The Analysis and PDE Seminar will take place on Monday, April 10, in room 740, Evans Hall, from 4:10-5:00 pm.

Title: The Helicoidal Method II

Abstract: The helicoidal method is a new, extremely efficient way, of proving multiple vector valued inequalities in harmonic analysis. About a month ago, we gave a talk at MSRI, in which we explained some consequences of this method, such as the proof of sparse domination results for various multilinear operators, and their multiple vector valued extensions.

The main task of the current talk will be different (hence the II from the title) as we plan to discuss the ideas that led us to the method. One specific application that we also plan to present, is the proof of mixed norm estimates for paraproducts on the bidisk, in the full possible range of Lebesgue spaces, answering completely an open question of Kenig, from the early 90’s. Joint work with Cristina BENEA.

The Analysis and PDE seminar will take place on Monday, April 3rd, in Evans 740 from 16:10 to 17:00.

Title: Models for Rayleigh-Taylor mixing and interface turnover

Abstract: The instability of a heavy fluid layer supported by a light one is generally known as Rayleigh-Taylor (RT) instability. It can occur under gravity and, equivalently, under an acceleration of the fluid system in the direction toward the denser fluid. Whenever the pressure is higher in the lighter fluid, the differential acceleration causes the two fluids to mix.

The Euler equations serve as the basic mathematical model for RT instability and mixing between two fluids. This highly unstable system of conservation laws is both difficult to analyze (as it is ill-posed in the absence of surface tension and viscosity) and simulate; DNS of RT can be prohibitively expensive. In this talk, I will describe a novel framework to derive a hierarchy of asymptotic models that can be used to predict the location and shape of the RT interface as well as the mixing of the two fluids.

The models are derived in two very different asymptotic regimes. The first regime assumes that the fluid interface is a graph with size restrictions on the slope of the interface. The model PDE inherits the RT stability condition from the Euler equations, and in the stable regime, it is both locally and globally well-posed with precise asymptotic behavior that predicts nonlinear saturation for bubble growth. In the second asymptotic regime the interface can turnover, and there are no size restrictions on the amplitude or slope of the interface.

I will describe these models and show numerical simulations and comparisons with well-known RT experiments and simulations. I will then show results of fluid mixing, and discuss current work, advancing both modeling strategies. This is joint work with Rafa Granero.

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.