The seminar meets regularly on wednesday at 4pm in Eckhart 202. We also have special seminars during other days.
In this talk I will report recent advances on some classes of nonvariational free boundary problems as well as singular PDEs governed by fully nonlinear elliptic operators.
The vortex-stretching term in the vorticity-velocity formulation of the 3D Navier-Stokes equations holds a key to understanding the mechanism of possible formation of singularities in the model. It is known that on sufficiently small scales, the vortex-stretching term is strongly dominated by the diffusion. The goal of this lecture is to show that, in a statistically significant sense, outside of the diffusion range and across a range of scales extending from a power of a modified Kraichnan scale to the integral scale, the vortex-stretching term is in fact comparable to a modified diffusion term. The proof is based on a suitable ensemble averaging process recently introduced in a joint work with R. Dascaliuc concerning the study of turbulent cascades in physical scales of the incompressible flows. The lower bound on the (averaged) vortex-stretching term may seem like bad news; however, it also indicates (again, in a statistically significant sense) a geometric scenario resulting in closing the scaling gap in the regularity problem.
We study the regularity of almost minimizers for the types of functionals analyzed by Alt, Caffarelli and Friedman. Although almost minimizers do not satisfy an equation using appropriate comparison functions we prove several regularity results. For example in the one phase situation we show that almost minimizers are Lipschitz. Our approach is reminiscent of the one used in geometric measure theory to study the regularity of almost minimizers for area. This project is joint work with Guy David.
The addition of white noise driven terms to the fundamental equations of physics and engineering are used to model numerical and empirical uncertainties. In this talk we will discuss some recent results for the Stochastic Navier-Stokes and Euler Equations as well as for the Stochastic Primitive Equations, a basic model in geophysical scale fluid flows. For all of the above systems our results cover the case of a general nonlinear multiplicative stochastic forcing.
I will discuss several aspects of Branching random walks and their relation with the KPP equation on the one hand, and the maximum of certain (two dimensional) Gaussian fields on the other. I will not assume any knowledge about either of these terms.
I shall present some recent work in collaboration with V. Banica about the stability of the selfsimilar solutions to the vortex filament equation. In this equation a curve (filament) moves in the direction of the binormal with a speed proportional to its curvature.