On the Cauchy problem for degenerate dispersive equations

The HADES seminar on Tuesday, January 21st will be given by Sung-Jin Oh in Evans 732 from 3:40 to 5 pm.

Speaker: Sung-Jin Oh

Abstract: In plasma physics or fluid dynamics, one sometimes encounters a degenerate dispersive equation, i.e., a nonlinear dispersive equation whose dispersion relation is degenerate (i.e., vanishes at some points). A satisfactory understanding of the Cauchy problem for such equations is still missing, largely due to the appearance of challenging (and interesting!) phenomena from degenerate dispersion, such as the strong focusing of bicharacteristics near the degeneracy.

The purpose of this talk is to provide an introduction to this topic, by focusing on simple examples. In the first part of my talk, I’ll work with simple linear models, namely linear degenerate Schrödinger equations on the line, to demonstrate some key phenomena related to degenerate dispersion. Then in the second part of my talk, I’ll describe some nonlinear illposedness results for a quasilinear degenerate Schrödinger equation on the line, whose proof builds off of the understanding of the linear models. This talk is based on joint work with In-Jee Jeong.

Sub-Riemannian limit of the differential form heat kernels of contact manifolds

The HADES seminar on Tuesday, December 10th will be given by Hadrian Quan in Evans 740 from 3:40 to 5 pm.

Speaker: Hadrian Quan, UIUC

Abstract: In this talk I will report on joint work with Pierre Albin in which we study heat kernel of the Hodge Laplacian on a contact manifold endowed with a family of Riemannian metrics that blow-up the directions transverse to the contact distribution. We apply this to analyze the behavior of global spectral invariants such as the eta-invariant and the determinant of the Laplacian. Time permitting, I will discuss connections with the Rumin complex, and the associated limit of Analytic Torsion.

Propagation of singularities for gravity-capillary waves

The HADES seminar on Tuesday, December 3rd will be given by Hui Zhu in Evans 740 from 3:40 to 5 pm.

Speaker: Hui Zhu, MSRI

Abstract: The surface tension makes free surfaces of fluids instantaneously smooth. For 2D gravity-capillary waves, this phenomenon has been justified by Christianson–Hur–Staffilani and Alazard–Burq–Zuily as local smoothing effects. In this talk, I will present a microlocal justification of this phenomenon for gravity-capillary waves in arbitrary dimensions. My main results are two propagation theorems for some quasi-homogeneous wavefront sets of gravity-capillary waves.

Small data global regularity for simplified 3-D Ericksen-Leslie’s compressible hyperbolic liquid crystal model

The HADES seminar on Tuesday, November 26th will be given by Jiaxi Huang in Evans 740 from 3:40 to 5 pm.

Speaker: Jiaxi Huang, USTC

Abstract: In this talk, we will consider the Ericksen-Leslie’s hyperbolic system for compressible liquid crystal model in three spatial dimensions. Global regularity for small and smooth initial data near equilibrium is proved for the case that the system is a nonlinear coupling of compressible Navier-Stokes equations with wave map to $\mathbb{S}^2$. Our argument is a combination of vector field method and Fourier analysis. The main strategy to prove global regularity relies on an interplay between the control of high order energies and decay estimates, which is based on the idea inspired by the method of space-time resonances. In particular, the different behaviors of the decay properties of the density and velocity field for compressible fluids at different frequencies play a key role. Joint work with Ning Jiang, Yi-Long Luo, and Lifeng Zhao.

Some New Prodi-Serrin Type Regularity Criteria for the in-Compressible Navier-Stokes Equations

The HADES seminar on Tuesday, November 19th will be given by Benjamin Pineau in Evans 740 from 3:40 to 5 pm.

Speaker: Benjamin Pineau, Berkeley

Abstract: It is a classical result of Leray from the 1930s, that for appropriate initial data and domain, there exists a global weak solution (now known as a Leray-Hopf solution) to the n-dimensional, incompressible Navier-Stokes equations. For n ≥ 3, the question of uniqueness, and regularity of Leray-Hopf solutions remains open. On the other hand, by imposing certain “integrability” conditions on a weak solution, one can often establish global regularity using energy-type arguments. These types of conditions are often referred to as Prodi-Serrin type criteria. In this talk, I will present a relatively simple method for establishing global regularity of a weak solution, provided a certain quantity (e.g. velocity, pressure, etc.) satisfies a particular weak-Lebesgue “integrability” condition. This allows one to generalize several regularity criteria in the literature.

Geodesic stretch, pressure metric and the marked length spectrum rigidity conjecture

The HADES seminar on Tuesday, November 12th will be given by Thibault Lefebvre in Evans 740 from 3:40 to 5 pm.

