# Seminars & Events for Analysis Seminar

##### A Hyperbolic Free-Boundary Problem for 3D Compressible Euler Flow in Physical Vacuum

We prove well-posedness for compressible flow with free-boundary in physical vacuum, modeled by the 3D compressible Euler equations. The vanishing of the density at the vacuum boundary induces degenerate hyperbolic equations that become characteristic, requiring a separate analysis of time, normal, and tangential derivatives to handle the manifest 1/2-derivative loss. Unfortunately, the methods for incompressible flow do not work for the degenerate compressible regime; a priori nonlinear estimates are obtained using the geometric structure of the Euler equations, and an existence theory is developed using a novel approximation scheme employing an artificial phase. The result is in collaboration with Coutand and Lindblad.

##### Multi-parameter Carnot-Carathéadory balls

We discuss multi-parameter Carnot-Carathéadory balls. In particular, we discuss questions motivated by multi-parameter singular integrals. These results generalize results due to Nagel, Stein, and Wainger in the single parameter setting.

##### Global Schrodinger maps: small data in the critical Sobolev spaces

I will discuss recent work on the global regularity of the Schrodinger map initial-value problem with small data, in all dimensions $d\geq 2$.

##### A general unique continuation theorem for the Einstein equations

I will discuss a unique continuation result for the vacuum Einstein equations across bifurcate horizons. The main result uses a recent Carleman estimate of Ionescu and Klainerman, together with some geometric gauge constructions. More broadly I will indicate how Carleman estimates for the wave operator can be used to derive unique continuation for the Einstein equations.

##### On the asymptotic behavior of solutions to the Einstein equations

##### Almost global wellposedness of the 2-D full water wave problem

We consider the problem of global in time existence and uniqueness of solutions of the 2-D infinite depth full water wave equation. It is known that this equation has a solution for a time period $[0, T/\epsilon]$ for initial data of form $\epsilon\Psi$, where $T$ depends only on $\Psi$. We show that for such data there exists a unique solution for a time period $[0, e^{T/{\epsilon}}]$. This is achieved by better understandings of the nature of the nonlinearity of the full water wave equation

##### Fifth order KdV equations

We study the fifth-order KdV equations, which arise in the KdV hierarchy. In this talk, we discuss the initial value problem in Sobolev spaces with low regularity. In the linear part the fifth-order equation has stronger dispersion effect and so better smoothing than KdV equation. But it comes with stronger nonlinear parts compared to dispersion. As a result, unlike KdV equation, the fifth-order equation in the hierarchy has too strong low-high frequency interaction. We exploit this to show a negative result. We will discuss both positive and negative results, local well-posedness in the standard sense (existence, uniqueness and continuous dependence of data) for rough data, but ill-posedness in the sense of failure of uniformly continuous dependence on data on a bounded set.

##### Global Existence for Nonlinear Dispersive Equation

Starting from small data, when does a nonlinear dispersive PDE have global solutions? A classical approach, just like for ODE, is to study resonances. But I will show that for PDE a new kind of resonances arises, that I call space resonances. This is the basis of a new method, that I will present; I will also show how it applies to a variety of equations of Mathematical Physics: non-linear Schrödinger, water waves, Euler-Maxwell... This is joint work with Nader Masmoudi and Jalal Shatah.

##### The cubic fourth order Schrödinger equation

We will discuss on which dimensions the cubic fourth-order Schrödinger equation is globally wellposed in the natural energy space. We will mainly concentrate on the case when the equation becomes energy-critical.

##### Regularity of singular harmonic maps and axially symmetric stationary electrovacuum spacetimes

According to the Ernst-Geroch reduction, to each axially symmetric stationary vacuum/electro-vacuum spacetime, one can associate an axially symmetric harmonic map with singular boundary behavior. This idea has been exploited in the literature to construct asymptotically flat, axially symmetric stationary spacetimes with disconnected horizons, i.e. having multiple black holes. This family of spacetimes is uniquely parameterized by the “masses”, the “momenta”, the “charges” of the black holes and the “distances” between them. I’ll discuss the regularity of the corresponding reduced harmonic maps and its implication on the regularity of those spacetimes.

