The analysis seminar covers a wide range of topics in analysis with particular emphasis on partial differential equations. Many of the speakers are Courant Institute visitors and postdocs. A seminar talk may cover original research or report on an interesting paper. The seminar meets on Thursdays at 11:00 am in room 1302 of Warren Weaver Hall at 251 Mercer Street, New York. Talks generally last an hour. A few special analysis seminars may be held at other times and locations.
The most reliable and inclusive list of weekly seminars and
events is to be found in the weekly bulletin that is posted on
a day-by-day basis on the CIMS home page.
January 19
Tianling Jin (Rutgers)
A fractional Yamabe flow and some applications.
In this talk, we introduce
a fractional Yamabe flow involving nonlocal conformally
invariant operators on the conformal infinity of
asymptotically hyperbolic manifolds, and show that on the
conformal spheres $(S^n, [g_{S^n}])$, it converges to the
standard sphere up to a Mobius diffeomorphism. This result
allows us to obtain extinction profiles of solutions of some
fractional porous medium equations. In the end, we use this
fractional fast diffusion equation, together with its
extinction profile and some estimates of its extinction time,
to improve a Sobolev inequality via a quantitative estimate of
the remainder term. This is joint work with Jingang Xiong.
January 25 ** Special
Wednesday Seminar**
Frederic Bernicot
Differential inclusions
describing unilateral constraints
We will
present existing and new results about differential
inclusions, involving proximal normal cone. This tool aims to
encode unilateral constraints in some differential equations
of order 1 and 2. Using some convex analysis, we presnet
results of global existence for solutions of such equations
and we will discuss about the uniqueness of them
January 26
Lazhar Tayeb (University of Tunis)
Approximation by
diffusion and homogenization for Fermi-Dirac-Poisson
Statistics
We study the
diffusion approximation of a Boltzmann-Poisson system
dealing with Fermi-Dirac statistics in the presence of
an extra external oscillating electrostatic potential.
The relative entropy disspation and the two-scale Young
measures are used to prove a two-scale strong convergence
leading to a nonlinear Drift-diffusion with a effective
potential and coupled to Poisson equation.
February 2
Hideyuki Miura
On fundamental
solutions for fractional diffusion equations with divergence
free drift
We are
concerned with fractional diffusion equations in the presence
of a divergence free drift term. By using the Nash approach,
we show the existence of fundamental solutions, together with
the continuity estimates, under weak regularity assumptions on
the drift. Our results give the alternative proof of
Caffarelli-Vasseur's theorem on the regularity for the
critical 2D dissipative quasi-geostrophic equations. This is a
joint work with Yasunori Maekawa.
February 9
Alessio Figalli, UT Austin ** this talk is at 10am**
Di Perna-Lions theory, with
application to semiclassical limits for the Schrodinger
equation
At the beginning of the '90, DiPerna and Lions studied in detail the connection between transport equations and ordinary differential equations. In particular, by proving an existence and uniqueness result at the level of the transport equation, they obtained (roughly speaking)
existence and uniqueness of solutions for ODEs with Sobolev vector-fields for a.e. initial condition. Ten years later, Ambrosio has been able to extend such a result to BV vector fields. In some recent works we have investigated this theory in a more general setting, which allows us to show the semiclassical convergence of the quantum dynamics to the Liouville dynamics for the linear Schrodinger equations, under very weak regularity assumptions on the potential. In analogy to the classical DiPerna-Lions' theory, the price to pay for allowing singular potential is that the convergence result holds true only for "a.e. initial data", where "a.e." is with respect to a suitable family of reference measures in the space of the initial data. The aim of this talk is to give an overview of these results.
Mahir Hadzic
(MIT)
The Classical Stefan problem and the vanishing surface
tension limit
We develop a
new unified framework for the treatment of well-posedness for
the Stefan problem with and without surface tension. We
provide new estimates for the regularity of the moving surface
in the absence of surface tension. We conclude by proving that
solutions of the Stefan problem with positive surface tension
converge to solutions of the
Stefan problem without surface tension. This is joint work
with S. Shkoller.
February 16
Paul Feehan (Rutgers)
Degenrate Obstacle
Problems
Degenerate elliptic and parabolic obstacle problems arise in mathematical finance when valuing American-style options on an underlying asset modeled by a degenerate diffusion process. We will describe our work on existence, uniqueness, and regularity of solutions to stationaryand evolutionary variational inequalities and associated obstacle problemswhen the underlying asset is modeled by a degenerate diffusionprocess.This is joint work with Panagiota Daskalopoulos (Department ofMathematics, Columbia University) and Camelia Pop (Department ofMathematics, Rutgers University).
