# PDE Seminar Fall 2016

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

Date: **Friday, August 26, 2016**

- Speaker:
**Chenyun Luo, Johns Hopkins University** - Title: On the motion of the free surface of a compressible liquid with vorticity.

- Abstract: I would like to go over some recent results on the compressible Euler equations with free boundary. We first provide a new apriori energy estimates which are uniform in the sound speed, which leads to the convergence to the solutions of the incompressible Euler equations.This is a joint work with Hans Lindblad. On the other hand, the energy estimates can be generalized to the compressible water wave problem, i.e., the domain that occupied by the fluid is assumed to be unbounded. Our method requires the detailed analysis of the geometry ofthe moving boundary.

Date: **Friday, September 9, 2016**

- Speaker:
**Igor Kukavica, University of Southern California** - Title: On the existence and uniqueness of solutions to a fluid-structure system.

- Abstract: We address the system of partial differential equations modeling motion of an elastic body inside an incompressible fluid. The fluid is modeled by the incompressible Navier-Stokes equations while the structure is represented by the damped wave equation with interior damping. We will review the local for large and global existence results for small data. The global existence result is obtained for small initial data in a suitable Sobolev space and is based on an exponential decay of solutions. The results are joint with M. Ignatova, I. Lasiecka, and A. Tuffaha.

Date: **Friday, September 16, 2016**

- Speaker:
**Yixiang Wu, Vanderbilt University** - Title: Coexistence of competing species for intermediate dispersal rates in a reaction-diﬀusion chemostat model.

- Abstract: In this talk, we consider a diﬀusive chemostat model with two competing species. We ﬁrst present various results on the single species model, and show that small diffusion rate is beneﬁcial to the species. We then prove the existence of stable steady states of the two species model within certain parameter ranges. We explore the dynamics of the model, including proving that there is no coexistence steady state when the diﬀusion rate is small. Our result demonstrates that coexistence is possible only for intermediate diﬀusion rates. This is joint work with Junping Shi from College of William and Mary and Xingfu Zou from University of Western Ontario.

Date: **Friday, September 23, 2016**

- Speaker:
**Abbas Moameni, Carleton University (Canada)** - Title:New variational principles, convexity and supercritical semi-linear Elliptic problems.

- Abstract: The object of this talk is to present new variational principles for certain differential equations. These principles provide new representations and formulations for the superposition of the gradient of convex functions and symmetric operators. They yield new variational resolutions for a large class of hamiltonian partial differential equations with variety of linear and nonlinear boundary conditions including many of the standard ones. This approach seems to offer several useful advantages: It associates to a boundary value problem several potential functions which can often be used with relative ease compared to other methods such as the use of Euler-Lagrange functions. These potential functions are quite flexible, and can be adapted to easily deal with both nonlinear and homogeneous boundary value problems. Additionally, in some cases, this new method allows dealing with problems beyond the usual locally compactness structure (problems with a supercritical Sobolev nonlinearity).

Date: **Thursday, October 6, 2016 (Notice the date: Colloquium on a PDE topic).**

- Speaker:
**Lydia Bieri, University of Michigan** - Title: The Einstein Equations and Gravitational Waves.

- Abstract: In Mathematical General Relativity (GR) the Einstein equations describe the laws of the Universe. This system of hyperbolic nonlinear pde has served as a playground for all kinds of new problems and methods in pde analysis and geometry. A major goal in the study of these equations is to investigate the analytic properties and geometries of the solution spacetimes. In particular, fluctuations of the curvature of the spacetime, known as gravitational waves, have been a highly active research topic. Last year, gravitational waves were observed for the first time by LIGO. Understanding gravitational radiation is tightly interwoven with the study of the Cauchy problem in GR. I will talk about geometric-analytic results on gravitational radiation and the memory effect of gravitational waves. We will connect the mathematical findings to experiments. I will also address recent work with David Garfin- kle on gravitational radiation in asymptotically flat as well as cosmological spacetimes.

Date: **Friday, October 7, 2016**

- Speaker:
**Giusy Mazzone, Vanderbilt University** - Title:Asymptotic Behavior of Rigid Bodies with a Liquid-Filled Gap.

- Abstract: Click here.

Date: **Friday, October 28, 2016**

- Speaker:
**Georgi Kapitanov, University of Iowa** - Title: Linking Cellular and Mechanical Processes in Articular Cartilage Lesion Formation: A Mathematical Model.

- Abstract: Post-traumatic osteoarthritis affects almost 20% of the adult US population. An injurious impact applies a significant amount of physical stress on articular cartilage and can initiate a cascade of biochemical reactions that can lead to the development of osteoarthritis. In our effort to understand the underlying biochemical mechanisms of this debilitating disease, we have constructed a multiscale mathematical model of the process with three components: cellular, chemical, and mechanical. The cellular component describes the different chondrocyte states according to the chemicals these cells release. The chemical component models the change in concentrations of those chemicals. The mechanical component contains a simulation of a blunt impact applied onto a cartilage explant and the resulting strains that initiate the biochemical processes. The scales are modeled through a system of partial-differential equations and solved numerically. The results of the model qualitatively capture the results of laboratory experiments of drop-tower impacts on cartilage explants. The model creates a framework for incorporating explicit mechanics, simulated by finite element analysis, into a theoretical biology framework. The effort is a step toward a complete virtual platform for modeling the development of post-traumatic osteoarthritis, which will be used to inform biomedical researchers on possible non-invasive strategies for mitigating the disease.

Date: **Friday, November 11, 2016**

- Speaker:
**Elaine Cozzi, Oregon State University** - Title: Incompressible Euler equations and the effect of changes at a distance.

- Abstract: Because pressure is determined globally for the incompressible Euler equations, a localized change to the initial velocity will have an immediate effect throughout space. For solutions to be physically meaningful, one would expect such effects to decrease with distance from the localized change, giving the solutions a type of stability. One can easily show that this is the case for sufficiently smooth solutions having spatial decay. In this talk, we consider a broader class of weak solutions with vorticity lacking spatial decay, and we show that such stability still holds. This is based on joint work with James Kelliher.

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