ABSTRACTS FOR TALKS IN THE ANALYSIS SEMINAR


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Alexander Kiselev, University of Chicago

Talk I: Scattering for Schr\"odinger operators with long-range potentials.
Colloquium, 
Wednesday, January 30, 2002, 4:00 p.m., Van Vleck B239

Talk II:  Enhancement and Quenching of Combustion by Fluid Flow
Thursday, January 31, 2002, 2:30 p.m.,  Room TBA


Abstract for the Wednesday talk:

We prove WKB-type asymptotics of solutions and existence of modified wave
operators for one-dimensional Schr\"odinger operators
with potential in $L^p(\reals)$, $p<2$. If in addition the potential is
conditionally integrable, then the usual M\"oller wave operators exist.
All previous results in this class required strong additional
assumptions on the potential. The results are close to optimal in a sense
that there are examples of potentials in $\cap_{p>2} L_p$ for which
the spectrum is purely singular and hence no wave operators can exist.   

The methods involve application of Fourier analysis to study
asymptotic behavior of eigenfunctions. In particular we prove new
results on almost everywhere convergence of certain oscillatory
multilinear operators. The $p=2$ case remains open and appears to be
linked to a nontrivial generalization of Carleson's a.e. convergence
theorem.



 
Abstract for the Thursday talk:
Influence of the fluid flow can lead to drastic speed up, or,
in some situations, quenching of burning. Many important engineering
applications of combustion operate in the presence of turbulent advection, 
and therefore the influence of advection on burning has been studied
extensively by physicists, engineers and mathematicians. Nevertheless,
rigorous results on the subjects were scarce until recently.

We study a well-established model of premixed fluid combustion: the
reaction-diffusion equation with passive advection. We introduce in a
rigorous way the quantity measuring the  rate of combustion, the bulk
burning rate, which is well-defined in any burning regime. For various
types of reaction non-linearities we show that there is a class of the
flows, which we call percolating, that are very effective in speeding up
the reaction. These flows are characterized by the presence of the long
tubes of streamlines connecting the burned and unburned material, and in
particular include shear flows in a direction perpendicular to the front.
On the other hand, the flows with closed streamlines, such as cellular,
are shown to produce weaker burning enhancement. 

We also study in detail the opposite effect of quenching the combustion in
a shear flow, discovering connections between the structure of the flow
and its efficiency in helping extinguish the compactly supported flame. 

The methods involve a new functional inequality, a new procedure for
estimation of higher order derivatives, maximum principle based
techniques,  and a study of hypoellipticity of 
certain degenerate  parabolic PDE.



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February 5, 2:30 p.m.:

Speaker: Chris Miller, Ohio State

Title: Hausdorff dimension of Borel subrings

Abstract: Any subring of the real numbers R that is Borel (as a set)
either has Hausdorff dimension zero or is equal to R. Extensions of the
method of proof yield (among other things) that every Borel subring of 
the complex numbers C either has Hausdorff dimension zero or is equal to 
R or C.  (Joint work with G.A. Edgar.)