# Seminars & Events for Analysis Seminar

##### Scattering for the 3d Zakharov system

We prove global existence and scattering for small localized solutions of the Cauchy problem for the Zakharov system in 3 space dimensions. Joint work with Zaher Hani and Jalal Shatah.

##### On the loss of continuity for supercritical drift-diffusion equations

We consider the (linear) drift-diffusion equation $\partial_t \theta + u \cdot \nabla \theta + (-\Delta)^s \theta = 0$. Here the divergence free drift $u$ belongs to a supercritical space, and $0 < s \leq 1$. We prove that starting with smooth initial data solutions may become discontinuous in ﬁnite time. For $s < 1$ this may even be achieved with autonomous drift. On the other hand, for $s = 1$ and autonomous drift, in two space dimensions we obtain a modulus of continuity for the solution depending only on the $L^1$ norm of the drift, which is a supercritical quantity. This is joint work with L. Silvestre and A. Zlatos.

##### A Large Data Regime for non-linear Wave Equations

For semi-linear wave equations with null form non-linearities on $\mathbb{R}^{3+1}$, we exhibit an open set of initial data which are allowed to be large in energy spaces, yet we can still obtain global solutions in the future. This is a joint work with Jinhua Wang.

##### Breakdown criterion in general relativity: spherically symmetric spacetimes

At the heart of the (weak and strong) cosmic censorship conjectures is a statement regarding singularity formation in general relativity. Even in spherical symmetry, cosmic censorship seems, at the moment, mathematically intractable. To give a framework in which to address these very difficult problems, we will introduce a notion of spherically symmetric ‘strongly tame’ Einstein-matter models, examples of which are given by Einstein-Maxwell-Klein-Gordon (charged scalar fields) and Einstein-Maxwell-dilaton (solutions of which appear in the low-energy limit of string theory). We will demonstrate that for any ‘strongly tame’ model there is an a priori characterization of the spacetime boundary.

##### Low-regularity local wellposedness of Chern-Simons-Schroedinger

The Chern-Simons-Schroedinger model in two spatial dimensions is a covariant NLS-type problem that is $L^2$ critical. We prove that, with respect to the heat gauge, this problem is locally well-posed for initial data that is small in $H^s$, $s > 0$. This work is joint with Baoping Liu and Daniel Tataru.

##### The large box limit for 2D NLS

I will report on work in progress, in collaboration with Zaher Hani and Erwan Faou. Starting from 2D NLS set on the torus, we derive, in the appropriate weakly nonlinear regime, a new asymptotic model set in the whole space. The limiting system has very striking properties and is related to weak turbulence questions.

##### Fourier Law and Non-Isothermal Boundary in the Boltzmann Theory

In the study of the heat transfer in the Boltzmann theory, the basic problem is to construct solutions to the steady problem for the Boltzmann equation in a general bounded domain with diffuse reflection boundary conditions corresponding to a non-isothermal temperature of the wall. Denoted by \delta the size of the temperature oscillations on the boundary, we develop a theory to characterize such a solution mathematically. We construct a unique solution F_s to the Boltzmann equation, which is dynamically asymptotically stable with exponential decay rate. We remark that this solution differs from a local equilibrium Maxwellian, hence it is a genuine non-equilibrium stationary solution.

##### Scalar equations as asymptotic models for internal waves in Oceanography

We propose to derive rigorously scalar asymptotic models for the propagation of gravity waves at the interface between two layers of immiscible fluids of different densities (modeling fresh and salt water interface).

The KdV equation is known to accurately describe such a system in the long wave regime, but the rigorous justification of this fact is recent. We will present and extend this result to a regime allowing greater nonlinearities, yielding higher order equations, of Camassa-Holm type.

##### Global results for linear waves on expanding Schwarzschild de Sitter cosmologies

In this talk I will present recent results for solutions to the linear wave equation on Schwarzschild de Sitter black hole spacetimes. We focus here on the expanding region of the spacetime, and exhibit a stability mechanism which manifests itself in a global redshift effect. I shall describe how this can be combined with earlier results to obtain a global linear stability result. If time permits, applications to Kerr de Sitter spacetimes will be discussed.

