# Seminars & Events for Analysis Seminar

##### Ground states of the $L^2$-critical boson star equation

The boson star equation $\sqrt{-\Delta} u - (|x|^{-1} * |u|^2) u = -u$ in $R3$ involves both a non-local kinetic and potential energy and is $L^2$-critical. We establish uniqueness, radial symmetry (up to translations) and analyticity of non-negative solutions. We also prove the nondegeneracy of the linearization. Our proof of uniqueness blends variational arguments with the harmonic extension, and our proof of radial symmetry extends the moving planes method to our non-local setting. This is joint work with E. Lenzmann.

##### Counterexamples to the Strichartz estimates for the wave equation in domains

We prove that the Strichartz estimates for the wave equation inside a strictly convex domain $\Omega$ of dimension $2$ suffer losses when compared to the usual case $\mathbb{R}2$, (at least for a subset of the usual range of indices) and this is due to microlocal phenomena such as caustics generated in arbitrariIly small time near the boundary.

##### Cocompact imbeddings and critical nonlinearity revisited

We introduce a notion of cocompact imbeddings relative to a group of linear isometries.We discuss the notion of critical Sobolev nonlinearity in connection with the usual dilation actions that make the (non-compact) limit Sobolev imbedding co-compact and yield solutions of Talenti type for semilinear elliptic equations with self-similar autonomous nonlinearities of critical growth.

We then consider similar dilation and translations groups for $H_01(B)$, where $B$ is a unit disk on a plane, which preserve the Sobolev norm, but do not preserve the Trudinger-Moser functional $\int e^{4\pi u^2}$. We give then two examples of invariant critical nonlineairites that are stronger than Trudinger-Moser nonlinearity and lack the weakly continuity properties of the latter.

##### The stability of the irrotational Euler-Einstein system with a positive cosmological constant

The irrotational Euler-Einstein system models the evolution of a dynamic spacetime containing a perfect fluid with vanishing vorticity. In this talk, which is a summary of recent joint work with Igor Rodnianski, I will discuss the stability of a family of background cosmological solutions to the irrotational Euler-Einstein system in 1 + 3 dimensions with a positive cosmological constant $\Lambda$. The background solutions describe an initially uniform quiet fluid of positive energy density evolving in a spacetime undergoing accelerated expansion. Our main result is a proof that under the equation of state $p = c^2_s\rho, 0 < c^2_s < 1/3$; the background solutions are globally future-stable under small irrotational perturbations.

##### Strongly Focused Gravitational Waves

Christodoulou (2008) proved that trapped spheres can form in evolution, through the focusing of gravitational waves. Recently, with E. Trubowitz, we considered the same physical problem, using very different mathematical methods. Our approach is based on a systematic use of formal expansions, scaling symmetries and energy estimates. We give a direct construction of vacuum solutions by a far field expansion and exhibit trapped spheres that first appear deep inside the far field region.

##### Astala's conjecture on Hausdorff measure distortion under planar quasiconformal mappings and related removability problems

In his celebrated paper on area distortion under planar quasiconformal mappings (Acta 1994) (for which he received the Salem prize), Astala proved that if $E$ is a compact set of Hausdorff dimension $d$ and $f$ is $K$-quasiconformal, then $fE$ has Hausdorff dimension at most $d' = \frac{2Kd}{2+(K-1)d}$, and that this result is sharp. He conjectured (Question 4.4) that if the Hausdorff measure $\mathcal{H}^d (E)=0$, then $\mathcal{H}^{d'} (fE)=0$. This conjecture was known to be true if $d'=0$ (obvious), $d'=2$ (Ahlfors), and $d'=1$ (Astala, Clop, Mateu, Orobitg and UT, Duke 2008.) The approach in the last mentioned paper does not generalize to other dimensions. UT showed that Astala's conjecture is sharp in the class of all Hausdorff gauge functions (IMRN, 2008).

##### Quasi-local horizons

##### Radiation field for Einstein Vacuum equations

##### On the structure of singularities of solutions to the Einstein equations with toroidal symmetry

I will present recent results concerning the study of the global Cauchy problem in relativity, without restrictions on the size of the data but under certain symmetry and topological assumptions. More specifically, I will focus on the issue of the structure of singularities for solutions with toroidal symmetry, in relation to the so-called strong cosmic censorship conjecture. In particular, I shall present several global existence results for the T2-symmetric Einstein equations in areal coordinates.

##### White noise for KdV, mKdV, and cubic NLS on the circle

We discuss two methods for establishing the invariance of the white noise for the periodic KdV. First, we briefly go over the basic theory of Gaussian Hilbert spaces and abstract Wiener spaces and show that the Fourier-Lebesgue space $\mathcal{F}L^{s, p}$ captures the regualrity of the white noise for $sp < -1$. We then establish local well-posedness (LWP) of KdV via the second iteration introduced by Bourgain. This in turn provides almost sure global well-posedness (GWP) of KdV as well as the invariance of the white noise. Then, we discuss how one can use the same idea to obtain LWP of the stochastic KdV with the additive space-time white noise in the periodic setting.

