# Analysis Seminar

## Spring 2013

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
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 1302Tak Kwong WongLocal existence and uniqueness of Prandtl equationsThe 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

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

We will examine the IVP for equations of 2d incompressible isotropic elastodynamics.
We show that classical solutions exist “almost globally" in time, for small initial perturbations of the identity.
We use the energy method with a ghost weight. This is combined with decay estimates, obtained via generalized Sobolev inequalities and weighted L^2 bounds. The argument exploits the inherent null structure of the nonlinearities.
This is joint work with Zhen Lei and Yi Zhou of Fudan University.

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 IrvineMay 9 Philip Rosenau, Tel-Aviv University On a Well-Tempered DiffusionThe 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.


## Fall 2012

September 6
Edriss Titi, University of California - Irvine
On the Loss of Regularity for the Three-Dimensional Euler Equations

A basic example of shear flow was introduced by DiPerna and Majda to study the weak limit of oscillatory solutions of the Euler equations of incompressible ideal fluids. In particular, they proved by means of this example that weak limit of solutions of Euler equations may, in some cases, fail to be a solution of Euler equations. We use this shear flow example to provide non-generic, yet nontrivial, examples concerning the immediate loss of smoothness and ill-posedness of solutions of the three-dimensional Euler equations, for initial data that do not belong to C1; . Moreover, we show by means of this shear  flow example the existence of weak solutions for the three-dimensional Euler equations with vorticity that is having a nontrivial density concentrated on non-smooth surface. This is very diferent from what has been proven for the two-dimensional Kelvin-Helmholtz problem where a minimal regularity implies the real analyticity of the interface. Eventually, we use this shear flow to provide explicit examples of non-regular solutions of the three-dimensional Euler equations that conserve the energy, an issue which is related to the Onsager conjecture. In addition, we will use this shear  flow to provide a nontrivial example for the use of vanishing viscosity limit, of the Navier-Stokes solutions, as a selection principle for uniqueness of weak solutions of the 3D Euler equations.
This is a joint work with Claude Bardos.

September 20
Philippe LeFloch, University of Paris 6
Finite energy method for compressible fluids

I will discuss the initial value problem for the Euler system of compressible fluid flows governed by a general pressure law, when solutions enjoy a certain symmetry (for instance, plane symmetry) and have solely finite total energy. In a recent series of papers with Pierre Germain (Courant Institute), I have established an existence and compactness theory for this problem and analyzed the vanishing viscosity--capillarity limit in weak solutions with finite energy to the Navier-Stokes-Korteweg system. The proposed approach is referred to as the Finite Energy Method for compressible fluid flows and leads to a generalization of DiPerna's theorem (bounded solutions) and LeFloch-Westdickenberg's theorem (finite energy solutions) to the Euler system of polytropic fluids.

September 27
Andrew Comech, Texas A&M
Linear instability of solitary waves in nonlinear Dirac equation

We study the linear instability of solitary wave solutions  to the nonlinear Dirac equation (known to physicists as the Soler model). That is, we linearize the equation at a  solitary wave and examine the presence of eigenvalues with  positive real part.

We show that the linear instability of the small amplitude  solitary waves is described by the Vakhitov-Kolokolov stability
criterion which was obtained in the context of the nonlinear  Schroedinger equation: small solitary waves are linearly
unstable in dimensions 3, and generically linearly stable in 1D.

A particular question is on the possibility of bifurcations  of eigenvalues from the continuous spectrum; we address it
using the limiting absorption principle and the Hardy-type estimates.

The method is applicable to other systems, such as the  Dirac-Maxwell system.

Some of the results are obtained in collaboration with  Nabile Boussaid, Université de Franche-Comté, and
Stephen Gustafson, University of British Columbia.

October 4
Tej-eddin Gouhl, CIMS

October 11
Arie Israel, CIMS
The Whitney Extension Problem in Sobolev Spaces

October 18
Christophe Lacave  (Paris 7)
Large time behavior for the two-dimensional motion of a disk in a viscous fluid

October 25
Wang Changyou, University of Kentucky
Well-posedness of the nematic liquid crystal flow in  $L3_{uloc}(R3)$

In this talk, I will consider a simplified Ericksen-Leslie system modeling the incompressible nematic liquid crystal flow.  For any initial data $(u_0,d_0)$, with $(u_0,\nabla d_0)$ belonging to   $L3_{uloc}(R3)$-- the space of uniformly locally $L3$-integrable functions, we will show the   local well-posedness provided that the $L3_{uloc}(R3)$-norm of $(u_0,\nabla d_0)$ is small.    A characterization of the first singular time for such a solution will also be given. This is a joint  work with Jay Hineman.

