# PDE and Applied Mathematics Seminar

## Spring 2014

### February 4, (CAS 220D, 4:30-5:30pm)
Dmitry Ryabogin (Kent State University)
will speak on "Several problems of uniqueness of convex bodies".

####
Abstract: Assume that we are given two convex bodies $K$ and $L$ in
${\mathbb R^n}$, $n\ge 2$, such that the corresponding projections
(onto every hyperplane passing through the origin) are congruent
(there is a translation and a rotation bringing one projection into
another). Do the bodies $K$ and $L$ coincide (up to reflection and
translation)?
We will discuss this open question and other related problems of
convex geometry.
This talk should be clear for graduate students.

### March 13, (CAS 220D, 4:30-5:30 pm) Fedor Nazarov (Kent State University)
will give "Two remarks on the dissipative surface quasi-geostrophic equation".

####
Abstract: We discuss the results concerning the well-posedness of the
2D dissipative quasi-geostrophic equation in the critical regime
(Laplacian to the power 1/2) and the eventual regularity in the
supercritical (any small positive power of the Laplacian) regime.
I do not expect to be able to present much more than just rough
outlines of the main ideas of the proofs this time, but should
enough interest arise, I'll be happy to give a series of
lectures with full details.
The presentation is based on the
papers "Global well-posedness for the critical 2D dissipative
quasi-geostrophic equation" by A. Kiselev, F. Nazarov, A. Volberg and
"Eventual Regularity of the Solutions to the Supercritical
Dissipative Quasi-Geostrophic Equation" by M. Dabkowski (both
available on arXiv).

###
April 8, (CAS 220D, 3:30-4:30pm) Truyen Nguyen (University of Akron)
will speak on "Gradient estimates and global existence of smooth solutions
to a cross-diffusion system. "

####
Abstract: We study the initial value problem for a cross-diffusion
system in bounded domains
of any dimension. This system was proposed by Shigesada, Kawasaki and
Teramoto in 1979 to
describe the habitat segregation phenomena between two species which
are competing in the
same domain. A global-time existence of smooth solutions is
established by deriving global
$W^{1,p}$-estimates for weak solutions to a class of nonlinear
parabolic equations with nonlinear
diffusion. A perturbation argument and a new double scaling technique
are introduced to obtain
the estimates for large reaction terms. This is joint work with Luan
Hoang and Tuoc Phan.

###
April 22, (CAS 220D, 2:30-3:30pm)
Irene Fonseca (Carnegie Mellon University)
will speak on
"Variational Methods for Crystal Surface Instability."

####
Abstract:
Using the calculus of variations it is shown that important qualitative
features of the equilibrium shape of a material void in a linearly
elastic solid may be deduced from smoothness and convexity properties of
the interfacial energy.
In addition, short time existence, uniqueness, and regularity for an
anisotropic surface diffusion evolution equation with curvature
regularization are proved in the context of epitaxially strained
two-dimensional films. This is achieved by using the $H^{-1}$-gradient
flow structure of the evolution law, via De Giorgi's minimizing
movements. This seems to be the first short time existence result for a
surface diffusion type geometric evolution equation in the presence of
elasticity.

Previous seminars:

Fall 2013

Spring 2013

Fall 2012