Analysis Seminar


Time Location Seminar Chair
Tuesdays 9:45am AP&M 7421 Andrej Zlatos

Fall 2017

Date Speaker Title + Abstract
September 27
(Wed) at 3pm
Yuan Yuan
Syracuse University
Diameter rigidity for Kahler manifolds with positive bisectional curvature
I will discuss the recent work with Gang Liu on the diameter rigidity for Kahler manifolds with positive bisectional curvature.
October 24 Igor Kukavica

Postponed to November 21

October 31 In-Jee Jeong
Princeton University
Finite time blow-up for strong solutions to the 3D Euler equations
We show finite time blow-up for strong solutions to the 3D Euler equations on the exterior of a cone. The solutions we construct have finite energy, and velocity is axisymmetric and are Lipschitz continuous before the blow up time. We achieve this by first analyzing scale-invariant (radially homogeneous) solutions, whose dynamics is governed by a 1D system. Then we make a cut-off argument to ensure finiteness of energy.
November 7 Hantaek Bae
Regularity and decay properties of the incompressible Navier-Stokes equations
In this talk, I will consider the incompressible Navier-Stokesequations in the mild solution setting. Using this setting, I will show how to obtain analyticity of mild solutions using the Gevrey regularity technique. This regularity enables to get decay rates of weak solutions of the Navier-Stokes equations. This idea can be applied to other dissipative equations with analytic nonlinearities. I will finally consider the regularity of the flow map of mild solutions using the Log-Lipschitz regularity of the velocity field.
November 14 Hung Tran
On some selection problems for fully nonlinear, degenerate elliptic PDEs
I will describe some interesting selection problems for fully nonlinear, degenerate elliptic PDEs. In particular, I will focus on the vanishing discount procedure and show the convergence result via a new variational technique.
November 14
at 3pm
Nikolay Shcherbina
University of Wuppertal
Squeezing functions and Cantor sets
We construct "large" Cantor sets whose complements resemble the unit disk arbitrarily well from the point of view of the squeezing function, and we construct "large" Cantor sets whose complements do not resemble the unit disk from the point of view of the squeezing function.
November 21 Igor Kukavica
On the size of the nodal sets of solutions of elliptic and parabolic PDEs
We present several results on the size of the nodal (zero) set for solutions of partial differential equations of elliptic and parabolic type. In particular, we show a sharp upper bound for the $(n-1)$-dimensional Hausdorff measure of the nodal sets of the eigenfunctions of regular analytic elliptic problems in ${\mathbb R}^n$. We also show certain more recent results concerning semilinear equations (e.g. Navier-Stokes equations) and equations with non-analytic coefficients. The results on the size of nodal sets are connected to quantitative unique continuation, i.e., on the estimate of the order of vanishing of solutions of PDEs at a point. The results on unique continuation are joint with Ignatova and Camliyurt.
November 28 Yifeng Yu
UC Irvine
Some properties of the mysterious effective Hamiltonian: a journey beyond well-posedness
A major open problem in the periodic homogenization theory of Hamilton-Jacobi equations is to understand "deep" properties of the effective equation, in particular, how the effective Hamiltonian depends on the original Hamiltonian. In this talk, I will present some recent progress in both the convex and non-convex settings.
December 5 Georgios Moschidis
Princeton University
A proof of the instability of AdS spacetime for the Einstein's null dust system
The AdS instability conjecture is a conjecture about the initial value problem for Einstein vacuum equations with a negative cosmological constant. Proposed by Dafermos and Holzegel in 2006, the conjecture states that generic, arbitrarily small perturbations to the initial data of the AdS spacetime, under evolution by the vacuum Einstein equations with reflecting boundary conditions on conformal infinity, lead to the formation of black holes. Following the work of Bizon and Rostworowski in 2011, a vast amount of numerical and heuristic works have been dedicated to the study of this conjecture, focusing mainly on the simpler setting of the spherically symmetric Einstein--scalar field system. In this talk, we will provide the first rigorous proof of the AdS instability conjecture in the simplest possible setting, namely for the Einstein--null dust system, allowing for both ingoing and outgoing dust. This system is a singular reduction of the spherically symmetric Einstein--massless Vlasov system, in the case when the Vlasov field is supported only on radial geodesics. In order to overcome the "trivial" break down occurring once the null dust reaches the center $r=0$, we will study the evolution of the system in the exterior of an inner mirror with positive radius $r_{0}$ and prove the conjecture in this setting. After presenting our proof, we will briefly explain how the main ideas can be extended to more general matter fields, including the regular Einstein--massless Vlasov system.

