Wednesdays at 4 PM in Eckhart 206.
We examine the asymptotic behavior of the eigenvalue $\mu(h)$ and corresponding eigenfunction associated with the variational problem $$ \mu(h)\equiv\inf_{\psi\in H^{1}(\Omega;{\bf C} )} \frac{\int_{\Omega } \abs{(i\nabla+h{\bfA})\psi}^{2}\,dx\,dy} {\int_{\Omega }\abs{\psi}^{2}\,dx\,dy} $$ in the regime $h>>1$. Here ${\bf A}$ is any vector field with curl equal to 1. The problem arises within the Ginzburg-Landau model for superconductivity with the function $\mu(h)$ yielding the relationship between the critical temperature vs. applied magnetic field strength in the transition from normal to superconducting state.
We study nonlinear patterns on sandy beds that appear in shallow tidal seas. The interaction between moving water and sediment is modeled by using the equations of the fluid motion coupled with the mass conservation law for the sediment transport. We investigate various instabilities that occur as a result of this interaction. Sand banks and sand waves are the two types of sand structures that are commonly observed on a sea bed. There are at least two mechanisms responsible for the growth of sand banks and sand waves. One is linear instability, and the other is nonlinear coupling between long sand banks and short sand waves. It turns out that the latter is more important for the generation of sand banks. It is possible to derive nonlinear amplitude equations governing the coupled dynamics of sand waves and sand banks. Based on these equations, one can estimate characteristic features for sand banks and find that the estimates are consistent with measurements.
Suppose that in a "dark" region with piecewise smooth boundary there is a flashlight which emittes a narrow parallel beam of rays. The rays are reflected from the boundary according to the law "angle of reflection equals angle of incidence". Will the entire region be illumunated? This question arises naturally e.g. in optics, acoustics and in a quantum problem in this region. It was known for a long time that the answer is "yes" in a boundary is dispersing. Obviously the answer is "no" for some simple focusing boundaries (e.g. for a circle). It has been discovered though at the early 70th that some domains with focusing boundaries can be illuminated. However all these examples were in 2D. The obstacle to illumination of domains with focusing components of the boundary is the well known in optics phenomenon of astigmatism. It has been shown recently that still such domains can be illuminated in D>2 as well. However the prise paid to astigmatism is the extra (compared to 2D) restriction that focusing components are not allowed to be big.
Front propagation occurs in many applied problems, such as chemical kinetics, combustion, transport in porous media, and in biology. The basic phenomena can often be described by reaction-diffusion-advection equations. In homogeneous media front propagation has been studied for a long time. However, the study of fronts in inhomogeneous media has begun more recently. Understanding the influence of heterogeneities on the location of fronts, on their profile, and on their speed is of great importance. We study the effects of a periodically varying environment on the speed of fronts. We give a general existence proof of fronts in periodcally varying media close to the homogenization limit. Then we derive a variational principle for their speed. This allows for the calculation of several asymptotic estimates of the velocity. We apply this to the situation of an underlying given fluid flow. In the case of shear flows we show the enhancement of the velocity. Our proofs are based on the maximum principle and therefore further extensions are possible: periodically perforated domains, domains with variable cross section, monotone systems in higher dimensions, discretized diffusion, time dependent coefficients.
The multivariable variational problems with non-convex Lagrangians are commonly met in structural optimization, phase transitions in solids, and other problems of optimal layouts of several materials. The minimizing sequences in these problems tend to infinitely often oscillating functions that correspond to media with special microstructures. The talk discusses several approaches to such problems: A sufficient conditions method (Translation Method), an approximation by special sequences (Laminates of High Rank), and Minimal Extension method based on Weierstrass-type local conditions. We illustrate these methods demonstrating bounds for thermal expansion tensor of composites and optimal microstructures of multimaterial composites.
We introduce a general framework for studying the localization of classical waves in inhomogeneous and random media, which encompasses acoustic waves with position dependent compressibility and mass density, elastic waves with position dependent Lam\'{e} moduli and mass density, and electromagnetic waves with position dependent magnetic permeability and dielectric constant. We also allow for anisotropy. Localization is shown for local or random perturbations of periodic media with gap in the spectrum.
We povide a test for numerical simulations for several 2D incompressible flows that appear to develop sharp fronts, and for the collapse of tubes carried by a 3D incompressible flow. In particular, we obtain necessary conditions for 3D Euler to have a vortex tube collapse in finite time. Similar conditions are necessary for sharp fronts for the Quasi-geostrophic equation.
We propose a scenario in which sparse geometry of the vorticity super-level sets, and in particular quasi-lowdimensionality of coherent vortex structures, depletes the effect of the nonlinearity in the 3D NSE. An analogous scenario is proposed for the 3D MHD system; namely, the depletion of the nonlinearities induced by the sparse geometry of the high-intensity magnetic field.
In recent years there has been a tremendous interest in the application of Discontinuous Galerkin methods to problems where the diffusion is not negligible and more recently to pure elliptic problems. In this talk, we will present a brief overview of such methods in a general framework. In particular we describe the so called Local Discontinuous Galerkin method. Theoretical as well as practical issues will be discussed.
We consider a family of rectangular billiards and ensemble of particles that move inside the billiards. A similar problem arises in fluid and magneto-fluid dynamics for the field lines winding the surfaces with handles. It is shown that the main asymptotic behavior of particles corresponds to the fractional kinetics with space-time self-similarity. The dynamics possesses an algebraic complexity with power-wise distribution of the Poincare recurrences and displacements. The important part of the understanding of such type of systems is log-periodicity in time for the distribution of particles, recurrences, and moments. Just this property is linked to the finite approximates of the continuous fractions and their scaling properties. It is shown that some universal scaling can be established for the considered type of systems.
I examine decadal predictability in the North Atlantic. A region near Cape Code and the Gulf of Maine is identified as featuring the highest North Atlantic persistence, with a unique combination of high-amplitude, persistent sea surface temperature anomalies, associated with substantial upper ocean heat content anomalies. A mechanism is advanced, whereby that region's surface ocean represents deep upper ocean dynamics, with thermal evolution that tracks heat flux divergence by the slowly adjusting subtropical gyre. This is consistent with the subtropical gyre playing a central role in NA climate variability. The proposed interpretation is supported by broad, high lag correlations between the Bermuda potential vorticity timeseries (representing the subtropical gyre state) and surface anomalies. Successive lag correlation patterns reveal surface oscillations that are not apparent in the raw surface data. Finally, the correlation patterns are used for NAOI forecasting, using a new forecasting method that is presented and explained. Forecasts are robust and reproducible, outperforming alternative methods. At lead times of 25 and 12 years, cross validated skills over 1927-64 (38 yrs) are 0.44 and 0.53, respectively. It is suggested that the combination of skills and lead times may prove societally useful.
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