• Harmonic Analysis and Partial Differential Equations
• Closing Workshop

• Schedule
• Participants

We establish global well-posedness and scattering for the cubic Dirac equation for small data in the critical space . The theory we develop is the Klein-Gordon equivalent of the Wave Maps / Schroedinger Maps theory. This is joint work with Sebastian Herr.

A timelike minimal surface in Minkowski space satisfies a quasilinear wave equation. I will explain how the equation exhibits a null structure that allows to lower the regularity compared to the sharp general result of H. Smith and Tataru. I will also explain how to obtain the appropriate bilinear space-time estimates for solutions of a variable rough coefficients wave equations.

Given a compact Lie group and a principal bundle over a closed Riemannian manifold, the quotient space of connections, modulo the action of the group of gauge transformations, has fundamental significance for algebraic geometry, low-dimensional topology, the classification of smooth four-dimensional manifolds, and high-energy physics. The quotient space of connections is equipped with the Yang-Mills energy functional and Atiyah and Bott (1983) had proposed that its gradient flow with respect to the natural Riemannian metric on the quotient space should prove to be an important tool for understanding the topology of the quotient space via an infinite-dimensional Morse theory. The critical points of the energy functional are gauge-equivalence classes of Yang-Mills connections. However, thus far, smooth solutions to the Yang-Mills gradient flow have only been known to exist for all time and converge to critical points, as time tends to infinity, in relatively few cases, including (1) when the base manifold has dimension two or three (Rade, 1991 and 1992, in dimension two and three; G. Daskalopoulos, 1989 and 1992, in dimension two), (2) when the base manifold is a complex algebraic surface (Donaldson, 1985), and (3) in the presence of rotational symmetry in dimension four (Schlatter, Struwe, and Tahvildar-Zadeh, 1998). Global existence of solutions with up to finitely many point singularities (caused by the ``bubbling’’ phenomenon) was proved independently by Struwe (1994) and Kozono, Maeda, and Naito (1995). However, the question of global existence of smooth solutions over general compact, Riemannian, four-dimensional base manifolds has thus far remained unresolved. In this talk we shall describe our proof of the following result: Given a compact Lie group and a smooth initial connection on a principal bundle over a compact, Riemannian, four-dimensional manifold, there is a unique, smooth solution to the Yang-Mills gradient flow which exists for all time and converges to a smooth Yang-Mills connection on the given principal bundle as time tends to infinity.

I will talk about some generalized space-time estimates for the Schrodinger equation. The first one is the generalized Strichartz estimates with weaker angular integrability. We obtain sharp estimate except some endpoints. The second one is spherical averaged maximal function estimates. These estimates are useful in the scattering problem for the nonlinear PDEs. Some applications will be given to 3D Zakharov system and 2D cubic derivative Schrodinger equation and Schrodinger map.

We review the method of testing by wave packets, developed by Ifrim and Tataru in the context of the 1d NLS and 2d water waves, and present an application.

We consider the exterior wave maps in 3d. Each such finite energy equivariant wave map has a fixed integer-valued topological degree, and in each degree class there is a unique harmonic map that minimizes the energy. We show that any arbitrary equivariant exterior wave map with finite energy will scatter to the unique harmonic map in its degree class. This completely revolves a recent conjecture of Bizon, Chmaj and Maliborski, who observed this asymptotic behavior numerically.

This is a joint work with C. Kenig, A. Lawrie and W. Schlag.

We consider a class of defocusing nonlinear Schrödinger equations for which the nonlinearity is neither mass-critical nor energy-critical. Following a concentration compactness approach, we prove that any solution that remains bounded in the critical Sobolev space must be global and scatter.

This talk is based on joint work with Nader Masmoudi. The nonlinear Schrodinger equation (NLS) is a universal approximation for PDE's, describing long-time dynamics of nonlinear dispersive waves with almost single frequency in space. Our goal is to extend this theory to waves with multiple frequencies, thereby revealing the dynamical effects of nonlinear resonances. As a model equation, we take the dispersion relation in two dimensional water wave with finite depth, surface tension and gravity, because it has concave-convex structure, leading to non-trivial resonances besides that in the standard NLS approximation. Taking the initial data from the so-called modulation space, we can prove long-time existence in the NLS scaling of solutions with infinitely many frequencies. If the initial data is concentrating around a finite number of frequencies, the asymptotic behavior is described by a system of NLS with some modifications and initial layers. The main idea is a simple but exhaustive use of time/space non-resonance, in space-time norms associated with the modulation space, which seems hard to implement in the weighted Sobolev space.

I will describe a recent joint work with Herbert Koch and Arshak Petrosyan on the analyticity of the regular part of the free boundary in the thin obstacle problem.  The main method we use is a generalized partial hodograph-Legendre transformation, which we show to be invertible by analyzing the precise asymptotic behavior of the solution near the free boundary. The corresponding Legendre function will satisfy a fully nonlinear PDE, which can be viewed as an appropriate perturbation of the Grushin operator. By using the existing Lp theory for such operators, we are able to show the smoothness and even the analyticity of the Legendre function and hence the analyticity of the free boundary.

The goal of this talk will be to find the sharp forms and characterize the complex-valued extremizers of the adjoint Fourier restriction inequalities on the sphere:

in the cases with and satisfying:

• Κ = 2, q ≥ 4 and 3 ≤ d ≤ 7
• Κ = 2, q ≥ 4 and d ≤ 8
• Κ = 3, q ≥ 2Κ and d ≥ 2

Tools include a spherical harmonic decomposition, the study of the Cauchy-Pexider functional equation for sumsets of the sphere, and a sharp multilinear weighted restriction inequality with weight related to the -fold convolution of the surface measure.

This topic builds up on recent results by D. Foschi, and is joint work with E. Carneiro.

As we all know, the threshold space of well/ill posedness to the two dimensional cubic Schrödinger equation is L², no matter on Euclidean space, torus or partial torus. As to the Hyperbolic cubic Schrödinger equation, the threshold space is also L² for Euclidean case, but there are no clues to imply this is also true for torus or partial torus. Here we give a negative answer to this question by constructing a periodic ill-posed initial data in H^s once s < 1/2. The well-posedness for s > 1/2 is also set up. We will also discuss the partial periodic case.

I will introduce some aspects of Host-Kra structure theory and sketch the role that they play in the proof of the multiple term return times theorem.