November 1
Jeremie Szeftel, ENS, France
Bounded L2 curvature conjecture

November 8
Gilles Francfort, University of Paris 13
Elasto-plasticity: heterogeneity and homogenization

November 15
Sylvia Serfaty, CIMS
Large vorticity local minimizers to the Ginzburg-Landau functional

November 29
Andrew Lawrie, University of Chicago ** This seminar is at 11am
Classification of large energy equivariant wave maps

I will discuss some recent joint work with Raphael Cote, Carlos Kenig, and Wilhelm Schlag. We consider energy critical 1-equivariant wave maps, R1+2 ! S2. This problem admits a unique (up to scaling) harmonic map, Q, given by stereographic projection. We show that every topologically trivial (degree 0) solution with energy less than twice the energy of Q exists globally in time and scatters. Next we establish a classi cation, in the spirit of recent work by Duyckaerts, Kenig, and Merle, of all degree 1 solutions with energy below three times the energy of Q.

Vedran Sohinger, University of Pennsylvania. **This seminar is at 11:55am
The Boltzmann equation, Besov spaces, and optimal decay rates in R^n

In this talk we will study the large-time convergence to the global Maxwellian of perturbative classical solutions to the Boltzmann equation on R^n, for n geq 3, without the angular cut-off assumption. We prove convergence of the k-th order derivatives in the norm L^r_x (L^2_v), for any 2 leq r leq infty, with optimal decay rates, in the sense that they are equal to the rates which one obtains for the corresponding linear equation. The initial data is assumed to lie in a mixed norm space involving the negative homogeneous Besov space of order geq -n/2 in the space variable, without a smallness assumption on the appropriate norm. The space for the initial data is physically relevant since it contains spaces of the type L^p_x (L^2_x), by the use of Besov-Lipschitz space embeddings. Due to the nature of the vector valued spaces, we need to use a vector analogue of the Calderon-Zygmund theory to prove the necessary nonlinear energy estimates. These results hold both in the hard and soft potential case. Furthermore, in the hard potential case, we prove additional optimal decay results if the order of the Besov space belongs to [-(n+2)/2,-n/2). The latter result requires a closer study of the spectrum of the linearized Boltzmann operator for small frequencies dual to the spatial variable. This is a joint work with Robert Strain.

December 4

Frederic Bernicot, University of Nantes  **Please note special day and place of this event 10am room 1314
Transport of BMO-type spaces by a measure-preserving map and applications to 2D Euler equation

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.


December 6
Jonathan Luk, Princeton University **Please note this seminar is at 11am
Impulsive Gravitational Waves

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.


Remi Schweyer, Toulouse**Please note this seminar is at 11:55am
Stable blow-up dynamics for the parabolic-elliptic Keller-Segel model of chemotaxis

December 13
Clement Mouhot, Cambridge

Spring 2012

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.

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 stationary and evolutionary variational inequalities and associated obstacle problems when the underlying asset is modeled by a degenerate diffusion process. This is joint work with Panagiota Daskalopoulos (Department of Mathematics, Columbia University) and Camelia Pop (Department of Mathematics, Rutgers University).

February 23
Vitaly Moroz (Swansea, UK)
Existence and concentration for nonlinear Schroedinger equations with fast decaying potentials

We discuss the existence of positive stationary solutions for a class of nonlinear Schrödinger equations. Amongst other results, we prove the existence of semi-classical solutions which concentrate around a positive local minimum of the potential. The novelty is that no restriction is imposed on the rate of decay of the potential at infinity. In particular, we cover the case where the potential is compactly supported. This is joint work with Jean Van Schaftingen(Louvain-la-Neuve, Belgium)

Erwan Faou

March 1
Frederic Rousset (Rennes)

March 8
Masashi Aiki

March 9 ** this talk is at 1pm in WWH 512
Pengfei Guan (McGill University)
On a uniqueness problem in classical geometry and the maximum principle

March 15
P-E Jabin

March 22
Tuomas Hytonen
How much, or little, is needed for Harmonic Analysis?

One aspect of my recent work has been developing harmonic analysis under minimal assumptions on the space on which the considered functions are defined. Perhaps surprisingly, some classical methods, which at first sight seem to rely heavily on the structure and symmetries of the Euclidean space, can actually be extended to very general settings. On the other hand, some methods developed to tackle with abstract spaces, have shown to be instrumental for getting sharp results for classical inequalities on the Euclidean space.

March 27 ** Special Seminar 3/27 @ 11:30a.m.**
Zhen Lei
On Incompressible Euler and navier-Stokes Equations

In this talk I will report our recent results on finite time finite energy singularities of a 3D incompressible inviscid model of Euler and Navier-Stokes equations and a Liouville theorem for axi-symmetric navier-Stokes equations.