February 23
Erwan Faou
March 1
Frederic Rousset, Rennes
Uniform regularity and inviscid limit for free surface Navier-Stokes
March 8
Masashi Aiki
March 15
Cyrill Muratov
April 12
David Gerard-Varet, Paris 7
April 19
Dehua Wang (University of Pittsburgh)
April 26
Alberto Bressan (Penn State)
May 3
Anne-Sophie de Suzzoni (CERGY)
On statistical description of the flow of dispersive PDEs
May 10
Ben Schweitzer, University of Dortmund
Homogenization of Maxwell equations in complex geometries: on the counter-intuitive behavior of meta materials
Optically
active meta-materials can nowadays be constructed as
physical objects. They can have astonishing properties or lead
to striking effects, the key-words are negative
refraction, perfect imaging, and cloaking. I will present the
effect of a negative magnetic permeability of the
effective material and perfect light transmission
through small holes in a metallic structure.
Mathematically, we analyze the time-harmonic Maxwell equations
in a heterogeneous medium, the coefficients of the
equation can oscillate on a small spatial scale and the
oscillations of the values can be very large. The
heterogeneity of the optical medium is prescribed by
specifying the permittivity, which varies on a small length
scale \eta. The electric and magnetic fields are determined by
the time-harmonic Maxwell system. We analyze the weak limits
of the electric and magnetic fields as \eta tends to
zero, obtaining an "effective equation" that characterizes the
limits. The coefficients of the effective equation
describe the behavior of the metamaterial.
This is joint work with G. Bouchitte and with A. Lamacz.
September 22
Tuesday, September 27, 10:00 a.m. in room 1314
September 29
October 6
October 13
October 20
Special Analysis Seminar Tuesday October 25
October 27
November 3
I will discuss some recent work, joint with S. Klainerman, on the problem of extension of Killing vector-fields in manifolds that satisfy the Einstein vacuum equations. This problem is motivated by the black hole rigidity conjecture, concerning the uniqueness of the Kerr family among regular, stationary black hole solutions of the Einstein vacuum equations.
November 10November 17
The well-posedness of the Euler system
has been of course the matter of many works, but a common point
in all the previous studies is that the boundary is at least
$C^{1,1}$. In a first part, we will establish the existence of
global weak solutions of the 2D incompressible Euler equations
for a large class of non-smooth open sets. These open sets are
the complements (in a simply connected domain) of a finite
number of connected compact sets with positive capacity.
Existence of weak solutions with $L^p$ vorticity is deduced from
an approximation argument, that relates to the so-called
$\gamma$-convergence of domains. In a second part, we will prove
the uniqueness if the open set is the interior or the exterior
of a simply connected domain, where the boundary has a finite
number of corners. Although the velocity blows up near these
corners, we will get a similar theorem to the Yudovich's result,
in the case of an initial vorticity with definite sign, bounded
and compactly supported. The key point for the uniqueness part
is to prove by a Liapounov energy that the vorticity never meets
the boundary. The existence part is a work in collaboration with
David Gerard-Varet.
SPECIAL ANALYSIS SEMINAR,
November 22, 11 a.m., room 1314
Shen Zhongwei, University of Kentucky
The Periodic Homogenization of
Green's and Neumann Functions
We consider the non-linear wave
equation on the real line iu_t-|D|u=|u|^2u. Its resonant
dynamics is given by the Szego equation, which is a completely
integrable non-dispersive non-linear equation. We show that the
solution of the wave equation can be approximated by that of the
resonant dynamics for a long time. The proof uses the
renormalization group method introduced by Chen, Goldenfeld, and
Oono in the context of theoretical physics. As a consequence, we
obtain growth of high Sobolev norms of certain solutions of the
non-linear wave equation, since this phenomenon was already
exhibited for the Szego equation.
December 8
Aynur Bulut (IAS)
The defocusing Cubic Nonlinear Wave Equation in the Energy
Super-critical Regime
In this talk, we will discuss a series of recent works on the
global well-posedness and scattering conjecture for the
defocusing cubic nonlinear wave equation in the energy
super-critical regime, that is dimensions five and higher.
More precisely, using a concentration compactness approach we
show that if a solution remains bounded in the critical Sobolev
space throughout its maximal interval of existence then it is
global and scatters.
We present
two results on the regularity of viscosity solutions of
Hamilton-Jacobi equations obtained in collaboration with
Professor Stefano Bianchini. When the Hamiltonian is strictly
convex viscosity solutions are semiconcave, hence their gradient
is BV. First we prove the SBV regularity of the gradient of a
viscosity solution of the Hamilton-Jacobi equation u_t+
H(t,x,D_x u)=0 in an open set of R^(n+1), under the hypothesis
of uniform convexity of the Hamiltonian H in the last variable.
Secondly we remove the uniform convexity hypothesis on the
Hamiltonian, considering a viscosity solution u of the
Hamilton-Jacobi equation u_t+ H(D_x u)=0 in an open
set of R^(n+1) where H is smooth and convex. In this case the
viscosity solution is only locally Lipschitz. However when the
vector field d(t,x):=H_p(D_xu(t,x)), here H_p is the gradient of
H, is BV for all t in [0,T] and suitable hypotheses on the
Lagrangian L hold, the divergence of d(t, ) can have
Cantor part only for a countable number of t's in [0,T]. These
results extend a result of Bianchini, De Lellis and Robyr for a
uniformly convex Hamiltonian which depends only on the
spatial gradient of the solution.
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