# Seminars & Events for Analysis Seminar

##### Gauge choice for the Yang-Mills equations using the Yang-Mills heat flow

In this talk, I will describe an approach to the problem of gauge choice for the Yang-Mills equations on the (d+1)-dimensional Minkowski space ($d \geq 2$), which reveals the special structure (e.g. null structure) of the equations and is applicable to arbitrarily large data. The key ingredient is the Yang-Mills heat flow, which is a parabolic analogue of the Yang-Mills equations. Several applications of this approach will be described, including an alternative proof of the finite energy global well-posedness in $d=3$ (a classical result of Klainerman-Machedon '95) and a proof of almost optimal local well-posedness in $d=4$ for arbitrarily large data.

##### Euler equations and endpoint elliptic regularity in nonsmooth domains

The planar Euler equations describe the motion of a 2-D inviscid incompressible fluid, and also arise as a model problem for the study of the barotropic mode (to put it simply, the vertical average) of the Primitive equations of the ocean. PLEASE CLICK ON SEMINAR TITLE FOR COMPLETE ABSTRACT.

##### Resonances for Normally Hyperbolic Trapped Sets

(PLEASE NOTE SPECIAL TIME.) Resonances are complex analogs of eigenvalues for Laplacians on noncompact manifolds, arising in long time resonance expansions of linear waves. CLICK ON SEMINAR TITLE FOR COMPLETE ABSTRACT.

##### Weighted Integrability of Polyharmonic Functions

A function $u$ is is said to be $N$-harmonic if it solves the PDE $\Delta^N u=0$. We shall consider $N$-harmonic functions on the unit disk, which is a rather special case, as a result of quadrature-type properties. PLEASE CLICK ON SEMINAR TITLE FOR COMPLETE ABSTRACT.

##### Soliton stability for Schroedinger and Wave Maps

I will discuss the stability of the equivariant ground states for Schroedinger and Wave Maps in two dimension.

##### On A Well-Tempered Diffusion

PLEASE NOTE SPECIAL DATE, TIME AND LOCATION. The classical transport theory as expressed by, say, the Fokker-Planck equation, lives in an analytical paradise but, in sin. CLICK ON SEMINAR TITLE FOR COMPLETE ABSTRACT.

##### On the stability of breathers

In this talk I will discuss stability properties of a class of very special solutions of some well-known integrable equations, called ''breathers''. Our proof does not rely on the inverse scattering transform, so it applies to very general initial data. This is a joint work with Miguel A. Alejo.

##### Navier-Stokes regularity criteria

PLEASE NOTE DIFFERENT DATE, TIME AND LOCATION. We generalize a well-known result due to Escauriaza-Seregin-Sverak by showing that Navier-Stokes solutions cannot develop a singularity if certain scale-invariant spatial Besov norms remain bounded in time. CLICK ON SEMINAR TITLE FOR COMPLETE ABSTRACT.

##### Statistical Mechanics for gKdV.

We will discuss ideas from statistical mechanics applied to the analysis of the periodic generalized KdV equation (gKdV). We first discuss invariance of the Gibbs measure for gKdV.

##### Maximal hypoellipticity for the $\overline{\partial}$-Neumann problem

We establish maximal hypoellipticity (in $L^p$-Sobolev and Lipschitz norms) for the $\overline{\partial}$-Neumann problem on smooth, bounded pseudoconvex domains in $\mathbb{C} ^n$ under the weakest possible condition on the Levi form. In particular, maximal hypoellipticity holds on the level of $(n-1)$-forms for all smooth, bounded pseudoconvex domains of finite commutator type. These results are new in dimensions $n \ge 3$.