Speaker: Thibault Lefebvre, Paris XI

Abstract: In 1985, Burns and Katok conjectured that the marked length spectrum of a negatively-curved Riemannian manifold (namely the collection of lengths of closed geodesics marked by the free homotopy of the manifold) should determine the metric up to isometries. This conjecture was independently proved for surfaces in 1990 by Croke and Otal but since then little progress has been accomplished in higher dimensions until our recent proof of the local version of the conjecture, obtained in collaboration with C. Guillarmou. Considering a geometric point of view in the moduli space of isometry classes, I will explain a new proof of this local version of the conjecture which relies on the notion of geodesic stretch. If time permits, I will show that this fits into a more general framework which generalizes Thurston’s distance and the pressure metric (initially defined on Teichmuller space) to the setting of variable curvature and higher dimensions. Joint work with C. Guillarmou, G. Knieper

Semiclassical resolvent bound for compactly supported Hölder continuous potentials

The HADES seminar on Tuesday, November 5th will be given by Jacob Shapiro in Evans 740 from 3:40 to 5 pm.

Speaker: Jacob Shapiro, ANU

Abstract: We prove a weighted resolvent estimate for the semiclassical Schrödinger operator $-h^2 \Delta + V : L^2(\mathbb{R}^n) \to L^2(\mathbb{R}^n)$, $n \ge 3$. We assume the potential $V$ is compactly supported and $\alpha$-Hölder continuous, $0< \alpha < 1$. The logarithm of the resolvent norm grows like $h^{-1-\frac{1-\alpha}{3 + \alpha}}\log(h^{-1})$ as the semiclassical parameter $h \to 0^+$. This bound interpolates between the previously known $h$-dependent resolvent bounds for Lipschitz and $L^\infty$ potentials. To key step is to prove a suitable global Carleman estimate, which we establish via a spherical energy method. This is joint work with Jeffrey Galkowski.

Semiclassical defect measures and observability estimate for Schrödinger operators with homogeneous potentials of order zero

The HADES seminar on Tuesday, October 29th will be given by Keita Mikami in Evans 740 from 3:40 to 5 pm.

Speaker: Keita Mikami, RIKEN

Abstract: In this talk, we will consider asymptotic behavior as $|x| \to \infty$ of Schrödinger operators with homogeneous potentials of order zero. Localization in direction was known as a property of Schrödinger operators with homogeneous potentials of order zero or corresponding Hamiltonian flow. We will introduce this known localization in direction first. We then introduce the localization of defect measures. We then give necessary conditions for observability of Schrödinger operators with homogeneous potentials of order zero which is related to the localization result.

Local energy, resolvents, and wave decay in the asymptotically flat setting

The HADES seminar on Tuesday, October 1st will be given by  Katrina Morgan in Evans 740 from 3:40 to 5 pm.

Speaker: Katrina Morgan, MSRI

Abstract: Asymptotically flat spacetimes (i.e. Lorentzian manifolds whose metric coefficients tend toward the flat metric as $|x| \to \infty$) arise in General Relativity, which has motivated many mathematical questions about wave behavior on such spacetimes. The dispersive estimate local energy decay has proven to be a powerful tool for studying these questions. It has been used to establish Strichartz estimates (global, mixed norm estimates often used in existence proofs) and pointwise estimates. The estimate holds if the underlying geometry allows waves to spread out enough to get decay of energy within compact sets. Local energy decay is connected to the presence of trapped geodesics and resolvent behavior.
This talk will provide a brief overview of local energy estimates and the use of resolvents
in studying wave behavior. The application of these tools in establishing the relationship between how quickly the background geometry tends toward flat and the pointwise decay rate of waves will be discussed. We find that a solution u to the wave equation on a spacetime which tends toward flat at a rate of $|x|^{-k}$ satisfies the pointwise bounds $|u|\le C_x t^{-k-2}$.
This result extends the work of Tataru 2013 which proved a $t^{-3}$ pointwise decay rate for
waves when the background geometry tends toward flat at a rate of $|x|^{-1}$.

Wave invariants and inverse spectral theory

The HADES seminar on Tuesday, October 22nd will be given by  Amir Vig in Evans 740 from 3:40 to 5 pm.

Speaker: Amir Vig, UCI & MSRI

Abstract: The wave trace is a distribution on $\mathbb{R}$ given by $\sum_{j = 1}^\infty e^{it \lambda_j}$, where $\lambda_j^2$ are the (positive) eigenvalues of the Laplacian on a compact domain. In general, two linear waves can be superimposed to give another solution to the wave equation. When we add up a bunch of waves at different frequencies, the peak singularities appear at points with substantial constructive interference. On a manifold, the famous “propagation of singularities” tells us that waves propagate along geodesics, so the constructive interference is most pronounced along orbits which are traversed infinitely often (i.e. periodic orbits). On the trace side of things, this phenomenon is reflected in the Poisson relation, which says that the singular support of the wave trace is contained in the length spectrum (the collection of lengths of all periodic orbits). For planar domains, the geodesic flow is replaced by the billiard (or broken bicharacteristic) flow and we see an interesting connection between geometric, dynamical and spectral properties of the domain. In this talk, we introduce some simple cases of wave trace formulas before discussing recent work on explicit formulas for wave invariants associated to periodic orbits of small rotation number in a smooth, strictly convex bounded planar domain. This involves proving a dynamical theorem on the structure of such orbits and then constructing an explicit oscillatory integral representation, which microlocally approximates the wave propagator in the interior.