Winter 2018

Date Speaker Title + Abstract
January 9
at 9:55am
Gautam Iyer
Winding of Brownian trajectories and heat kernels on covering spaces
We study the long time behaviour of the heat kernel on Abelian covers of compact Riemannian manifolds. For manifolds without boundary work of Lott and Kotani-Sunada establishes precise long time asymptotics. Extending these results to manifolds with boundary reduces to a "cute" eigenvalue minimization problem, which we resolve for a Dirichlet and Neumann boundary conditions. We will show how these results can be applied to studying the "winding" / "entanglement" of Brownian trajectories.
January 16 Peter Hintz
UC Berkeley

Postponed to May 1

January 23 Michele Coti-Zelati
Imperial College
Vortex axisymmetrization, inviscid damping, and vorticity depletion in the linearized 2D Euler equations, I
We study the 2D Euler equations linearized around smooth, radially symmetric vortices with strictly decreasing vorticity profiles. Under a trivial orthogonality condition, we prove that the perturbation vorticity winds up around the vortex and weakly converges to a radially symmetric configuration, as time goes to infinity. This process is known as "vortex axisymmetrization" in the physics literature and is thought to stabilize vortex structures such as hurricanes and cyclones. Additionally, the velocity field converges strongly in L2 to the corresponding equilibrium (as time goes to infinity) and we give optimal decay rates in weighted L2 spaces. Interestingly, the rate of decay is faster for the linearized 2D Euler equations than for the passive scalar equation. The passive scalar rate is degraded by the slow mixing at the vortex core, but the linearized 2D Euler equations expel vorticity from the origin leading to a faster decay rate.
January 30 Huy Nguyen
Princeton University
Global Regularity for 1D Viscous Compressible Fluid Models with Degenerate Viscosity
We will discuss a class of one-dimensional compressible Navier-Stokes type equations in which the viscosity depends on the density and vanishes with the density. We prove large data global regularity for a class of models covering viscous shallow water equations. Another result proves a conjecture of Peter Constantin on singularity formation for a model describing slender axisymmetric fluid jets. The proofs of these results rely on a new equation for the flux in the momentum equation.
February 6 Ben Dodson
Global well-posedness for Schrodinger maps with small Besov norm
In this talk we will show a scattering-type result for Schrodinger maps with small Besov norm. The proof extends an earlier result of Bejenaru, Ionescu, and Kenig.
February 13
at 3pm in 6218
Bingyuan Liu
UC Riverside
Geometric analysis on the Diederich–Fornæss index
In this talk, we discuss the Diederich–Fornæss index in several complex variables. A domain $\Omega\subset \mathbb C^n$ is said to be pseudoconvex if $−\log(−\delta(z))$ is plurisubharmonic in $\Omega$, where $\delta$ is a signed distance function of $\Omega$. The Diederich–Fornæss index has been introduced since 1977 as an index to refine the notion of pseudoconvexity. After a brief review of pseudoconvexity, we discuss this index from the point of view of geometric analysis. We will find an equivalent index associated to the boundary of domains and with it, we are able to obtain accurate values of the Diederich–Fornæss index for many types of domains.
February 20 Klaus Widmayer
EPFL Lausanne
Global Stability of Solutions to a Beta-Plane Equation
We study the motion of an incompressible, inviscid two-dimensional fluid in a rotating frame of reference. There the fluid experiences a Coriolis force, which we assume to be linearly dependent on one of the coordinates. This is a common approximation in geophysical fluid dynamics and is referred to as beta-plane. In vorticity formulation the model we consider is then given by the Euler equation with the addition of a linear anisotropic, non-degenerate, dispersive term. This allows us to treat the problem as a quasilinear dispersive equation whose linear solutions exhibit decay in time at a critical rate. Our main result is the global stability and decay to equilibrium of sufficiently small and localized solutions. Key aspects of the proof are the exploitation of a “double null form” that annihilates interactions between waves with parallel frequencies and a Lemma for Fourier integral operators, which allows us to control a strong weighted norm and is based on a non-degeneracy property of the nonlinear phase function associated with the problem. Joint work with Fabio Pusateri; prior work with Tarek Elgindi.
February 20
at 2pm in 7218
Laurent Stolovitch
University of Nice
Non analytic hypoellipticity of sum of squares through complex analysis
I will present work that aim at understanding the failure of analytic hypoellipticity of special differential operators, namely sums of squares of (analytic) vector fields. According to Hörmander, given such an operator P that satisfies the "bracket condition" and given a smooth function f, if u is distribution solution to Pu=f then u is also a smooth function. P is then said to be hypoelliptic. But if f is real analytic, then u need not to be analytic but merely smooth Gevrey for some indices, usually to be guessed. Examples were built by Metivier, Matsuzawa, Bove, Baouendi-Goulaouic..., using real variables methods. In this joint work with Paulo Cordaro, by using methods of complex analysis, we show that this failure of analytic hypoellipticity due to the presence of irregular singularity of some holomorphic ODEs the analysis of which defines the best Gevrey indices to be expected. The theory of summability of formal solutions of holomorphic ODEs as developped by Ramis, Malgrange, Braaksma is a fundemental tool here.
February 27 Hao Jia
University of Minnesota
Channel of energy inequality and absence of null concentration of energy for wave map equations
Wave maps are natural hyperbolic analogue of harmonic maps. The study of the wave maps over the last twenty plus years has led to many beautiful and deep ideas, culminating in the proof of the "ground state conjecture" by Sterbenz and Tataru (with independent proof by Tao, and by Krieger&Schlag when the map has hyperbolic plane as target). The understanding of wave maps is now quite satisfactory. There are however still some remaining interesting problems, involving more detailed dynamics of wave maps. In this talk, we shall look at the problem of ruling out the so called "null concentration of energy" for wave maps. We will briefly review the history of wave maps, and explain why the null concentration of energy is relevant, and why the channel of energy inequality seems to be uniquely good in ruling out such possible energy concentration in the presence of solitons.
March 6 Hanlong Fang
Rutgers University


