57th Midwest Partial Differential Equations Seminar April 21-23,
2006, at UIC
* Antonio Bove (Bologna)
TITLE:
Analytic Hypoellipticity in the Presence of Lower Order Terms
ABSTRACT:
We consider a second order operator with analytic coefficients whose
principal symbol vanishes exactly to order two on a symplectic real
analytic manifold. We assume that the first (non degenerate)
eigenvalue vanishes on a symplectic submanifold of the
characteristic manifold. In the C¥ framework this
situation would mean a loss of 3/2 derivatives (as Helffer
has shown). We prove that this operator is analytic hypoelliptic.
* Jonathan Cohen (UIC and DePaul)
TITLE: Global existence for a coupled system of KdV-like equations
with rough initial data
ABSTRACT: This talk concerns coupled systems of Korteweg-deVries
type equations. Such systems arise in various contexts including
internal wave propagation. We outline a theory of global
well-posedness for such systems. This is done in two natural
steps- a local well posedness theory set in Hs(R) where s > -(3/4) and a corresponding global extension. The extension relies on
approximation of energy inequalities.
* Luca Capogna (Arkansas)
TITLE: Generalized mean curvature flow in Carnot group
ABSTRACT: The sub-Riemannian mean curvature flow consists in
flowing hypersurfaces in the direction of the "horizontal normal"
with velocity depending on the horizontal mean curvature. The
presence of so-called characteristic points makes this study much
harder than its Riemannian counterpart. The level set formulation
of the flow, coupled with the notion of viscosity (or generalized)
solution extends well to the sub-Riemannian setting and allows to
prove analogues of the results by Evans-Spruck, Chen-Giga-Goto. I
will describe ongoing joint work with Giovanna Citti (U. Bologna)
where we study viscosity solutions of the mean curvature flow of
level sets in Carnot groups.
* Loredana Lanzani (Arkansas)
TITLE: Hodge systems and the d-bar problem
ABSTRACT: In the first part of this talk I will give an overview
of joint work with E. M. Stein concerning Lr-estimates of the
Hodge system for forms in RN. In the second part I will
discuss the following question: can these results be used to
obtain new estimates of the d-bar problem for (p,q)-forms in
Cn?
* Andreea Nicoara (Harvard)
TITLE: The Kohn Algorithm in More General Classes of Functions
ABSTRACT: In 1979 Joseph J. Kohn introduced an algorithm that
yields ideals of subelliptic multipliers, which measure whether
there is gain in regularity of the solution of the
[`(¶)]-Neumann problem on a pseudoconvex domain in
Cn. If such a domain is defined by a real analytic
function, subellipticity of the [`(¶)]-Neumann problem is
equivalent to the Kohn algorithm generating the entire ring of
real analytic functions. I will discuss what happens to this
equivalence for local rings of functions that are strictly larger
than the local ring of real analytic functions, in particular
those of the Denjoy-Carleman classes of functions.
* Wolfgang Staubach (Chicago)
TITLE: Y-pseudodifferential operators and estimates for maximal
oscillatory integrals
ABSTRACT: We define a class of pseudodifferential operators with
symbols a(x,x) without any regularity assumptions in the x
variable and explore their Lp and H1 boundedness
properties. The results are used to obtain estimates for certain
maximal operators associated with oscillatory singular integrals.
(Joint work with Carlos Kenig)
* Jie Shen (Purdue)
TITLE: Fast Spectral-Galerkin Method: Algorithms, Analysis and
Applications
ABSTRACT: In recent years, spectral methods have become
increasingly popular among computational scientists and engineers
because of their superior accuracy and efficiency when properly
implemented. In this talk, I shall present fast
spectral-Galerkin algorithms for some prototypical partial
differential equations. These spectral-Galerkin algorithms have
computational complexities which are comparable to those of finite
difference and finite element algorithms, yet they are capable of
providing much more accurate results with a significantly smaller
number of unknowns. A key ingredient for the efficiency and
stability of the spectral-Galerkin algorithms is to use (properly
defined) generalized Jacobi polynomials as basis functions.
I shall present applications of these fast spectral-Galerkin
algorithms to a number of scientific and engineering problems,
including KdV type equations, Navier-Stokes equations, multiphase
incompressible flows and bounded obstacle scattering.
* Catherine Sulem (Toronto)
TITLE: Water waves over a varying bottom
ABSTRACT: We will discuss the problem of nonlinear wave motion of
the free surface of a body of fluid with a varying bottom. The
object is to describe the character of wave propagation in a
long-wave asymptotic regime for two and three dimensional flows.
We consider bottom topography which is periodic in horizontal
directions on a short length-scale and may in addition exhibit
slow variations. For two dimensional flows, we will also consider
the case of a random rough bottom. The purpose is to understand
how the varying bottom affects the effective Boussinesq equations
and in the appropriate unidirectional limit, KdV or KP type
equations.
