Analysis Seminar

This talk is concerned with solutions of the 3D incompressible Navier-Stokes equations that are bounded in a critical space. From small initial data, these solutions are known to be globally well-posed due to classical work of Fujita-Kato and others...

The compressible Euler equation can lead to the emergence of shock discontinuities in finite time, notably observed behind supersonic planes. A very natural way to justify these singularities involves studying solutions as inviscid limits of Navier...

Let $M(n,m)$ denote the real $m\times n$ matrices. A continuous function $f\colon M(n,m)\to \R$ is called {\it Morrey quasi-convex} if  $$\int_{\R^n}(f(A+\nabla\vf(x))-f(A))\,dx\ge 0$$ for each smooth, compactly supported $\vf\colon\R^n\to\R^m$ and...

In 1962, Yudovich established the well-posedness of the two-dimensional incompressible Euler equations for solutions with bounded vorticity. However, uniqueness within the broader class of solutions with L^p vorticity remains a key unresolved...

Semiclassical measures are a standard object studied in quantum chaos, capturing macroscopic behavior of sequences of eigenfunctions in the high energy limit. They have a long history of study going back to the Quantum Ergodicity theorem and the...

The lateral periodic condition has been imposed canonically and investigated extensively in the study of a channel flow, which unfortunately is not compatible with either the celebrated Reynolds' experiment or the self-similar Blasius boundary layer...

New results on quantum tunneling between deep potential wells, in the presence of a strong constant magnetic field are presented. This includes a family of double well potentials containing examples for which the low-energy eigenvalue splitting...

An evolving surface is a mean curvature flow if the normal component of its velocity field is given by the mean curvature. First introduced in the physics literature in the 1950s, the mean curvature flow equation has been studied intensely by...

Gamow introduced the liquid drop model in 1928 as a model for the nucleus. I will discuss some recent work (with Ian Ruohoniemi) concerning roundness of minimizers. 

Extremal black holes are special solutions of Einstein’s equations which have absolute zero temperature in the thermodynamic analogy of black hole mechanics. In this talk, I will present a proof that extremal black holes arise on the critical...