Meets Tuesdays at 3:00 pm in Skiles 255 unless otherwise indicated
08/28/07
Title: Single and multitransition solutions for a class of nonlinear elliptic PDE's
Abstract: The equation -(Lapacian)u + F_u(x,u) = 0, x in R^n where F is 1-periodic in its arguments, provides a simple model for the study of phase transition behavior. We will survey recent existence results for solutions displaying transition behavior, namely spatial homoclinics and heteroclinics, and will discuss the methods used to obtain these results.
08/30/07
Title: Exponentially Decaying Solutions of Schrodinger Equations
Abstract: Are
there solutions of the one-dimensional variable coefficient Schrodinger
equation that are square integrable
with an exponential weight?
If we consider the initial value problem, there is certainly data for which the
solution does not have this exponential decay at any later time.
Instead, this talk will show how to construct such solutions by solving a
non-standard boundary value problem, as well as discuss why we might want to
do so! This construction relies strongly on a new commutator
estimate for the projections onto the positive and negative frequencies.
09/04/07
Title: On contact waves for Jin-Xin relaxation model
Abstract: We
first construct contact waves for Jin-Xin relaxation model. Such wave is
approximating the corresponding contact discontinuity of equilibrium conservation
laws. We then prove the contact waves are nonlinear stable under small
initial perturbation.
09/11/07
Prof. Ronghua Pan ,
Georgia
Tech. ( I will be away. Someone else
chairs that day)
09/18/07 Prof.
Ronghua Pan ,
Georgia
Tech.
( I
will be away. Someone else chairs that day)
09/25/07
Title: Existence and regularity of
solutions to shock reflection problem
10/02/07
Title: Asymptotic behavior of infinity harmonic functions near an isolated singularity
Abstract: I will
discuss an asymptotic behavior of the infinity harmonic
function near a non-removable isolated
10/09/07
(Fall
2007 Recess: No talk)
10/16/07
Title: Steady Water Waves: Theory
and Computation
10/25/07
(
Note the
unusual date: thursday at 3pm instead of tuesday at 3pm)
Title:
The porous medium
equation and the rate of relaxation for the critical case in the
Keller--Segal model
10/30/07
Title: The sharp constant in the
Hardy-Sobolev-Maz'ya inequality
11/06/07
Title: Drift diffusion equations with fractional diffusion and the quasi-geostrophic equation
Abstract: Motivated
by the critical dissipative quasi-geostrophic equation, we
prove that drift-diffusion equations with L2
initial data and
minimal assumptions on the drift are locally Holder continuous. As an
application we show that solutions of the quasi-geostrophic equation
with initial L2 data
and critical diffusion are
locally smooth for any space dimension. This problem was
proposed by previous
authors as a toy problem for the global regularity of solutions to 3D
Navier-Stokes equations.
11/13/07
Title: Curvature, continuity and uniqueness of optimal transportation maps
Abstract: Despite years of study, surprisingly little is understood about the optimal transportation of a mass distribution from one manifold to another, where optimality is measured against a cost function on the product space. I shall present a uniqueness criterion subsuming all previous criteria, yet which is among the very first to apply to smooth costs on compact manifolds, and only then when the topology is simple. I shall review the regularity theory of Ma, Trudinger, Wang and Loeper for optimal maps, and the counterexamples of Loeper, before explaining my surprising discovery with Young-Heon Kim (University of Toronto) that this
theory is based on a hidden pseudo-Riemannian structure, which leads to
a simple direct proof of a key result in the theory, and opens several new research directions.
11/20/07
Title: Surfactants in Foam Stability : a Phase Field Model
Abstract: 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, what is in agreement with experimentation.
Title: Liouville theorems for the Navier-Stokes equations
Abstract: The classical Liouville theorem for the heat equation says that any bounded solution defined in all space and for all negative times is constant. We will investigate the validity of a suitable version of this statement for the Navier-Stokes equations and discuss its connection to the regularity theory. Although the main questions arising in this context seem to be out of reach of the present-day methods, it is still possible to prove some partial results and formulate some plausible conjectures.
Title: Forward Backward and Elliptic Harnack Estimates for Singular
Parabolic Equations
Title: Pertubation of Hamiltonian systems via the Moreau-Yosida approximation
Abstract: The general theory of Hamiltonian ODE's for non-smooth Hamiltonian H, is well understood only in finite-dimensional spaces. In infinite-dimensional spaces such as manifolds or Hilbert spaces, very little appears to be known even at the level of existence of solutions. In this talk, following a work by Ambrosio and Gangbo, we propose a perturbation approach based on the Moreau-Yosida approximation, to solve the Hamiltonian system in the Wasserstein space. For the sake of illustration, we keep our focus on a specific example which appears in semigeostrophic systems.
02/05/08
Title: Unstable surface water waves
02/12/08
Title: The zero-electron-mass limit in the
hydrodynamic model (Euler-Poisson system)
02/19/08
Title: Free-
02/26/08
Title: Free-
Title: Mathematical Analysis and Numerical Simulation of Bose-Einstein Condensation
Title: Free-
04/08/08
Title: Free-
04/15/08
Title: Free-
04/22/08
Title: Free-
05/06/08
Title: Free-
Please contact me to volunteer to talk or recommend speakers.