Leonardo Abbrescia, Michigan State University
Location: Stevenson 1307

We will present a series of geometric ideas that are helpful to study the initial value problem of quasilinear wave equations satisfying the null condition on the (1+1)-dimensional Minkowski space. Using a double-null geometric formulation, we show how the conformal invariance of the equation semilinearizes it into a system that is decoupled from the equations governing the null geometry. This allows us to solve the wave equations independently, which we exploit to show that the null geometry is sufficiently regular to guarantee global existence. If time permits, I will explain how this ties into a global wellposedness result with “large” initial data. This is joint work with Willie Wong.

Location: Stevenson Center 1 (Math Building)

Location: Stevenson Center 1 (Math Building)

Rares Rasdeaconu, Vanderbilt University
Location: Stevenson 1320

Marek Kaluba, Adam Mickiewicz University, Poland
Location: Stevenson 1308

Peter Bonventre, University of Kentucky
Location: Stevenson 1320

Zack Tripp, Vanderbilt University
Location: Stevenson 1206

Max Gunzburger, Florida State University
Location: Stevenson 1206

Nonlocal continuum models allow for interactions between a point and other points separated by a nonzero distance, in contrast with PDE models for which interactions occur only in infinitesimal neighborhoods surrounding the point. If the extent of interactions are limited to be no greater than a finite distance, then a length scale is introduced into the models that renders them as being multiscale mono-models, by which we mean that depending on the size of the viewing window used relative to the extent of nonlocal interactions, a single model can display very different behaviors. Some nonlocal models that are spatial-derivative free also allow for discontinuous solutions which make them well suited for simulations of fracture and other settings. We discuss theories for the analysis and numerical analysis of the nonlocal models considered, relying on a nonlocal vector calculus to define weak formulations in function space settings. Brief forays into examples and extensions are made, including obstacle problems and wave problems.

Michael Hull, University of North Carolina, Greensboro
Location: Stevenson 1308

Julio Caceres, Vanderbilt University
Location: Stevenson 1432

Ben Hayes, University of Virginia
Location: Stevenson 1432

Juhi Jang, University of Southern California
Location: Stevenson 1206

Thomas Sinclair, Purdue University
Location: Stevenson 1432

Casey Rodriguez, Massachusetts Institute of Technology
Location: Stevenson 1307

Ellen Weld, Purdue University
Location: Stevenson 1432

Jonathan Campbell, Duke University
Location: Stevenson 1320

Dan Ginsberg, Princeton University
Location: Stevenson 1307

Ravshan Ashurov, Institute of Mathematics, National University of Uzbekistan
Location: Stevenson 1307

Historically progress with solving the Luzin conjecture has been made by considering easier problems. For multiple Fourier series one of such easier problems is to investigate convergence almost everywhere of the spherical sums on TN\ supp(f) (so called the generalized localization principle). For the spherical partial integrals of multiple Fourier integrals the generalized localization principle in classes Lp(RN) was investigated by many authors. In particular, in the remarkable paper of A. Carbery and F. Soria the validity of the generalized localization was proved in Lp(RN) when 2 <= p < 2N/(N – 1). In this paper the generalized localization principle for the spherical partial sums of the multiple Fourier series in L2 – class is proved. It was previously known that the generalized localization was not valid in classes Lp(TN) when 1 <= p = 1: if p >= 2 then we have the generalized localization and if p < 2, then the generalized localization fails.