##### Generalized principal eigenvalue of elliptic operators in unbounded domains and applications

In this talk, I will discuss several notions that are extensions of the principal eigenvalue of a linear elliptic operator to the framework of unbounded domains along with some of their properties. I will describe applications to semi-linear elliptic equations and to propagation in reaction-diffusion equations of the KPP type. These equations are considered in unbounded domains and for general heterogeneous settings. I report here on joint works with Luca Rossi, with François Hamel and with Grégoire Nadin.

##### Quantized Poincarè maps in chaotic scattering

I will sketch a recent approach to study the resonance spectrum of scattering Schrödinger operators, in cases where the trapped set of the corresponding classical dynamics (near some positive energy) is a fractal chaotic repeller. In that situation, we are interested in the distribution of resonances in the vicinity of the real axis, in the semiclassical limit. Our approach tends to mimic the Poincarè section method used to study the corresponding classical flows. Namely, we show that resonances can be defined through an implicit equation involving an appropriately defined quantized Poincarè map. The subsequent study of this "quantum map" allows to recover known properties of the resonance spectrum (fractal bounds on the number of resonances in strips, spectral gap), and hopefully more.

##### Complex variables are not dead

Our lecture will focus on two problems in pde which are solvable by ideas in holomorphic functions of complex variables. The first problem is called the strip theorem. Let $f$ be a function defined in the strip in the complex plane $|Im z| \leq 1$. Suppose $f$ agrees on the boundary of each unit circle centered on the real axis, radius $1$, with the solution (depending on the circle) of a suitable elliptic pde, the agreement being to order one greater than the order of the Dirichlet data. Then $f$ satisfies this pde. If the equation is the Cauchy-Riemann equation then equality suffices. The second type of problems we discuss are the Phragmen-Lindelof theorem for pde and a form of the Heisenberg uncertainty for pde. These were introduced in Kenig's lecture at Fefferman's birthday bash. We shall put them in a general framework.

##### Radiative decay of bubble oscillations

We consider the dynamics of a gas bubble in an unbounded, inviscid and compressible fluid with surface tension. Kinematic and dynamic (Young-Laplace) boundary conditions couple the dynamics of bubble surface deformations to the dynamics of waves in the fluid. We study the linear decay estimates for the fluid and deforming bubble near the spherical equilibrium. The local energy decay is exponential in time, $exp(-\Gamma t)$. $\Gamma$ is determined by a non-selfadjoint scattering resonance spectral problem. The scattering resonances which limit the time-decay rate are of a high order multipole character and are due to surface tension. The decay rate for general solutions ($\Gamma$, exponentially small in the Mach number) is much, much slower than for spherically symmetric solutions ($\Gamma_{radial}$, linear in the Mach number).

##### Global Classical Solutions of the Boltzmann Equation with Long-Range Interactions

In this talk we explain our recent proof of global stability for the Boltzmann equation (1872) with the physically important collision kernels derived by Maxwell 1867 for the full range of inverse power intermolecular potentials, $r^{-(p-1)}$ with $p > 2$ and more generally. Our solutions are perturbations of the Maxwellian equilibrium states, and they decay rapidly in time to equilibrium as predicted by Boltzmann's celebrated H-Theorem.

This equation provides a basic example where a wide range of geometric fractional derivatives occur in a physical model of the natural world. We are able to characterize these non-isotropic fractional differentiation effects precisely using in part the "geometric Littlewood-Paley'' theory of Stein and Klainerman-Rodnianski. This is joint work with P. Gressman.

##### The space time resonance method and global existence of small surface waves.

We will present a new approach to proving global existence of small solutions to dispersive equations. This approach combines the well established methods of vector fields and normal forms and extend them to show global existence of small amplitude surface water waves. We will also show how many of the well known results of existence of small solutions of dispersive equations can be simply established by the space time resonance method.

##### Interaction of Light with Arbitrarily Shaped Dielectric Media: Compactness and Robustness in Electromagnetic Scattering

The scattering of electromagnetic waves by homogeneous dielectric media is characterized by a strongly singular integral equation, corresponding to the identity operator perturbed by a non-compact Green operator. Using the Kondrachov-Rellich compact imbedding and the Calderon-Zygmund theory, we prove that the Green operator is polynomially compact if the dielectric boundary is a compact smooth manifold. We then show that the electromagnetic scattering problem admits a robust solution for all non-accretive media ($\mathrm{Im}\chi\leq0$) satisfying certain geometric and topological constraints, except for the critical point $\chi=-2$, where unbounded electromagnetic enhancement may occur.

##### On the soliton dynamics under a slowly varying medium for generalized KdV equations

We consider the problem of the soliton propagation, in a slowly varying medium, for a generalized Korteweg - de Vries equations (gKdV). We study the effects of inhomogeneities on the dynamics of a standard soliton. We prove that slowly varying media induce on the soliton solution large dispersive effects in large time. Moreover, unlike gKdV equations, we prove that there is no pure-soliton solution in this regime.