Seminars are listed in reverse chronological order. The
top of the list is subject to change, since more seminars
are still being planned. All seminars are held at 3:10p.m.
in 1431 Stevenson Center unless otherwise noted. For further
information on events in the department, you may also
consult the
colloquia schedule, the
weekly
calendar and past
calendars.
Wednesday, April 16th, 2003, 4:10p.m.
Bo Su, of the
School of Mathematics,
Georgia Institute of Technology.
Viscous Approximation for a Multidimensional Unsteady Euler Flow:
Existence Theorem for Potential Flow.
We study a nonlinear system of partial differential equations
that is a viscous approximation for a multidimensional
unsteady Euler potential flow governed by the
conservation of mass and the Bernoulii law. The system
consists of a transport equation for the density and
the viscous nonhomogeneous Hamilton-Jacobi equations
for the velocity potential. This is a simplified model
of compressible Navier-Stokes equations. We establish
the existence and regularity of global solutions for
the nonlinear system with arbitrary large periodic
initial data. We also show that the density in our
global solutions has a positive lower bound, that is,
our solutions always stay away from the vacuum, as
long as the initial density has a positive lower
bound. The steady case was studied by Gamba-Morawetz
in 1996. The proof in the unsteady case can also be
applied to show steady case.
Wednesday, April 16th, 2003, 3:10p.m.
Tilak Bhattacharya, of the
Department of Mathematics,
Bishop's University, Lennoxville,
Quebec, Canada.
Some Recent Results for Infinity-Harmonic Functions.
We present some recent results for
infinity-harmonic functions in Rn. These are
solutions of the partial differential equation (PDE)
(*)
Δ∞u
=∑ni,j=1Diu Dj u
Diju=0.
This is nonlinear, elliptic and degenerate, and by solutions we will
mean viscosity solutions in the sense of Crandall and Lions. This
PDE appears quite naturally in the study of minimal Lipschitz
extensions and in a sense reflects calculus of variations in
L∞. The same PDE also arises in image
processing and its 'parabolic counterpart'
ut=Δ∞u
has found applications. Our talk will focus only on the elliptic
PDE in $(*)$. The PDE in $(*)$ is also referred to the
infinity-Laplacian since more often works have employed
approximating sequences that are weak solutions of the
p-Laplacian
Δpup
=div(|Dup|p-2Dup)=0,
1<p<∞
This approach captures solutions of (*) by studying
up as p→∞. In a way we abandon
this point of view and work directly with (*) and interpret this
in a viscosity sense. We discuss direct proofs of the Harnack
inequality, boundary Harnack inequality near flat boundaries,
comparison, ∞-capacitary functions and some questions of
more classical nature such as Picard's principle. A key fact we
will state and use is that being solutions of (*) is same as
being cone-like locally.
Wednesday, April 9th, 2003.
José Miguel Urbano, of the
Universidade de Coimbra, Portugal.
Regularity in Sobolev Spaces for Doubly Nonlinear Parabolic Equations.
The doubly nonlinear parabolic equation
ut=div [ |∇(|u|m-1u)|
p-2
∇(|u|m-1u) ]
with m>1 and m(p-1)>1 is considered in several dimensions and
regularity results in fractional order Sobolev spaces are
obtained. The main tools in the proof are a difference quotient
technique and the imbedding theorem of Nikolskii spaces into
Sobolev spaces.
Joint work with Carsten Ebmeyer (University of Bonn).
Wednesday, April 2nd, 2003.
Luca Capogna, of the
Department of Mathematics,
University of Arkansas.
Wave Maps in Heisenberg Groups.
In a joint work with Jalal Shatah (Courant), we study critical points
of the energy functional for maps from the Minkowski space to
odd dimensional Euclidean spaces. We impose the constraint that such
critical points are Legendrian with respect to the standard contact
structure. We address questions of existence, uniqueness and continuous
dependence from the initial data. The problem is motivated by the
study of geodesics in Carnot-Caratheodory
spaces, and the behavior of sequences of harmonic or wave maps when
the target Riemannian metrics degenerate.
Wednesday, March 26th, 2003.
Andrea Bertozzi, of the
Department of Mathematics,
Duke University.