March 6
at 2pm in 6218
Nordine Mir
Texas A&M at Qatar
Regularity of CR maps in positive codimension
I will discuss recent joint results with B. Lamel addressing the $C^\infty$ regularity problem for CR mappings between smooth CR submanifolds in complex spaces of possibly different dimension. We essentially show that a nowhere smooth CR map must have its image contained in the set of D’Angelo infinite type points of the target manifold, from which we derive a number of new regularity results, even in the hypersurface case. Applications to the boundary regularity of proper holomorphic mappings will also be mentioned.
March 6
at 3pm in 6218
Ilya Kossovskiy
Masaryk University
The equivalence theory for infinite type hypersurfaces
Holomorphic classification of real submanifolds in complex space is one of the central goals in complex analysis in several variables. This classification is well understood and approaches are well developed for submanifolds satisfying certain bracket generating conditions of Hormander type, while very little is known in more degenerate setting. In particular, somewhat surprizingly, the classification problem is still completely open for hypersurfaces in complex 2-space. The class of hypersurfaces bringing conceptual difficulties here is the class of (Levi-nonflat) infinite type hypersurfaces. In our joint work with Ebenfelt and Lamel, we develop the equivalence theory for infinite type hypersurfaces in $C^2$. We do so by providing a normal for for such hypersurfaces. We extensively use the newly developed approach of Associated Differential Equations. The normal form construction is performed in two steps: (i) we provide a normal for for associated ODEs; (ii) we use the normal form of ODEs for solving the equivalence problem for hypersurfaces. Somewhat similarly to the Poincare-Dulac theory in Dynamical System, our classification theory exhibits resonances, convergence and divergence phenomena, Stokes phenomenon and sectorial regularity phenomena.
March 13
at 3:30pm in 6402
Joint with CCOM
Stefan Steinerberger
Yale University
Spherical Designs and the Heat Equation
Spherical Designs are finite sets of points on the sphere with the property that the average of low-degree polynomials over the sphere coincides with the average over the finite set. These objects are very beautiful, very symmetric and have been studied since the 1970s. We use a completely new approach that replaces delicate combinatorial arguments with the a simple application of the heat equation; this approach improves the known results and extends to other manifolds. We also discuss some related issues in Combinatorics, Irregularities of Distribution and Fourier Analysis.

Spring 2018

Date Speaker Title + Abstract
April 10 Katya Krupchyk
UC Irvine
April 17 Anna Mazzucato
Penn State
May 1 Peter Hintz
UC Berkeley
May 22 Theodore Drivas
Princeton University