The Analysis and PDE seminar will take place on Monday, March 20, in room 740, Evans hall, from 16:10 to 17:00.
Title: Homogenization: Beyond well-posedness theory.
Abstract: I will describe some recent progress on going beyond the well-posedness theory in homogenization of Hamilton-Jacobi equations. In particular, I will focus on the decomposition method to find the formula of the effective Hamiltonian in some situations. Joint work with Qian and Yu.
The Analysis and PDE Seminar will take place on Monday, March 13, in room 740, Evans Hall, from 4:10-5:00 pm.
Title: Stationary solutions to the 2d Euler equation
Abstract: The two dimensional Euler equation has a large number of stationary solutions. Distribution functions of the vorticity are preserved under the flow.
I will explain a parametrization of Arnold stable stationary solutions by distribution functions of their vorticity. This is joint work with Antoine Chiffrut.
The Analysis and PDE Seminar will take place on Monday, March 6, in room 740, Evans Hall, from 4:10-5:00 pm.
Title: Asymptotic analysis of Fourier transform on the Heisenberg group when the vertical frequency tends to 0
Abstract: In this joint work with Jean-Yves Chemin and Raphael Danchin, we propose a new approach of the Fourier transform on the Heisenberg group. The basic idea is to take advantage of Hermite functions so as to look at Fourier transform of integrable functions as mappings on the set $\tilde{\mathbb{H}}^d=\mathbb{N}^d\times\mathbb{N}^d\times\mathbb{R}\setminus\{0\}$ endowed with a suitable distance $\hat{d} $ (whereas with the standard viewpoint the Fourier transform is a one parameter family of bounded operators on $L^2(\mathbb{R}^d)$). We prove that the Fourier transform of integrable functions is uniformly continuous on $\tilde{\mathbb{H}}^d$ (for distance $\hat d$), which enables us to extend $\hat f_\mathbb{H}$ to the completion $\hat {\mathbb{H}}^d$ of $\tilde {\mathbb{H}}^d,$ and to get an explicit asymptotic description of the Fourier transform when the `vertical’ frequency tends to $0.$ We expect our approach to be relevant for adapting to the Heisenberg framework a number of classical results for the $\mathbb{R}^n$ case that are based on Fourier analysis.
The Analysis and PDE Seminar will take place on Monday, February 13, in room 740, Evans Hall, from 4:10-5:00 pm.
Speaker: Laurent Michel
Title: On the small eigenvalues of the Witten Laplacian
Abstract: The Witten Laplacian, introduced (by E. Witten) in the early 80’s to give an analytic proof of the Morse inequalities, also models the dynamics of the over-damped Langevin equation. The understanding of the so-called metastable states goes through the description of its small eigenvalues. The first result in this direction was obtained in 2004 (Bovier-Gayrard-Klein, Helffer-Klein-Nier) under some generic assumptions on the landscape potential. In this talk, we present the approach of Helffer-Klein-Nier and show some recent progress to get rid of the generic assumption.
The Analysis and PDE Seminar will take place on Monday, December 5, in room 740, Evans Hall, from 4:10-5:00 pm.
Speaker: Casey Jao
Title: Mass-critical inverse Strichartz theorems for 1d Schr\”{o}dinger operators
Abstract: I will discuss refined Strichartz estimates at $L^2$ regularity for a family of Schrödinger equations in one space dimension. Existing results rely on sophisticated Fourier analysis in spacetime and are limited to the translation-invariant equation $i\partial_t u = -\tfrac{1}{2} \Delta u$. Motivated by applications to mass-critical NLS, I will describe a physical space approach that applies in the presence of potentials including (but not limited to) the harmonic oscillator. This is joint work with Rowan Killip and Monica Visan.
The Analysis and PDE Seminar will take place on Monday, November 21th, in room 740, Evans Hall, from 4:10-5:00 pm.
Title: Scattering below the ground state for the radial focusing NLS
Abstract: We consider scattering below the ground state for the radial cubic focusing NLS in three dimensions. Holmer and Roudenko originally proved this via concentration compactness and a localized virial estimate. We present a simplified proof that avoids the use of concentration compactness, relying instead on the radial Sobolev embedding and a virial/Morawetz hybrid. This is joint work with Ben Dodson.
The Analysis and PDE Seminar will take place on Monday, November 14th, in room 740, Evans Hall, from 4:10-5:00 pm.
Title: Finite depth gravity water waves In holomorphic coordinates
Abstract: In this article we consider irrotational gravity water waves with finite bottom. Our goal is two-fold. First, we represent the equations in holomorphic coordinates and discuss the local well-posedness of the problem in this context. Second, we consider the small data problem and establish cubic lifespan bounds for the solutions. Our results are uniform in the infinite depth limit, and match our earlier infinite depth paper.
The Analysis and PDE Seminar will take place on Monday, November 7th, in room 740, Evans Hall, from 4:10-5:00 pm.
Title: Global well-posedness for the energy critical Massive Maxwell-Klein-Gordon equation with small data
Abstract: We discuss the global well-posedness and modified scattering for the massive Maxwell-Klein-Gordon equation in the Coulomb gauge on $ R^{1+d}$ $(d \geq 4)$ for data with small critical Sobolev norm. This extends to $ m^2 > 0 $ the results of Krieger-Sterbenz-Tataru ($d=4,5 $) and Rodnianski-Tao ($ d \geq 6 $).
The proof is based on generalizing the global parametrix construction for the covariant wave operator and the functional framework from the massless case to the Klein-Gordon setting. The equation exhibits a trilinear cancelation structure identified by Machedon-Sterbenz. To treat it one needs sharp $ L^2 $ null form bounds, which are proved by estimating renormalized solutions in null frames spaces similar to the ones considered by Bejenaru-Herr.
To overcome logarithmic divergences we rely on an embedding property of $ \Box^{-1} $ in conjunction with endpoint Strichartz estimates in Lorentz spaces.
The Analysis and PDE Seminar will take place on Monday, October 24th, in room 740, Evans Hall, from 4:10-5:00 pm.
Title: The essential spectrum of the Neumann-Poincare operator
Using an idea of Poincare one can realize the Neumann-Poincare operator on a space of square integrable fields. In two variables this leads to a precise estimate of the essential spectrum, for domains with corners.
The Analysis and PDE Seminar will take place on Monday, September 26, in room 740, Evans Hall, from 4:10-5:00 pm.
Speaker: Satoshi Masaki
Title: Minimization problems on non-scattering solutions to NLS equation
Abstract: We consider global dynamics of focusing nonlinear Schrodinger equations. A first step in this direction is small data scattering which tells us that solutions around the zero solution asymptotically behave like free solutions. On the other hand, there exists non-scattering solutions such as standing waves and blowing-up solutions.
In this talk, we will seek threshold solutions between scattering solutions around zero and solutions with other behaviors, by introducing two minimization problems on non-scattering solutions. In particular, our main interest is the analysis of mass-subcritical case, in which the ground states are stable. The analysis of the minimization problems are based on concentration compactness/rigidity argument initiated by Kenig and Merle.