Mathematical Aspects of Continuum Physics: Analysis, Computation, and Modeling
Org: Rustum Choksi (SFU) and Mary Pugh (Toronto)
- STAN ALAMA, McMaster University, Hamilton, Ontario, Canada
Vortices in multiply connected Bose-Einstein condensates
We study minimizers of the Gross-Pitaevskii energy, introduced to
model Bose-Einstein condensates (BEC) which are subject to a uniform
rotation. This energy is very closely related to the Ginzburg-Landau
energy of superconductivity, and an essential feature of the model is
the formation of quantized singularities (vortices) in an appropriate
singular limit. Following some recent experiments in BEC, we consider
condensates with annular (planar) or toroidal (3D) geometry and
examine minimizers to determine the presence and location of vortices
in the condensate as the rotational speed increases. These questions
involve singularly perturbed elliptic systems, and we will use
variational methods with sharp estimates on the energy together with
some tools from geometric measure theory to study the 3D case.
These results have been obtained in collaboration with L. Bronsard,
A. Aftalion and J. A. Montero.
- FERNANDO BRAMBILA, Departamento de Matemáticas, Facultad de Ciencias, UNAM,
Complete Vectorial Radon Transform
We will talk about the complete inversion formula for the Vectorial
It is well known how to recover the curl of a vector field using
Doppler effect (G. Sparr, 1995). Using more information and Helmholtz,
it is possible to have a complete inversion.
- IRENE FONSECA, Department of Mathematical Sciences, Carnegie Mellon
University, Pittsburgh, PA 15213, USA
Surfactants in Foam Stability: a Phase Field Model
The role of surfactants in stabilizing the formation of bubbles in
foams is studied using a phase-field model. The analysis is centered
on a van der Walls-Cahn-Hilliard-type energy with an added term
accounting for the interplay between the presence of a surfactant
density and the creation of interfaces. In particular, it is
concluded that the surfactant segregates to the interfaces, and that
the prescription of the distribution of surfactant will dictate the
locus of interfaces, which is in agreement with experimentation.
This is joint work with Massimiliano Morini and Valeriy Slastikov.
- CARLOS GARCIA-CERVERA, Mathematics Department, University of California, Santa
Barbara, CA 93106, USA
An Efficient Real Space Method for Orbital-Free Density
I will describe an efficient implementation of the truncated-Newton
method for energy minimization in the context of orbital-free density
functional theory. I will illustrate the efficiency and accuracy of
the method with numerical simulations in an aluminium FCC lattice.
- JOY KO, Brown University, Providence, Rhode Island, USA
Steady rotational water waves near stagnation
Two-dimensional finite-depth periodic water waves with general
vorticity and large amplitude are computed. The mathematical
formulation and numerical method that allow us to compute a continuum
of such waves with arbitrary vorticity are described. The
computations in the case of constant vorticity show that there are
only two points of stagnation and that the qualitative nature of the
free surface depend on the vorticity. For variable vorticity,
modelling surface shears and undertows, phenomena such as internal
stagnation can occur.
This is joint work with Walter Strauss (Brown University).
- JONATHAN MATTINGLY, Department of Mathematics, Duke University, Durham, NC
Challenges in the analysis of degenerately forced stochastic
I will give a few example of degenerately forced PDEs which might be
of modeling interest, such as reaction diffusion or fluid equations.
Then I will discuss the mathematical difficulties which arise in the
stochastic PDE setting which do not arise for stochastic ODEs. Then I
will give some cases where we can circumvent these difficulties,
either by direct calculation or by some new mathematical tools.
- ROBERT MCCANN, University of Toronto Mathematics
Nonlinear diffusion from a delocalized source: affine
self-similarity, spacetime asymptotics, and focusing
A family of explicit solutions to the porous medium equation and its
fourth order generalizations is described, in the full range of
nonlinearities, in which the pressure is given by a quadratic
function of space at each instant in time. These include spreading
solutions whose source is concentrated on any conic region of
dimension lower than the ambient space, and solutions which focus at
conic regions. The singular limiting distributions are affine
projections of Barenblatt type profiles with arbitrary signature. A
time-reversal symmetry is revealed which transforms spreading
solutions to focusing solutions, and vice versa. This yields new
information about the long and short time asymptotics of finite-mass
solutions, about the instability of focusing, and about singularity
This work is joint with Jochen Denzler (University of Tennessee at
Knoxville). Preprints are found at www.math.toronto.edu/mccann.
