Lefschetz Center for Dynamical Systems Seminar
Joint Departmental Colloquium/LCDS Talk) | |
Abstract:
Diophantine conditions are well-known from time averaging
and Kolmogorov-Arnold-Moser theory. In contrast, we
consider semilinear reaction diffusion equations with
spatially quasiperiodic coefficients in the
nonlinearity, rapidly varying on spatial scale \epsilon.
Under Diophantine conditions on the spatial frequencies,
we derive quantitative homogenization estimates of order
\epsilon^{\gamma} on Sobolev spaces H^{\sigma} in the
triangle
0 < \gamma < min(\sigma - n/2, 2 - \sigma).
Here n denotes spatial dimension. The estimates measure
the distance to a solution of the homogenized equation
with the same initial condition, on bounded time
intervals. The same estimates hold for C1-convergence
of local stable and unstable manifolds of hyperbolic
equilibria. Our results apply to homogenization of
the Navier-Stokes equations with spatially rapidly
varying quasiperiodic forces in space dimensions 2 and 3.
Our results also extend to quantitative homogenization
of global attractors for near-gradient system. An
extension to damped hyperbolic wave equations will
be indicated. All results are joint work with
Mark I. Vishik. For references see
www.math.fu-berlin.de/~Dynamik/
Brown University Graduate School Dissertation Defense
Brain Science Program/Applied Mathematics Seminar No. 1
There will be a special joint "double header" seminar on the
biological significance of dendritic spines and some methods
from computer vision for detecting and analyzing them
automatically.
Scientific Computing Seminar
Abstract: A procedure is introduced for estimating the time necessary to produce the onset of fluid turbulence initiated by constant acceleration and impulsive (shock) acceleration-driven flow instabilities. The former are known as Rayleigh-Taylor instabilities (RTI). The latter are known as Richtmyer-Meshkov instabilities (RMI). The procedure involves application of some contemporary turbulence theory together with a phenomenological treatment. This provides a basis for the requisite space and time scaling. Specific interest here is on transitional mixing flow development from the early stages of RTI and RMI material interface penetration and continuing (given the appropriate flow and state conditions) through the fully developed turbulent mixing phase. Essentially the procedure introduced extends to unsteady flow situations the remarkably universal mixing transition criterion proposed by Dimotakis [J. Fluid Mech., 409, 69 (2000)] for stationary flows. Experimental comparisons with the procedural results are presented. Three different classes of acceleration driven instability experiments are used for experimental comparison with the predictions: (1) classical, relatively low speed, RTI experiments conducted at Cambridge University; (2) a set of shock tube RMI flow mixing experiments carried out at University of Arizona; (3) results of laser driven RTI and RMI mixing experiments involving instability growth at very high energy density. The last named experiments are of special interest as they provide scaleable flow conditions approaching those of interest in current astrophysical studies such as shock-driven hydrodynamic mixing in supernova evolution.
Brain Science Program/Applied Mathematics Seminar No. 2
Brain Science Program/Applied Mathematics Seminar No. 3
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