* Stephen Yau (UIC and Harvard)
TITLE: Real time solution of Duncan-Mortensen-Zakai equation in nonlinear
filtering
ABSTRACT: It is well known that filtering theory plays an
important role both in commercial and military industries. In this
talk we shall give a brief introduction of nonlinear filtering.
The central problem is to solve Duncan-Mortensen-Zakai (DMZ)
equation (which is a time dependent parabolic equation) in real
time. We shall describe some previous rigorous methods in solving
special DMZ equation. We shall also describe our new general
method to solve DMZ equation in real time which is a joint work
with S. T. Yau.
* Huiqiang Jiang and Wei-Ming Ni (Minnesota)
TITLE: On Steady States of Van Der Waals Force Driven Thin Film
Equations
ABSTRACT: Let W Ì RN, N ³ 2 be a bounded
smooth domain and a > 1. We are interested in the singular elliptic equation
Dh =
1
a
h-a - p in W
with Neumann boundary conditions. We gave a complete
description of all continuous radially symmetric solutions. In
particular, we constructed nontrivial smooth solutions as well as
rupture solutions. Here a continuous solution is said to be a
rupture solution if its zero set is nonempty. When N = 2 and
a = 3, the equation has been used to model steady states of
van der Waals force driven thin films of viscous fluids. We also
considered the physical problem when total volume of the fluid is
prescribed.
* Ivana
Alexandrova (Toronto)
TITLE: The Scattering Amplitude at a Maximum of the Potential
ABSTRACT: We consider the semi-classical scattering amplitude for
a short range perturbation of the Laplacian at an energy which is
a maximum of the Laplacian. We prove that the scattering amplitude
is a semi-classical Fourier integral operator associated to the
appropriately defined scattering amplitude.
* Maxim Zyskin (Bristol)
TITLE: Liquid Crystals on Polyhedral Domains.
ABSTRACT: We consider director fields (ie maps to RP2) on a
contractible polyhedra with tangent boundary conditions on faces.
Tangent boundary conditions prevent continuity at vertices;
assuming continuity elsewhere, there are many topologically
inequivalent configurations, classified by certain homotopy
invariants. We establish a lower bound for the Dirichlet energy of
such maps as a function of the homotopy invariants. For maps on a
rectangular prism, we establish an upper bound which differs from
the lower bound by a factor which depends only on the aspect
ratios of the prism but not on the invariants. We discuss a
smooth-to singular-on edge transition for a certain topologically
nontrivial solution on a prism, as a function of prism aspect
ratios. This work has applications to new multi-stable nematic
liquid crystal devices.
* Siddhartha P Chakrabarty (UIC)
TITLE: Cancer drug delivery in three dimensions for a PDE driven
model using finite elements
ABSTRACT: The Galerkin finite element method for a three
dimensional case in spherical so-ordinates is used to develop
procedures for the optimal drug delivery to brain tumors. The
mathematical model comprises of a system of three coupled reaction
diffusion models, involving the density of tumor cells and normal
tissue as also the drug concentration. An optimal control problem
is formulated with the goal of minimizing the tumor cell density
and reducing the side effects of the drug. The classical method of
calculus of variations is used to obtain a coupled system of
forward and backward PDE's. Galerkin finite element method is then
used to realistically represent the brain structure. The Galerkin
ODEs are solved by a combination of Crank-Nicolson and
predictor-corrector methods.
* Alireza Yazdani (Loughborough)
TITLE: Derivation of the residual free bubbles using the Method of
Least Squares
ABSTRACT: The
Galerkin’s method of weighted residuals is based on local
approximation of the solution of a given differential equation,
where this approximation is obtained from substitution of a
piecewise linear interpolation into equation. Nodal values of the
piecewise solution are obtained throughout a reduction of the
problem to a linear system of equations. The locality is then
resolved via an assembly process, which, along with other steps,
form up the very concept of Finite Element Method. The so called
“bubble function” method is developed from the idea of enrichment
of the interpolation base functions by addition a bubble function,
normally coming from an infinite dimensional augmented linear
space, so that this bubble function takes zero at element
boundaries which in turn, confines the generated approximation
error inside the element as well as relaxing the non-homogeneity
of the original boundary conditions [1- 2]. Ideally, the bubble
function is the analytical solution of the residual differential
equation, subject to homogeneous boundary conditions. However, the
analytical solution is hardly obtainable in general, and as is the
case in many practical situations, a simple polynomial approximate
form is needed for computational purposes. Many people adopt the
aforementioned approach to treat differential equations
numerically within the context of finite element modelling of
physical phenomena. In this work we assume a polynomial form of
the bubble function and derive the approximate polynomial form of
the practical bubble, using the method of least squares. This
method turns out to yield high degree of accuracy, capability of
generalization to higher dimensions and non-linear differential
equations as well as being computerizable, where suggest a
benchmark for dealing with different classes of differential
equations.
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