New Challenges for Hydrodynamics: Microfluidics,
Imaging Science, and Mobile Sensors.
This talk will showcase three
new research areas involving mathematical fluid dynamics.
Microfluidics is a rapidly growing field being driven by new
technological applications in the medical, materials, and chemical
sciences. Surface tension effects (Marangoni stresses)
are important on these scales. We consider the basic physics
of surface tension gradients (used to move liquids) in conjunction
with body forces on fluids and show that the ensuing dynamics can
yield multiple shock structures involving undercompressive waves.
In the field of imaging science,
Image inpainting involves filling in part of an image or video
using information from the surrounding area. We introduce a
class of automated methods for digital inpainting
using ideas from classical fluid dynamics.
The main idea is to think of the image intensity as a
'stream function' for a two-dimensional incompressible
flow. The method is directly based on the Navier-Stokes equations for
fluid dynamics, which has the immediate advantage of well-developed
theoretical and numerical results.
An emerging area of mobile sensor control is the design
of algorithms for multiple unmanned vehicles.
Taking ideas from mathematical biology, we consider swarming
algorithms for fluid-like motion based on simple rules for
self-propulsion and local interaction. Applications range from
mine detection algorithms to perimeter patrol and gradient
searching.
Wednesday, March 12th, 2003.
Jan Prüss, of the
Department of Mathematics,
Martin-Luther Universität Halle-Wittenberg.
Mass Transport Through Charged Membranes.
A modern technique for desalination or softening of water is the
so-called nanofiltration by means of membranes which carry
a fixed electric charge. For the mathematical modeling of such
processes, two features are particularly important. Firstly,
the distribution of ionic species generates an electric field which
in turn affects the fluxes of these irons. Therefore, besides
diffusion and convection, electromigration has to be taken
into account. This leads to a strong coupling of the concentrations
of all charged species, which can often be adequately incorporated
into the model via the assumption of electroneutrality.
Secondly, electrical double-layers (Donnan potential) build up at
the surface of the membrane, which cause discontinuities in the
ionic concentration profiles.
We deduce a mathematical model for such a nanofiltration process.
This leads to a strongly coupled quasilinear parabolic system with
nonlinear transmission and dynamical boundary conditions. By
means of degree theory we obtain existence of stationary solutions,
while Lp-maximal regularity is employed to get
local strong wellposedness of this model.
Monday, February 10th, 2003, 4:10p.m.
Darren Oldson, of the
Mathematics Department,
Duke University.
Dynamics of Feedback-Regulated Flow in the Nephrons of the Kidney:
Perturbations, Oscillations, and Compensation.
A mathematical model previously formulated by Layton et al. predicts
that limit-cycle oscillations (LCO) in nephron flow are mediated by
tubuloglomerular feedback (TGF) and that the LCO arise from a
bifurcation that depends heavily on the feedback gain magnitude
γ. We will use this model to show how sustained perturbations in
proximal tubule flow, a common experimental maneuver, can initiate
or terminate LCO by changing the value of γ. This result may
help explain experiments in which intratubular pressure oscillations
were initiated by the sustained introduction or removal of fluid from
the proximal tubule. In addition, this model predicts that sustained
perturbations that initiate or terminate LCO can yield substantial and
abrupt changes in both distal NaCl delivery and NaCl delivery
compensation, changes that may play an important role in the response
to physiological challenge. The linear stability analysis for an
ordinary differential equation will be compared with the linear
stability analysis for the delay partial differential equation that
arises in this model for TGF.
Wednesday, February 5th, 2003, 1206 Stevenson Center.
Gabriel Soto, of the
School of Mathematics,
University of Minnesota.
Modeling Calcium Dynamics During Synaptic Transmission.
Neurons form connections called "synapses" at which information
transfer from one to another(others) occurs, process which is called
"synaptic transmission". At these specialized sites, electrical impulses
are converted into chemical impulses, thus neurons have developed
complex mechanisms to modulate such changes. This is one of the most
important examples in bioloy where calcium plays a fundamental role in
modulating synaptic this information processing. Moreover, under the
"Calcium Hypothesis" calcium is the trigger for synaptic transmission.
Under this assumption, I will present a model for calcium dynamics
during synaptic transmission that integrates different mechanisms
utilized by neurons that modulates calcium dynamics and hence synaptic
transmission.
