Mathematics

Recent work on the regularity problem of some variation of 2D Boussinesq equations

Lizheng Tao (UIUC Math)

Abstract: In this talk, I will present my previous work for the 2D Boussinesq equations. Regularity results are achieved for a slightly super-critical variation of the standard equations. A modified Besov space will be introduced to handle the logarithmically super-critical operator.

Properties of minimizers of the Lawrence-Doniach energy with perpendicular magnetic fields

Guanying Peng (Purdue Math)

Abstract: We analyze minimizers of the Lawrence-Doniach energy for layered superconductors occupying a bounded generalized cylinder. For an applied magnetic field in the intermediate regime that is perpendicular to the layers, we prove an asymptotic formula for the minimum Lawrence-Doniach energy as the reciprocal of the Ginzburg-Landau parameter and the interlayer distance tend to zero. Under an appropriate assumption on the relationship between these two parameters, we establish comparison results between the minimum Lawrence-Doniach energy and the minimum 3D anisotropic Ginzburg-Landau energy.

Dispersive Estimates for Schrodinger and Wave Equations

Will Green (Rose-Hulman Institute of Technology)

Abstract: In this talk we will investigate recent work on Schrodinger and wave equations. In particular, we will discuss the time decay of the solution operators to these equations. We will survey recent results that show the effect of obstructions at zero energy, eigenfunctions and/or resonances, have on the time decay rates. We will also discuss some applications of these estimates to certain non-linear equations.

The zero set of a random Fourier polynomial on the sphere

Erik Lundberg   [email] (Purdue University)

Abstract: We start with a question motivated by the fundamental theorem of algebra: How many zeros of a random polynomial are real? We discuss three Gaussian ensembles that lead to three different answers. Of these, we emphasize the harmonic analyst’s model of choice which has the highest expected number of zeros (a fraction of the maximum) and reduces to a random trigonometric polynomial. The real section of the zero set of a polynomial in several variables is much more complicated. Hilbert’s sixteenth problem asks to study the possible arrangements of the connected components, and is especially concerned with the case of many components. I will describe a probabilistic approach to studying the topology, volume, and arrangement of the zero set (in real projective space) for a Gaussian ensemble of homogeneous polynomials. Again we will emphasize the harmonic analyst's model for random polynomials which is built out of a basis of spherical harmonics (eigenfunctions of the spherical Laplacian). This work is joint with Antonio Lerario.

Enhancement of biological reaction by chemotaxis

Yao Yao (U of Wisconsin-Madison)

Abstract: In this talk, we consider a system of equations arising from reproduction processes in biology, where two densities evolve under diffusion, absorbing reaction and chemotaxis. We prove that chemotaxis plays a crucial role to ensure the efficiency of reaction: Namely, the reaction between the two densities is very slow in the pure diffusion case, while adding a chemotaxis term greatly enhances reaction. While proving our main results we also obtain a weighted Poincare's inequality for the Fokker-Planck equation, which might be of independent interest. This is a joint work with A. Kiselev, F. Nazarov and L. Ryzhik.

Gibbs' measure and almost sure global well-posedness for one dimensional periodic fractional Schr\"odinger equation.

Seckin Demirbas (UIIUC Math)

Abstract: In this talk we will present recent local and global well-posedness results on the one dimensional periodic fractional Schr\"odinger equation. We will also talk about construction of Gibbs' measures on certain Sobolev spaces and how we can prove almost sure global well-posedness using this construction.

Tire tracks, the stationary Schrodinger's equation and forced vibrations.

Mark Levi   [email] (Penn State Math)

Abstract: I will describe a newly discovered equivalence between the first two objects mentioned in the title. The stationary Schrodinger's equation, a.k.a. Hill’s equation, is ubiquitous in mathematics, physics, engineering and chemistry. Just to mention one application, the main idea of the Paul trap (for which W. Paul earned the 1989 Nobel Prize in physics) amounts to a certain property of Hill's equation. As it turns out, Hill's equation is equivalent to a seemingly completely unrelated problem of “tire tracks”. In addition to this equivalence, I will describe a yet another connection between the ``tire tracks” problem and the high frequency forced vibrations.

A Morse index theorem for elliptic operators on bounded domain.

Graham Cox (UNC Chapel Hill)

Abstract: The Maslov index is a symplecto-geometric invariant that counts signed intersections of Lagrangian subspaces. It was recently shown that the Maslov index can be used to compute Morse indices of Schrodinger operators on star-shaped domains. We extend these results to general selfadjoint, elliptic operators on domains with arbitrary boundary geometry, and discuss some applications. (Joint work with C. Jones and J. Marzuola)

Layered Media Scattering: Fokas Integral Equations and Boundary Perturbation Methods

David Nicholls (UIC)

Abstract: In this talk we describe a class of Integral Equations to compute Dirichlet-Neumann operators for the Helmholtz equation on periodic domains inspired by the recent work of Fokas and collaborators on novel solution formulas for boundary value problems. These Integral Equations have a number of advantages over standard alternatives including: (i.) ease of implementation (high-order spectral accuracy is realized without sophisticated quadrature rules), (ii.) seamless enforcement of the quasiperiodic boundary conditions (no periodization of the fundamental solution, e.g. via Ewald summation, is required), and (iii.) reduced regularity requirements on the interface proles (derivatives of the deformations do not appear explicitly in the formulation). We show how these can be efficiently discretized and utilized in the simulation of scattering of linear acoustic waves by families of periodic layered media which arise in geoscience applications.

Rigorous justification of the modulation approximation to the full water wave problem

Nathan Totz (Duke)

Abstract: We consider solutions to the infinite depth water wave problem neglecting surface tension which are to leading order wave packets with small $O(\epsilon)$ amplitude and slow spatial decay that are balanced. Multiscale calculations formally suggest that such solutions have modulations that evolve on $O(\epsilon^{-2})$ time scales according to a version of a cubic NLS equation depending on dimension. Justifying this rigorously is a real problem, since standard existence results do not yield solutions to the water wave problem that exist for long enough to see the NLS dynamics. Nonetheless, given initial data suitably close to such a wave packet in $L^2$ Sobolev space, we show that there exists a unique solution to the water wave problem which remains within $o(\epsilon)$ to the formal approximation on the natural NLS time scales. The key ingredient in the proof is a formulation of the evolution equations for the water wave problem developed by Sijue Wu (U Mich.) with either no quadratic nonlinearities (in 2D) or mild quadratic nonlinearities that can be eliminated using the method of normal forms (in 3D).

Geometric theory of garden hoses, or string theory in your backyard

Vakhtang Putkaradze   [email] (University of Alberta, Mathematics)

Abstract: A garden hose inevitably wiggles and twists when water is rushing through it. We derive a fully three-dimensional, geometrically exact theory for this phenomenon. The theory also incorporates the change of the cross-section available to the fluid motion during the dynamics. Our approach is based on the symmetry-reduced, exact geometric description for elastic rods, coupled with the fluid transport and subject to the volume conservation constraint for the fluid. We analyze the linear stability, and show that the change of cross-section plays an important role. We derive and analyze several analytical, fully nonlinear solutions of traveling wave type in two dimensions. Time permitting, we shall also discuss the effects of the boundary conditions and experimental results. Partially supported by NSERC and the University of Alberta.