Steven Wise 
University of Tennessee at Knoxville

More information coming soon. 

Friday, March 22, 4pm-5:15pm
Rolla Building, Room G5

Kenji Nakanishi
Professor
Research Institute for Mathematical Sciences
Kyoto University

Title:  Global dynamics of the nonlinear Schrödinger equation with potential

Abstract: The nonlinear Schrödinger equation is a typical nonlinear disperisve PDE, which describes various wave phenomena in terms of dispersion and nonlinear interactions of the waves. Such PDEs are known to generate various types of solutions, typically scattering, solitons and blow-up. The goal of this study is to classify the solutions in terms of their behavior forward and backward in time, and to predict it from the initial data. In this talk, we consider the equation with a cubic nonlinearity and a linear potential, so that we can characterize the ground state solitons and the first excited solitons for small mass. The main result is classification into 9 sets of solutions with small mass under an energy constraint slightly above the first excited solitons.

Host: Murphy

Friday, February 1, 4pm-5:15pm
Rolla Building, Room G5


Abner Salgado
Associate Professor
Department of Mathematics
University of Tennessee

Title: Regularity and rate of approximation for obstacle problems for a class of integro-differential operators

Abstract: We consider obstacle problems for three nonlocal operators: 

A) The integral fractional Laplacian

B) The integral fractional Laplacian with drift

C) A second order elliptic operator plus the integral fractional Laplacian

For the solution of the problem in Case A, we derive regularity results in weighted Sobolev spaces, where the weight is a power of the distance to the boundary. For cases B and C we derive, via a Lewy-Stampacchia type argument, regularity results in standard Sobolev spaces. We use these regularity results to derive error estimates for finite element schemes. The error estimates turn out to be optimal in Case A, whereas there is a loss of optimality in cases B and C, depending on the order of the integral operator.

Biographical Sketch:  Dr. Abner J. Salgado is an Associate Professor at the Department of Mathematics at the University of Tennessee, Knoxville. His research revolves around the Numerical Analysis of partial differential equations and related questions. He is interested in the design, analysis and implementation of approximation schemes for complex fluids, nonlocal problems, degenerate and singular diffusion problems, and nonlinear partial differential equations in general.

Host: He

Friday, January 25, 4pm-5:15pm
Rolla Building, Room G5

Ari Stern
Associate Professor
Department of Mathematics
Washington University in St. Louis

Title: Hybrid finite element methods for geometric PDEs

Abstract: Many geometric PDEs have local properties, such as symmetries and conservation laws, that one might wish a numerical method to preserve. With classical finite element methods, it is difficult to make sense of such properties, except in a weak or averaged sense. I will discuss how hybrid finite element methods, based on non-overlapping domain decomposition, provide a natural framework for talking about such properties. Specifically, I will discuss some recent results obtained by applying this approach to the multisymplectic conservation law for Hamiltonian PDEs (joint with Robert McLachlan), as well as to charge conservation in Maxwell's equations (joint with Yakov Berchenko-Kogan).

Biographical Sketch:  Dr. Ari Stern is Associate Professor of Mathematics and Statistics at Washington University in St. Louis, where he has been since 2012. Before this, he received a BA and MA in Mathematics from Columbia University and a PhD in Applied and Computational Mathematics from Caltech, and he did postdoctoral work at UCSD. His research is primarily focused on structure-preserving numerical methods for differential equations, particularly for systems with symmetries, conservation laws, or other geometric structures.

Host: He