Wednesday, January 29th, 2003, 1206 Stevenson Center.
Dan Coombs, of the
Theoretical Biology Laboratory,
Los Alamos National Laboratory.
Modeling T Cell Activation.
In this talk I will present some ways in which mathematical modeling has
been helpful in studying T cell activation, and outline challenges for
the future. No prior knowledge of the immune system will be assumed.
The activation of T cells is an essential part of the immune response to
viruses and bacteria. Fragments ("antigens") of these enemies are
presented to T cells on the surfaces of antigen-presenting-cells, but an
individual T-cell carries receptors (TCR) that recognize only a few
possible antigens. After the T cell becomes activated, it may kill the
presenting cell (in the case of viral infection) or activate other
components of the immune system (in the case of bacterial infection).
Activation appears to rely on the formation of a stable region of close
apposition between the cells, termed the "immunological synapse". Within
the synapse, each TCR may individually be activated and labeled for
internalization by interaction with presented antigen. A key parameter
controlling individual TCR activation and internalization is the
lifetime of the bond between the TCR and a presented antigen.
We have developed a mathematical model consisting of reaction-diffusion
equations describing spatial and temporal changes that take place within
the synapse. From comparison of model predictions with experimental
data, we draw conclusions about the requirements for T cell activation
as well as the cellular internalization and degradation of TCR.
Wednesday, January 22nd, 2003.
Boris Belinskiy, of the
Department of Mathematics,
The University of Tennessee at
Chattanooga.
Some New Developments in the Method of Moments
in Connection with Exact Control Theory.
We study the exact controllability for a flexible elastic string fixed
at the end points under an axial stretching tension. The tension is a
sum of two terms, a constant tension and a slowly variable load. We say
that the string is controllable if, by suitable manipulation of the
transverse load, the string goes to the rest. We are looking for an
exterior transverse load g(x)f(t) that drives the state solution
to the rest. To prove our results we apply the method of moments.
This has been widely used in control theory of distributed parameter
systems since the classical papers of H.O. Fattorini and D.L. Russell
in the late 60s to early 70s. The problem of exact controllability is
reduced to a moment problem for the control f(t). The proof of
controllability is based on an auxiliary basis property result that is
of independent interest.
The results of this paper may be considered as generalization of the
classical results for one-dimensional wave equation. The main
difference
between our problem of control and the previous problems is that the
coefficient of the wave equation (tension in our model) is a function of
time. As a result, the functions that substitute non-harmonic
exponential functions even may not be found explicitly. This fact
sufficiently complicates the analysis of controllability. To our best
knowledge it is the first attempt to apply the method of moments for
equations with time dependent coefficients.
We outline some possible generalizations. One of them could be the
control of oscillations of a system (actually a graph) of connected thin
elastic cylinders (blood vessels). Another one could be a controlling
device that would stabilize unnecessary oscillations of a propeller of a
helicopter.
Wednesday, January 15th, 2003.
David Hachey, of the
Vanderbilt-Ingram Cancer
Center,
Vanderbilt University.
Compartmental Analysis of Complex Physiological Systems: Cholesterol
and Bile Acid Metabolism.
Cholesterol was first isolated by the French Chemist de la Salle in
1769 and further characterized by Marcel Chevreul in 1815. However,
it wasn't until dietary studies in rabbits in the early 20th
century that cholesterol in meat and dairy products was firmly
linked with elevated serum cholesterol and cardiovascular disease
in man. Regulation of serum cholesterol levels is an actively
controlled metabolic process that dynamically adjusts endogenous
biosynthesis and biliary sterol excretion to compensate for high
levels of dietary cholesterol. In order to study such intricate
feedback controlled systems, complex models are required which
describe the mechanisms of absorption, biosynthesis, catabolism and
excretion in an integrated model. In order to do so, complex
multi-isotopic tracer studies are required which independently
trace the different compartments of the system. In this work, five
stable isotopic tracers were used to define various aspects of
whole-body sterol metabolism using compartmental analysis. The
model defines the magnitude of the central vascular cholesterol
compartments, the fractional absorption of cholesterol in the
intestine, rates of hepatic sterol biosynthesis and metabolic
clearance of bile acids.
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