- GOVIND MENON, Brown University
Domain coarsening in a 1D bubble bath
We will study a mean-field model for coarsening in a 1D bubble bath.
An explicit solution formula found by Gallay and Mielke allows us to
provide a simple and complete characterization of the approach to
self-similarity in this model.
This is work with Barbara Niethmammer and Bob Pego.
- BOB PEGO, Carnegie Mellon Univ., Dept. Math. Sciences, Pittsburgh,
Scaling dynamics of coagulation equations with dust and gel
We study limiting behavior of rescaled size distributions that evolve
by Smoluchowski's rate equations for coagulation, with rate kernel
K=2, x+y or xy. We find that the dynamics naturally extend to
probability distributions on the positive half-line with zero and
infinity appended, representing populations of clusters of zero and
infinite size. The "scaling attractor" (set of subsequential
limits) is compact and has a Levy-Khintchine-type representation that
linearizes the dynamics and allows one to establish several signatures
of chaos. In particular, for any given solution trajectory, there is
a dense family of initial distributions (with the same initial tail)
that yield scaling trajectories shadowing the given one for all large
- MARIA G. REZNIKOFF, School of Mathematics, Georgia Institute of Technology,
686 Cherry Street, Atlanta, Georgia 30332
Slow Motion of Gradient Flows
Sometimes physical systems exhibit "metastability," in the sense
that states get drawn toward so-called metastable states and are
trapped near them for a very long time. A familiar example is the
one-dimensional Allen-Cahn equation: Initial data is drawn quickly to
a "multi-kink" state and the subsequent evolution is exponentially
slow. The slow coarsening has been analyzed by Carr & Pego, Fusco &
Hale, Bronsard & Kohn, and X. Chen.
In general, what causes metastability? Our main idea is to convert
information about the energy landscape (statics) into information
about the coarsening rate (dynamics). We give sufficient conditions
for a gradient flow system to exhibit metastability. We then apply
this abstract framework to give a new analysis of the 1-d Allen-Cahn
equation. The central ingredient is to establish a certain nonlinear
energy-energy-dissipation relationship. One benefit of the method is
that it gives a natural proof of the fact that exponential closeness
to the multi-kink state is not only propagated, but also generated.
This work is joint with Felix Otto, University of Bonn.
- SILVIA SERFATY, New York University
The Ginzburg-Landau energy close to the second critical
We are interested in the Ginzburg-Landau energy when the applied
magnetic field approaches the "second critical field" from below.
Then, bulk-superconductivity decreases and vortex lattices are
expected. I will present joint results with Etienne Sandier/Amandine
Aftalion, where we derive the uniform repartition of the energy and a
- THOMAS WANNER, George Mason University, Department of Mathematical
Sciences, Fairfax, VA 22030, USA
Complex Transient Patterns and their Topology
Many partial differential equation models arising in applications
generate complex time-evolving patterns which are hard to quantify due
to the lack of any underlying regular structure. Such models may
include some element of stochasticity which leads to variations in the
detail structure of the patterns and forces one to concentrate on
rougher common geometric features. In many of these instances, one is
interested in the geometry of sublevel sets of a function in terms of
their topology, in particular, their homology. In practice, however,
these sublevel sets are approximated using an underlying
discretization of the considered partial differential equation-which
immediately raises the question of the accuracy of the resulting
homology computation. In this talk, I will present a probabilistic
approach which gives insight into the suitability of this method in
the context of random fields. We will obtain explicit probability
estimates for the correctness of the homology computations, which in
turn yield a priori bounds for the suitability of certain
grid sizes. In addition, we present a computational approach to
homology validation in the above setting, and apply our results to
certain stochastic partial differential equations arising in materials