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
24
Yin, Z.Y., Sun Yat Sen University
February
7
Phil Isett, Princeton University ** This
seminar is at 10am in room 312
Frederic Rousset,
Construction of
multi-solitons solutions for the water waves
February
14
Kenji Nakanishi, Kyoto University
Center-stable manifold of
the ground state in the energy space for the critcal wave
equation
February
21
No seminar today, Abel Day
February
22
Walter Craig
Vortex filament interactions
February 28
Vlad Vicol, Princeton University
On the
inviscid limit for the stochastic Navier-Stokes equations
We discuss recent results on the behavior in the infinite Reynolds number limit of invariant measures for the 2D stochastic Navier-Stokes equations. We prove that the limiting measures are supported on bounded vorticity solutions of the 2D Euler equations. Invariant measures provide a canonical object which can be used to link the fluids equations to the heuristic statistical theories of turbulent flow. Motivated by 2D turbulence considerations we are lead to the problem of well-posedness for the stochastic 2D Euler equations. This is joint work with Nathan Glatt-Holtz and Vladimir Sverak.
March 5 ** This seminar is at 11am in room 1302
Tak Kwong Wong
Local existence and uniqueness of Prandtl equations
The Prandtl equations, which describe the boundary layer behavior of a viscous incompressible fluid near the physical wall, play an important role in the zero-viscosity limit of Navier-Stokes equations. In this talk we will discuss the local-in-time existence and uniqueness for the Prandtl equations in weighted Sobolev spaces under the Oleinik's monotonicity assumption. The proof is based on weighted energy estimates, which come from a new type of nonlinear cancellations between velocity and vorticity.
March
7
Emmanuel Hebey
Recent
advances on Klein-Gordon-Maxwell-Proca systems
Mihai Tohaneanu, Johns Hopkins University ** This seminar is at
12pm
The
Strauss conjecture on black holes
The Strauss conjecture for the Minkowski spacetime in three dimensions states that the semilinear equation wave (u) = u^p has a global solution for compactly supported and sufficiently small data if $p> 1+\sqrt 2$. We prove a similar result in the context of Schwarzschild and
Kerr with small angular momentum black holes. This is joint work with H. Lindblad, J. Metcalfe, C. Sogge, and C. Wang.
March 14
Camillo De Lellis, University of Zurich
/ Princeton University
Regularity theory for
area-minimizing currents
It was established by Almgren at the beginning of the eighties that area-minimizing $n$-dimensional currents in Riemannian manifolds are regular up to a singular set of dimension at most $n-2$. To reach this goal Almgren developed an entirely new regularity theory, which occupies a very large monograph, published posthumously.
This talk is based on a series of joint works with Emanuele Spadaro, where we give alternative proofs to all Almgren's main steps, resulting into a much more manageable approach to his entire theory.
March 21
Yannick Sire, University of Marseille
Fractional
Ginzburg-Landau equations and boundary harmonic maps
March 28
Thomas Sideris, U.C. Santa Barbara
Almost global
existence of small solutions for 2d incompressible Hookean
elastodynamics
April 4
Marius Beceanu, IAS
Strichartz Estimates
for Equations with Time-Dependent Potentials
In this talk I present some inequalities valid for Schroedinger's equation and for the wave equation, which hold for a more general class of time-dependent potentials than previously known estimates.
April 18
Yan Yan Li, Rutgers University
A compactness
theorem for a fully nonlinear Yamabe problem under
a lower Ricci curvature bound
April 23
Fabio Pusateri, Princeton University
Global existence
for two-dimensional water waves
April 25
Yoshikazu Giga, University of Tokyo
Analyticity of
the Stokes semigroup in space of bounded functions
The Stokes
system is a linearized system of the Navier-Stokes equations
describing the motion of incompressible viscous fluids. It is
believed that the nonstationary problem is very close to the
heat equation. (In fact, if one considers the Stokes system in a
whole space R^n, the problem is reduced to the heat equation.)
The solution operator S(t) of the Stokes system is called the
Stokes semigroup. It is well-known that S(t) is analytic in the
L^p setting for a large class of domains including bounded and
exterior domains with smooth boundaries provided that p is
finite and larger than 1. This property is the same as the heat
semigroup. Moreover, for the heat semigroup it is analytic even
when p equals the infinity.
The corresponding (p=infinity) result for the Stokes semigroup
S(t) has been open for more than thirty years even if the domain
is bounded. Using a blowup-argument, we have now solved this
long-standing problem for a large class of domains, including
bounded and exterior domains. A key step is to derive a harmonic
pressure gradient estimate by a velocity gradient. We give a
sketch of the proof as well as a few possible applications to
the Navier-Stokes equations. This is a joint work of my student
Ken Abe and the main paper is going to appear in Acta
Mathematica.
April 30
Isabelle Gallagher, University
Paris 7
The
diffusion limit from a hard spheres system, via the linear
Boltzmann equation
In this talk we report on a recent result with Laure Saint-Raymond and Thierry Bodineau in which we derive the linear Boltzmann equation from a Newtonian system of hard spheres, following Lanford's proof. This convergence holds for a very long time and enables us to obtain the heat equation in the diffusion limit.