##### Classical convolution inequalities and Boltzmann equations for integrable angular section

We study the integrability properties of the gain part of the Boltzmann collision operator using radial symmetrization techniques from harmonic analysis to show Young's inequality in the case of hard potentials and Hardy-Littlewood-Sobolev inequality for soft potentials. The contacts are given by exact formulas depending on the angular cross section. By applying these estimates we can revisit and obtain new results for existence and uniqueness to the corresponding space inhomogeneous equations with special initial data. This is work partly in collaboration with E. Carneiro and R. Alonso

##### Lower bounds on the blow-up rate of the axisymmetric Navier-Stokes equations

Consider axisymmetric strong solutions, v, of the incompressible Navier-Stokes equations in $\mathbb{R}^3$ with non-trivial swirl. Leray, in his 1934 Acta. Math paper, has shown that at a blow-up time, $T$, such solutions would have to satisfy for some fixed $\epsilon_1>0$ $$ \liminf_{t\uparrow T} \sqrt{T-t} \sup_x |v(x,t)| \ge \epsilon_1$ $$ We will discuss our recent proof which rules out this scale invariant blow-up rate: $$ |v(x,t)| \le C_*/\sqrt{T-t}. $$ Above $C_*$ is allowed to be large. This is joint work with Tsai, Chen and Yau.

##### On the evolution of solutions to a many-body Schrödinger equation

In part I, I will describe background material and a new proof for the uniqueness of solutions to the Gross-Pitaevskii hierarchy. This is joint work with S. Klainerman and is a simplification, based on space-time estimates, of an older proof of Erdös, Schlein and Yau.

In Part II (joint work with M. Grillakis and D. Margetis) I will discuss a new, highly non-linear but explicit NLS in two space variables, whose solutions, if they exist, provide a second order correction to the usual tensor product approximation, which works in the Fock space norm. This is inspired by recent work of Rodnianski and Schlein, as well as older work of Wu.

##### Landau damping

Sixty years ago, Landau discovered a paradoxical collisionless relaxation effect in plasmas. The Landau damping is now one of the cornerstones of classical plasma physics. From the mathematical point of view, it has remained elusive so far, since the best available results prove the existence of some damped solutions, without saying anything about their genericity. I shall report on new advances, and a whole new mathematical theory, for this problem. I will discuss the physical implications of these results. This is joint work with Clement Mouhot.

Note: This is part of a series of related talks, there is additionally a colloquium and a math physics seminar.

##### Entire functions and gap theorems

Several classical problems of Analysis can be translated into a universal language based on Hilbert spaces of entire functions and kernels of Toeplitz operators. Problems that can be treated this way include completeness/minimality problems for systems of exponentials or special functions in $L^2$ and spectral problems for second order differential operators. This approach was used to solve some of such problems in our recent papers with Nikolai Makarov.

In this talk I will show how the Toeplitz approach can be used to extend the so-called Beurling's Gap Theorem on the existence of gaps in the Fourier transform of a measure and to solve the Polya-Levinson problem on sampling sets for entire functions of exponential type zero.

##### $h$—Principle and fluid dynamics

Joint Princeton University and Institute for Advanced Study Analysis Seminar

##### New results for reaction-diffusion equations arising from reversible chemistry

Entropy/entropy dissipation methods have been used with success lately in the study of the large time behavior of kinetic equations, nonlinear diffusions, etc., and have led to the development of the concept of hypocoercivity. They are also very useful in the context of reaction-diffusion equations (especially when they are derived from reversible chemistry), where they lead to new results of convergence to equilibrium as well as new results of existence of weak and strongs solutions. We shall detail some of those results, together with their links with recent works on coagulation-fragmentation models, and the use of results of regularity for singular parabolic problems.

##### From Boltzmann equation to the incompressible Navier-Stokes-Fourier system with long-range interactions

Boltzmann's equation is known to converge, under a certain hydrodynamic regime, to an incompressible Navier-Stokes-Fourier system. It is only recently that the final steps to a mathematically rigorous and complete justification of this hydrodynamic convergence were provided. However, only certain types of intermolecular interactions, still physically unsatisfying, were considered.

We establish this hydrodynamic limit for the physically relevant case of long range intermolecular interactions. In this situation, the difficulty comes from the fact that the Boltzmann collision operator exhibits a rather complex nature due to a non-integrable singularity in the collision kernel.

##### Stefan Problem with Surface Tension

##### An Extension of the Stability Theorem of the Minkowski Space in General Relativity

We present a generalization of the celebrated results by D. Christodoulou and S. Klainerman for solutions of the Einstein vacuum equations in General Relativity. In 'The global nonlinear stability of the Minkowski space' they showed that every strongly asymptotically flat, maximal, initial data which is globally close to the trivial data gives rise to a solution which is a complete spacetime tending to the Minkowski spacetime at infinity along any geodesic. We consider the Cauchy problem with more general, asymptotically flat initial data. This yields a spacetime curvature which is no longer bounded in $L^{\infty}$. As a major result and as a consequence of our relaxed assumptions, we encounter in our work borderline cases, which we discuss in this talk as well. The main proof is based on a bootstrap argument.