Lefschetz Center for Dynamical Systems Seminar
Abstract: The pressure term has always created difficulties in treating the Navier-Stokes equations of incompressible flow, reflected in the lack of a useful evolution equation or boundary conditions to determine it. In joint work with Jian-Guo Liu and Jie Liu, we show that in bounded domains with no-slip boundary conditions, the pressure can be determined in such a way that it is strictly dominated by viscosity. As a consequence, in a general domain with no-slip boundary conditions, we can treat the Navier-Stokes equations as a perturbed vector diffusion equation instead of as a perturbed Stokes system. We illustrate the advantages of this view by providing simple proofs of (i) the stability of a difference scheme that is implicit only in viscosity and explicit in both pressure and convection terms, requiring no solutions of stationary Stokes systems or inf-sup conditions, and (ii) existence and uniqueness of strong solutions based on the difference scheme.
Brown University
Joint Materials/Solid Mechanics Seminar Series
Northwestern University, Evanston, IL | |
Abstract: Recent field and laboratory observations have shown that compaction in porous sandstones may not occur uniformly; instead, compaction can occur non-uniformly in one or more narrow, planar bands. In these bands, the porosity is significantly reduced compared to the surrounding material and, as a result, laboratory and field measurements have shown that the permeability is reduced by one or two orders of magnitude. Consequently, these bands form barriers to fluid flow. Their presence can dramatically alter the fluid flow properties of the formation and affect applications involving the injection or withdrawal of fluids from porous subsurface formations, such as aquifer management, energy recovery and storage, waste disposal and carbon sequestration. Because these features are localized, they are difficult to detect by surface geophysical or borehole measurements. This talk will summarize field and laboratory observations of these bands, present an analysis of conditions for the inception of these bands, similar to that used for the inception of shear bands, and discuss some possible models for their propagation.
Brown University Center for Statistical Sciences Seminar
Abstract: Technological changes that provide increasing opportunity to make observation on high-dimensional variables, such as gene expression and other high-throughput molecular data in genomics, raises many challenges to statistical science -- conceptual, methodological and computational challenges. Much of what we know works well in `standard' low-dimensional problems does not work well, if at all, as problems scale drastically. One key concept that seems to be fundamental to scaling methodology is sparsity. Our work with large-scale regressions and graphical models provides a number of examples of how coherent Bayesian models can be developed and applied in problems in high dimensions, underpinned by the emphasis -- through sparsity inducing priors -- on sparse structure in multivariate relationships. Questions of model search are addressed through distributed computational approaches including MCMC and our greedy "shotgun stochastic search" for rapid identification and evaluation of many, many models. This talk will review and discuss aspects of this work, with some examples and mention of core, open research issues.
Brown Analysis Seminar
PDE Seminar
Department of Mathematics Colloquium
Brown University -- Joint Materials/Solid Mechanics Seminar Series
Abstract: Morphological control down to the atomic scale is an important issue in fabrication of technologically significant nanostructures. In this talk, an attempt will be made to review some of the recent conceptual advances achieved in studies of the formation and stability of artificially structured materials of reduced dimensionality. Emphasis will be given to the role of various elemental atomic rate processes in morphological evolution, such as the Ehrlich-chwoebel barrier effect in different dimensions and the importance of true upward atom migration in nanocrystal formation and faceting. The need for bridging different length scales in predicting morphological evolution and stability will also be demonstrated using specific examples.
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