May 3
Edris Titi, UC Irvine
May 9
Philip Rosenau, Tel-Aviv University
On a Well-Tempered Diffusion
The classical transport theory as expressed by, say, the Fokker-Planck equation, lives in an analytical paradise but, in sin. Not only its response to initial datum spreads at once everywhere oblivious of the basic tenets of physics, but it also induces an infinite flux across a sharp interface. Attempting to overcome these difficulties one notices that the moment expansion of any of the micro ensembles of the kind that beget the equations of the classical mathematical physics, say the Chapman-Enskog expansion of Boltzmann Eq., if extended beyond the second moment, yields an ill posed PDE (the Pawla Paradox)!
We shall describe mathematical strategies to overcome these generic difficulties. The resulting flux-limited transport equations are well posed and capture some of the crucial effects of the original ensemble lost in moment expansion. For instance, initial discontinuities do not dissolve at once but persist for a while. There is a critical transition from analytical to discontinuous states with embedded sub-shock(s).
May 16
Neston Guillen, UCLA
Global
well-posedness for the homogeneous Landau equation
In joint work with Maria Gualdani we consider the homogeneous Landau equation from plasma physics. Both global well-posedness and exponential decay to equilibrium are proved assuming only boundedness and spatial decay of the initial distribution. In particular, we can
handle discontinuous initial conditions that might be far from equilibrium. Despite the equation not having a maximum principle the key steps of the proof rely on barrier arguments and parabolic regularity theory.
September
6
Edriss Titi, University of California - Irvine
On the Loss of Regularity
for the Three-Dimensional Euler Equations
We will present some sharp estimates about the transport of BMO-type spaces, via a bi-Lipschitz measure preserving map
in the Euclidean space. More precisely, we are interested in inequalities of the following type: $$ \| f(\phi) \|_X \lesssim C(\phi) \|f\|_X $$
where $X$ is a space like BMO, Lipschitz space, Hardy space, Carleson measure spaces .... and $\phi$ is a bi-Lipschitz measure preserving map.
The aim is to prove such inequalities with a sharp constant $C(\phi)$. We want to emphasize how the "measure preserving" property allows us to
get some improved inequalities. Then, we will explain how we can use this argument to describe a new framework for 2D Euler equations. We can define a space strictly containing $L^\infty$ where global well-posedness results can be proved with the vorticity living in this new space.
We study the problem of the propagation and nonlinear interaction of impulsive gravitational waves for the vacuum Einstein equations. The problem is studied in the context of a characteristic initial value problem with data given on two null hypersurfaces and containing curvature delta singularities. For a single impulsive gravitational wave, we show that in the resulting spacetime, the delta singularity propagates along a 3-dimensional characteristic hypersurface, while away from that hypersurface the spacetime remains smooth. We also construct spacetimes representing interaction of two impulsive gravitational waves in which the curvature delta singularities propagate along two 3-dimensional null hypersurfaces intersecting to the future of the data. To the past of the intersection, the spacetime can be thought of as containing two independent, non-interacting impulsive gravitational waves and the intersection represents the first instance of their nonlinear interaction. Our analysis extends to the region past their first interaction and shows that the spacetime still remains smooth away from the continuing propagating individual waves. This is joint work with Igor Rodnianski.
Spring 2012
January 19−Δu + εu−|u|p−2u + |u|q−2u = 0, x ∈ RN, where N ≥ 3, q > p > 2 and when ε > 0 is a small parameter. We give a complete characterization of all possible asymptotic regimes as a function of p, q and N. The behavior of solutions depends sensitively on whether p is less, equal or bigger than the critical Sobolev exponent p∗ = 2N/(N−2). For p < p∗ the solution asymptotically coincides with the solution of the equation in which the last term is absent. For p > p∗ the solution asymptotically coincides with the solution of the equation with ε = 0. In the most delicate case p = p∗ the asymptotic behavior of the solutions is given by a particular solution of the critical Emden–Fowler equation, whose choice depends on ε in a nontrivial way.
We study the leading order behavior of positive solutions of the equation
April 10I will discuss a non-local eigenvalue problem that arises as the Euler-Lagrange equation of Rayleigh quotients in the fractional Sobolev spaces. This can be seen as a non-local or fractional version of the eigenvalue problem for the p-Laplacian. In particular, I will talk about the limiting case when p goes to infinity for which the eigenvalues exhibit some strange behaviour that can be seen even in some one-dimensional examples.
Eric Lindgren (NTNU)
Fractional eigenvalues
May 31
Messoud Efendiev
Finite and infinite dimensional attractors for porous medium
equations
June 7
Slim Ibrahim, University of Victoria ***Please note this talk is at
10:00am
Existence of a ground state
and scattering for a nonlinear Schroedinger equation with
critical growth
This talk concerns the focusing energy-critical nonlinear
Schroedinger equation with a mass-supercritical and
energy-subcritical perturbation. In particular, we consider the
existence of a ground state and the scattering problem in
the spirit of Kenig-Merle.
This is a joint work with Akahori, Kikushi and Nawa
Adimurth Tifrcam, Bangalore **Please note
this talk is at 11:00 am
Structure Theorem for entropy
solutions of Conservation Law
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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