the metric $g$ depends on $u$ and equals the Schwarzschild metric when u is identically 0. Under a couple of extra assumptions on the metric $g$ near the trapped set and the light cone, we prove global existence of solutions. This is joint work with Hans Lindblad.

Date: ** Thursday, October 8, 2015 **(notice the date; this is a colloquium on a PDE topic)

- Speaker:
**Mihalis Dafermos, Princeton** - Title: TBA.

- Abstract: TBA.

Date: **Friday, October 30, 2015**

- Speaker:
**Brian Allen, University ot Tennessee at Knoxville** - Title: Non-Compact Inverse Mean Curvature Flow in Euclidean Space.

- Abstract: Inverse Mean Curvature Flow (IMCF) is an important geometric evolution equation that has been used to prove interesting geometric inequalities, most notably the Riemannian Penrose Inequality from General Relativity. In this talk we will discuss my result of long time existence of IMCF for bounded graphs over cylinders and explore the asymptotic properties we expect for the flow.

Date: ** Thursday, November 12, 3:10pm at SC 1312 **(notice the date and time)

- Speaker:
**Jozsef Farkas, University of Stirling (UK)** - Title: Mathematical Analysis of a Clonal Evolution Model of Tumour Cell Proliferation

- Abstract: We introduce and analyse a partial differential equation model of a cancer cell population, which is structured with respect to age and telomere length of cells. We assume a continuous telomere length structure, which corresponds to the clonal model of tumour cell growth. This assumption leads to a model with a non-standard non-local boundary condition. We establish global existence of solutions and study their qualitative behaviour. In particular, we study the effect of telomere restoration on cancer cell dynamics. Our results indicate that without telomere restoration, the cell population typically dies out. On the other hand, with telomere restoration, we may observe exponential growth in the linear model. We also study the effects of crowding induced mortality on the qualitative behaviour, and study the existence and stability of steady states of a nonlinear model with crowding effect.

Date: **Friday, December 4, 2015**

- Speaker:
**Christoph Walker, Leibniz Unversität Hannover (Germany)** - Title: A Free Boundary Problem Modeling MEMS.

- Abstract: Idealized electrostatic microelectromechanical systems (MEMS) consist of a fixed ground plate above which an elastic plate is suspended. The elastic plate deforms due to a voltage difference that is applied between the two components. The corresponding mathematical model involves the harmonic electrostatic potential in the free domain between the two plates along with an evolution equation for the displacement of the elastic plate. Of particular interest is the dynamics in dependence of the voltage difference applied between the plates. In this talk some results are presented on the number of stationary solutions and on the singularity that possibly occurs in the evolution when the elastic plate touches down on the ground plate.

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