Analysis Seminar

2017-2018

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 USC	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 UNIST	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 UW-Madison	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 USC	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 CMU	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 JHU	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	Cancelled
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 17 at 1pm	Anna Mazzucato Penn State	On the two-dimensional Kuramoto-Sivashinsky equation I will discuss recent results concerning the Kuramoto-Sivashinky equation in two space dimensions with periodic boundary conditions. In particular, I will present a global existence result in the Wiener algebra, when growing modes are absent, and bounds on the analyticity radius when the data is only $L^2$. This is joint work with David Ambrose (Drexel University).
April 24 at 1pm	Andrew Raich University of Arkansas	CR Invariants and Solvability of the $\bar\partial$-equation The main goal of this talk is to show that geometric information captured by certain invariant CR tensors provides sufficient information to establish the closed range property for $\bar\partial$ on a domain in $\mathbb{C}^n$. A secondary goal of the talk is to provide a general construction method for establishing when a domain (or its boundary) satisfies weak $Z(q)$.
May 1 at 1pm	Peter Hintz UC Berkeley	Stability of Minkowski space and asymptotics of the metric I will explain a new proof of the non-linear stability of the Minkowski spacetime as a solution of the Einstein vacuum equation. The proof relies on an iteration scheme at each step of which one solves a linear wave-type equation globally. The analysis takes place on a suitable compactification of $\mathbb{R}^4$ to a manifold with corners whose boundary hypersurfaces correspond to spacelike, null, and timelike infinity; I will describe how the asymptotic behavior of the metric can be deduced from the structure of simple model operators at these boundaries. This talk is based on joint work with Andras Vasy.
May 8	Xin Dong UC Riverside	Cancelled
May 15	Vincent Martinez Tulane University	Asymptotic coupling and uniqueness of invariant measures for the damped, stochastically-driven Korteweg de Vries equation In their 1967 seminal paper, Foias and Prodi captured precisely a notion of finitely many degrees of freedom in the context of the two-dimensional (2D) incompressible Navier-Stokes equations (NSE). In particular, they proved that if a sufficiently large spectral projection of the difference of two solutions converge to 0 asymptotically in time, then the corresponding complementary projection of their difference must also converge to 0 in the infinite-time limit. In other words, the high modes are "eventually enslaved" by the low modes. One could thus define the number of degrees of freedom of the flow to be the smallest number of modes needed to guarantee this convergence for a given flow, insofar as it is represented by a solution to the NSE. This property has since led to several developments in the understanding and application of the long-time behavior of solutions to the NSE, for instance, in the context of turbulence, data assimilation, and the existence of determining forms. In this talk, we will discuss this asymptotic enslavement property as it regards the issue of uniqueness of invariant measures for stochastically forced equations, particularly the damped, stochastic KdV equation.
May 17 (Thu) at 2pm in 5829 Joint with Complex Bio Systems	Shibin Dai University of Alabama	Degenerate FCH Functional and Defects in Amphiphilic Structures Amphiphilic structures such as cell membranes and lipid vesicles play essential roles in biological applications. In this talk, we will first introduce the functionalized Cahn-Hilliard (FCH) model for the free energy of amphiphilic mixtures. The FCH model admits local minimizers corresponding to amphiphilic bilayers, filaments, micelles, and other defect structures. We will describe the geometric motion of bilayers, filaments, and their competition. To capture the coexistence and localization of amphiphilic structures, we introduce a degenerate FCH functional as a modified model for the free energy of amphiphilic mixtures. We prove that the degenerate FCH functional admits geometrically localized minimizers, which correspond to localized amphiphilic structures. Specifically, we identify the leading order profile of bilayers under the assumption that the geometrically localized minimizers have bounded variations along the tangential directions.
May 22	Theodore Drivas Princeton University	Remarks on Onsager's Conjecture and Anomalous Dissipation on domains with and without boundaries We first discuss the inviscid limit of the global energy dissipation of Leray solutions of incompressible Navier-Stokes on the torus. Assuming that the solutions have Besov norms bounded uniformly in viscosity, we establish an upper bound on energy dissipation. As a consequence, Onsager-type "quasi-singularities" are required in the Leray solutions, even if the total energy dissipation is vanishes in the limit $\nu\to 0$. Next, we discuss an extension of Onsager's conjecture for domains with solid boundaries. We give a localized regularity condition for energy conservation of weak solutions of the Euler equations assuming (local) Besov regularity of the velocity with exponent $>1/3$ and, on an arbitrary thin layer around the boundary, boundedness of velocity, pressure and continuity of the wall-normal velocity. We also prove that the global viscous dissipation vanishes in the inviscid limit for Leray-Hopf solutions of the Navier-Stokes equations under the similar assumptions, but holding uniformly in a vanishingly thin viscous boundary layer. Finally, if a strong Euler solution exists, we show that equicontinuity at the boundary within a $O(\nu)$ strip alone suffices to conclude the absence of anomalous dissipation. The talk concerns joint work with G. Eyink and H.Q. Nguyen.
May 29	Guher Camliyurt USC	TBA
June 5	Sameer Iyer Princeton University	Postponed to Fall 2018