The talk is based on the preprint: http://arxiv.org/abs/1207.6055

##### On a class of super-critical quasi-geostrophic type models in fluid dynamics

I will discuss some recent results on a class of active scalars with non-local transport and super-critical dissipation. Part of the talk is based on some joint work with Hongjie Dong.

##### On inverse spectral problem for quasi-periodic Schrödinger equation

In this talk I will discuss the main ideas and new technology of our recent joint work with D.Damanik. We study the quasi-periodic Schr\"{o}dinger equation $$-\psi''(x) + V(x) \psi(x) = E \psi(x), \qquad x \in \mathbb{R} $$ in the regime of ``small'' $V$. PLEASE CLICK ON SEMINAR TITLE FOR COMPLETE ABSTRACT.

##### An integrability theorem for harmonic maps of interest in General Relativity

Einstein's Equations have been extensively studied in the context of integrability for several decades, drawing on results from inverse scattering to algebraic curves. In this talk, we will give a generalized notion of integrability for axially symmetric harmonic maps into symmetric spaces and prove that under some mild restrictions, all such maps are integrable. A primary application of the result involves generating $N$-soliton harmonic maps into the Grassmann manifold $SU(p,q) / S( U(p) x U(q) )$, a special case of which recovers the Kerr and Kerr-Newman family of solutions to the Einstein vacuum and Einstein-Maxwell equations, respectively. This is joint work with S. Tahvildar-Zadeh.

##### Local existence and uniqueness of Prandtl equations

The Prandtl equations, which describe the boundary layer behavior of a viscous incompressible fluid near the physical wall, play an important role in the zero-viscosity limit of Navier-Stokes equations. In this talk we will discuss the local-in-time existence and uniqueness for the Prandtl equations in weighted Sobolev spaces under the Oleinik's monotonicity assumption. The proof is based on weighted energy estimates, which come from a new type of nonlinear cancellations between velocity and vorticity.

##### Analysis and Partial Differential Equations on Polyhedral Domains

The classical theory of partial differential equations (PDEs) on smooth, bounded domains--a great achievement of modern mathematics--is well understood and has many applications in both pure and applied mathematics. Many domains that arise in applications are, however, not smooth (that is, they do not have a smooth boundary). PLEASE CLICK ON SEMINAR TITLE FOR COMPLETE ABSTRACT.

##### Global dynamics beyond the ground state for the energy critical Schrodinger equation

##### The Boltzmann equation, Besov spaces, and optimal decay rates in R^n

In this talk, we will study the large-time convergence to the global Maxwellian of perturbative classical solutions to the Boltzmann equation on R^n, for n geq 3, without the angular cut-off assumption. We prove convergence of the k-th order derivatives in the norm L^r_x(L^2_v), for any 2 leq r leq infinity, with optimal decay rates, in the sense that they are equal to the rates which one obtains for the corresponding linear equation. PLEASE CLICK ON SEMINAR TITLE FOR COMPLETE ABSTRACT.

##### Boundary regularity for Monge-Ampere equation

Boundary estimates for second derivatives of solutions to the Dirichlet problem for the Monge-Ampere equation were first obtained by Ivockina in 1980. PLEASE CLICK ON SEMINAR TITLE FOR COMPLETE ABSTRACT.

##### Quasilinear Schrödinger Equations

We discuss recent work with Jason Metcalfe and Daniel Tataru on local well posedness results for quasilinear Schrödinger equations. We will discuss both a natural functional framework, as well as the local smoothing, energy estimates and multilinear estimates required.

##### Probabilistic global well-posedness for radial nonlinear Schrodinger and wave equations on the ball via Gibbs measure evolution

We discuss recent works with Jean Bourgain in which we establish new global well-posedness results along Gibbs measure evolutions for the radial nonlinear wave and Schrodinger equations posed on the unit ball in $\mathbb{R}^N$. PLEASE CLICK ON SEMINAR TITLE FOR COMPLETE ABSTRACT.

##### Stable Big Bang Formation in Near-Flrw Solutions to the Einstein-Stiff Fluid System

(CLICK ON SEMINAR TITLE FOR COMPLETE ABSTRACT.) I will discuss some results that I recently obtained in collaboration with Igor Rodnianski. The results concern small perturbations of the well-known Friedmann-Lema\^{\i}tre-Robertson-Walker solution to the Einstein-stiff fluid system (stiff FLRW). PLEASE NOTE SPECIAL DAY AND TIME.