Events for 03/07/2024 from all calendars

Noncommutative Geometry Seminar

Time: 3:00PM - 4:00PM

Location: BLOC 628

Speaker: Jingwen Chen, University of Pennsylvania

Title: Mean curvature flow with multiplicity $2$ convergence

Abstract: Mean curvature flow (MCF) has been widely studied in recent decades, and higher multiplicity convergence is an important topic in the study of MCF. In this talk, we present two examples of immortal MCF in $\mathbb{R}^3$ and $S^n \times [-1,1]$, which converge to a plane and a sphere $S^n$ with multiplicity $2$, respectively. Additionally, we will compare our example with some recent developments on the multiplicity one conjecture and the min-max theory. This is joint work with Ao Sun. The talk is in person and also broadcast at https://tamu.zoom.us/s/94046447051.

The Foias Lectures

Time: 4:00PM - 5:00PM

Location: BLOC 117

Speaker: Professor Camillo De Lellis, Institute of Advanced Studies

Title: Anomalous dissipation and flows of rough vector fields

Abstract: Consider smooth solutions to the 3d Navier-Stokes for divergence-free vector fields $u^\varepsilon$: \[ \partial_t u^\varepsilon + {\rm div}\, (u^\varepsilon \otimes u^\varepsilon ) + \nabla p^\varepsilon = \varepsilon \Delta u^\varepsilon \] While the balance of the energy is \[ \frac{d}{dt} \int |u^\varepsilon|^2 (x,t)\, dx = - 2 \varepsilon \int |Du^\varepsilon|^2 (x,t)\, dx\, , \] it is a tenet of the theory of fully developed turbulence that in a variety of situations the size of the left hand side should typically be independent of $\varepsilon$. However this ``anomalous'' dissipation should not be triggered by an hypothetical initial datum $u^\varepsilon (\cdot, 0)$ in which we ignite lots of oscillations: a rule of thumb would be that, if the {\em linear} Stokes equations exhibit a similar behavior, then the roughness of the initial data is certainly excessive. Producing rigorous mathematical examples of this prediction is hard: a tight control on the smoothness of the initial data forces the solutions to converge to classical solutions of the Euler equation and therefore rules out any possible dissipation, unless one shows a quite severe blow-up for smooth solutions of Euler; for sufficiently rough initial data this obstruction is absent, but proving something about the solutions of Navier-Stokes becomes very challenging. In a recent joint work with Elia Bru\'e we study what happens if we introduce a forcing term $f^\varepsilon$. The problem is compatively easier and it can be proved that there is an exact regularity threshold below which anomalous dissipation happens in some examples and above which it would only be possible through a blow-up scenario. Surprisingly one side of the problem is linked with some fundamental questions about solving ODEs with rough coefficients.