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October 2019

Dean Baskin, Texas A&M University – Analysis & PDE Seminar

October 2, 2019 @ 4:00 pm - 5:00 pm

Title: Diffraction for the Dirac equation by Coulomb-like potentials Abstract: The Dirac equation describes the relativistic evolution of electrons and positrons. We consider the (time-dependent!) Dirac equation in three dimensions coupled to a potential with Coulomb-type singularities. We prove a propagation of singularities result for this equation and show that singularities are typically diffracted by the singularities of the potential. We finally compute the symbol of the diffracted wave and show it is typically non-zero. This talk is based on…

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Kiril Datchev, Purdue University – Analysis & PDE Seminar

October 9, 2019 @ 4:00 pm - 5:00 pm

Title: "Wave decay for star-shaped waveguides" Abstract: Let $X \subset \mathbb R^d$ be an unbounded open set. We wish to understand how decay of solutions to the wave equation on $X$ is related to the geometry of $X$. When $\mathbb R^d \setminus X$ is bounded, this is the celebrated obstacle scattering problem. Then a particularly favorable geometric assumption, going back to the original work of Morawetz, is that the obstacle is star shaped. We adapt this assumption to the study…

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January 2020

Annalaura Stingo, UC Davis – Analysis & PDE Seminar

January 22, 2020 @ 4:00 pm - 5:00 pm

Title: Almost global well-posedness for quadratic quasilinear 2D wave-Klein-Gordon systems with small and localized initial data. Abstract: In this talk we will discuss our recent result on the almost global existence of small solutions of strongly coupled wave-Klein-Gordon systems (WKG) in 2+1 space-time dimensions. The coupling we consider is quadratic, quasilinear, and satisfies the so-called null condition. No restriction is made on the support of the initial data, that are small and only mildly decaying at infinity. Wave-Klein-Gordon systems arise from physical models especially…

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February 2020

Andrew Lawrie, MIT – Analysis & PDE Seminar

February 19, 2020 @ 4:00 pm - 5:00 pm

Title: The dynamics of kink-antikink pairs for scalar fields on the line Abstract: We will discuss classical nonlinear scalar field models on the real line. If the potential is a symmetric double-well, such models admit static solutions called kinks and antikinks, which are perhaps the simplest examples of topological solitons. We will study pure multi-kinks, which are solutions that converge in one infinite time direction to a superposition of a finite number of kinks and antikinks, without radiation. Our main result is…

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Yannick Sire – Analysis & PDE Seminar

February 26, 2020 @ 4:00 pm - 5:00 pm

Title: Spectral analysis of Schrodinger operators and applications Abstract: I will report on recent results about quasimode, eigenfunction and spectral projection bounds for Schrodinger operators, $H_V=-\Delta_g+V(x)$, on compact Riemannian manifolds $(M,g)$ of dimension $n\ge2$ with critically singular potentials V. Using the spectral projection bounds  we can prove a number of natural $L^p\to L^p$ spectral multiplier theorems under the assumption that $V\in L^{\frac{n}2}(M)\cap {\mathcal K}(M)$. We will draw also several applications to some PDEs. I will also mention some results when…

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September 2021

Casey Rodriguez, UNC-Chapel Hill – Analysis & PDE Seminar

September 1 @ 4:00 pm - 5:00 pm

Phillips Hall, 332 Title: Simple motion of stretch-limited elastic strings Absract: Elastic strings are among the simplest one-dimensional continuum bodies and have a rich mechanical and mathematical theory dating back to the derivation of their equations of motion by Euler and Lagrange. In classical treatments, the string is either completely extensible (tensile force produces elongation) or completely inextensible (every segment has a fixed length, regardless of the motion). However, common experience is that a string can be stretched (is extensible),…

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Jian Wang, UNC-Chapel Hill – Analysis & PDE Seminar

September 15 @ 4:00 pm - 5:00 pm

Title: Mathematics of internal waves in a 2D aquarium Abstract: Internal waves are waves occurring in density stratified fluids. A fascinating feature of internal waves is the appearance of wave attractors in certain shapes of domains in the linear regime (Maas et al, 1997). A microlocal model for the formation of the wave attractors was introduced by Colin de Verdi\`ere -- Saint Raymond (2018) by using 0th order psuedodifferential operators over a torus. In the realistic case, where one considers…

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October 2021

Shi-Zhuo Looi, Kentucky – Analysis & PDE Seminar

October 6 @ 4:00 pm - 5:00 pm

Virtual talk over Zoom Projecting in PH 332 if anyone would like to attend in-person Title: Scattering and pointwise decay of some linear and nonlinear wave equations Abstract: We discuss the proof of sharp pointwise decay for linear wave equations, and then scattering and sharp pointwise decay for power-type nonlinear wave equations. These results hold on a general class of asymptotically flat spacetimes, which are allowed to be either nonstationary or stationary. The main ideas for the linear problem include…

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Zoe Wyatt, Cambridge – Analysis & PDE Seminar

October 27 @ 12:00 pm - 1:00 pm

Virtual and alternative time to be arranged to accommodate time difference Meeting in-person, 12:00 - 1:00 pm in Phillips Hall 228 Title: Coupled wave and Klein-Gordon equations in two and three spatial dimensions Abstract: Semilinear wave equations in three spatial dimensions with wave--wave nonlinearities exhibit interesting and well-studied phenomena: from John's famous blow-up examples, to the null condition of Christodoulou and Klainerman, and more recently to the weak null condition of Lindblad and Rodnianski. The study of coupled semilinear wave…

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November 2021

Jason Metcalfe, UNC-Chapel-Hill – Analysis & PDE Seminar

November 17 @ 4:00 pm - 5:00 pm

Mode: In-person TITLE: Variants of r^p weighted local energy estimates and their application to long time existence for nonlinear wave equations ABSTRACT:  We will examine long-time existence for nonlinear wave equations (1) satisfying the null condition, (2) containing dependence on the solution at the lowest order, and (3) satisfying a certain weak null condition.  The method of proof relies on a space-time Klainerman Sobolev estimate and a local energy estimate that is a variant of the r^p weighted estimates.  Efforts…

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