March 29
Victor Lie, Princeton University
Topics in the time-frequency analysis

This talk will be structured as follows: we start by discussing several facts about the history and evolution of the time-frequency area (Fourier Series, Calderon-Zygmund theory, wave-packet theory) and then refer to the general approach/tools for solving a time-frequency problem. Next we present two fundamental topics in this ¯eld: the pointwise con- vergence of the Fourier Series and the boundedness of the Bilinear Hilbert transform. Further we address several aspects of our work including the problem of the boundedness of the Bilinear Hilbert transform on smooth curves and the question regarding the boundedness of the Polynomial Car-leson operator

April 5

Cyrill Muratov
Asymptotic properties of ground states of scalar field equations with vanishing parameter

We study the leading order behavior of positive solutions of the equation
−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.

April 10Eric Lindgren (NTNU)Fractional eigenvalues
I 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.

April 12

David Gerard-Varet (Paris 7)
Dynamics of a rough body in a viscous fluid

April 19
Marcel Guardia, IAS **please note this seminar is at 10am**
Growth of Sobolev norms for the cubic defocusing nonlinear Schr\"odinger equation in polynomial time

We consider the cubic defocusing nonlinear Schr\"odinger equation in the two dimensional torus. Fix s>1. Colliander, Keel, Staffilani, Tao and Takaoka (2010) proved existence of solutions with s-Sobolev norm growing in time by any given factor R. Refining their methods in several aspects we find solutions with s-Sobolev norm growing in polynomial time in R. This is a joint work with V. Kaloshin.

Dehua Wang (University of Pittsburgh)

April 26
Alberto Bressan (Penn State)
Nash equlibria for a odel of traffic flow

In connection with the Lighthill-Whitham model of traffic flow, a cost functional can be introduced depending on the departure
time and on the arrival time of each driver. Under natural assumptions, there exists a unique globally optimal solution, minimizing the total cost to all drivers. In a realistic situation, however, the actual traffic is better described by the Nash equilibrium solution, where no driver can lower his individual cost by changing his own departure time. A characterization of the Nash solution can be provided, establishing its existence and uniqueness. Extensions to the case of several groups of drivers on a network of roads will also be discussed, together with open problems.

May 1 **Please note special day**
Yifeng Yu (UC Irvine)
G-equations in the modeling of turbulent flame speeds

Predicting turbulent flame speed ($s_T$) is a fundamental
problem in turbulent combustion theory. Several simplified models have been proposed to study $s_T$.  The G-equation (A Hamilton-Jacobi level set equation) is a very popular model in turbulent combustion. Two important projects are (1) establish the theoretical existence of $s_T$ and (2) determine the dependence of turbulent flame speeds on the turbulence intensity (think of the relation between the spreading velocity of wild fire and strength of the wind). In this talk, I will present some theoretical results under the G-equation model. If time permits, I will also compare it with predictions from a model introduced by Majda and
Souganidis.  These are joint works with Jack Xin. 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.

May 17
Benjamin Texier
The onset of instability in quasi-linear systems

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

Structure Theorem for entropy solutions of Conservation Law

## Fall 2011

September 15
Camil Muscalu, Cornell University
Triangular Fourier series and physical reality

September 22
Zaher Hani, CIMS
Long time strong instability and unbounded orbits for some nonlinear Schrodinger equations

Tuesday, September 27, 10:00 a.m. in room 1314
Xavier Cabre, Universitat Politecnica de Catalunya
Minimal surfaces and entire solutions of the Allen-Cahn equation

September 29
The energy critical Schrodinger equation on the 3D torus

October 6
George Hagstrom, CIMS
A Krein-Moser Theorem for Vlasov-Poisson

October 13
Federica Sani, CIMS and Univ of Milan

October 20
Jacob Bedrossian, CIMS
Global existence and finite time blow-up for critical Patlak-Keller-Segel models with spatially inhomogeneous diffusion

Special Analysis Seminar Tuesday October 25
2 pm room 1302 WWH Nassif Ghoussoub
University of British Columbia Selfdual polar decomposition

October 27
Vincent Duchene, Columbia University
Asymptotic models for internal waves in the long wave regime, and the dead water phenomenon

November 3
Alex Ionescu, Princeton
On the local extension of Killing vector-fields in Ricci flat manifolds

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 10
Comgming Li, University of Boulder
A degree theory approach for the shooting method

November 17
Christophe Lacave, Universite Paris-Diderot (Paris 7)
Well-posedness for Euler 2D in non-smooth domains

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

December 1st
Oana Pocovnicu (Imperial College, London)
Effective dynamics of a non-linear wave equation

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.

December 15
Daniela Tonon (ICERM and SISSA)
Regularity of viscosity solutions for Hamilton-Jacobi equations.

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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