The Analysis and PDE seminar will take place Monday, October 2nd, in 740 Evans from 4:10 to 5pm.

Title: Convergence of phase-field models and thresholding schemes for multi-phase mean curvature flow

Abstract: The thresholding scheme is a time discretization for mean curvature flow. Recently, Esedoglu and Otto showed that thresholding can be interpreted as minimizing movements for an energy that Gamma-converges to the total interfacial area. In this talk I’ll present new convergence results, in particular in the multi-phase case with arbitrary surface tensions. The main result establishes convergence to a weak formulation of (multi-phase) mean curvature flow in the BV-framework of sets of finite perimeter. Furthermore, I will present a similar result for the vector-valued Allen-Cahn equation.

This talk encompasses joint works with Felix Otto, Thilo Simon, and Drew Swartz.

The Analysis and PDE seminar will take place Monday, Sept 11, in Evans 740 from 4:10 to 5:00pm.

Title: Two-bubble dynamics for the equivariant wave maps equation.

Abstract: I will consider the energy-critical wave maps equation with values in the
sphere in the equivariant case, that is for symmetric initial data. It is
known that if the initial data has small energy, then the corresponding
solution scatters. Moreover, the initial data of any scattering solution
has topological degree 0. I try to answer the following question: what are
the non-scattering solutions of topological degree 0 and the least
possible energy? Such “threshold” solutions would have to decompose
asymptotically into a superposition of two ground states at different
scales, with no radiation.
In the first part I will show how to construct threshold solutions. In the
second part I will describe the dynamical behavior of any threshold
solution.
The second part is a joint work with Andrew Lawrie (MIT).

The Analysis and PDE seminar will take place Monday, May 8, in 740 Evans Hall, from 16:10-17:00.

Title: Strong cosmic censorship in spherical symmetry for two-ended asymptotically flat initial data

Abstract: Black holes are perhaps the most celebrated predictions of general relativity. Miraculously, stationary black hole spacetimes arise as explicit (i.e., exact expression can be written down!) solutions to the (vacuum) Einstein equation. Looking at these explicit solutions leads to an intriguing observation: While the black hole exterior looks qualitatively similar, the structure of the interior, in particular the nature of the `singularity’ inside the black hole, changes drastically depending on whether the black hole is spinning (Kerr) or not (Schwarzschild).

The topic of this talk is the strong cosmic censorship conjecture of R. Penrose, which is a proposed picture for what happens generically. In particular, I will present a recent work, joint with Jonathan Luk, where we establish a version of strong cosmic censorship for the Einstein-Maxwell-(real)-scalar-field system in spherical symmetry with two-ended asymptotically flat data, a toy model
that has been studied by physicists and mathematicians to understand issues connected to strong cosmic censorship in a simplified setting.

The Analysis and PDE Seminar will take place on Monday, April 17, in room 740, Evans Hall, from 4:10-5:00 pm.

Title: Control of water waves

Abstract: Water waves are disturbances of the free surface of a liquid. They are, in general, produced by the immersion of a solid body or by impulsive pressures applied on the free surface. The question we discuss in this talk is the following: which waves can be generated by blowing on a localized portion of the free surface. Our main result asserts that one can generate any small amplitude, periodic in x, two-dimensional, gravity-capillary water waves. This is a result from control theory. More precisely, we prove the local exact controllability of the incompressible Euler equation with free surface. This is a joint work with Pietro Baldi and Daniel Han-Kwan.

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, 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.