## PDE Real Analysis Seminar

Date, time & place Tuesday 10:30 - 11:30 Room #056 (Graduate School of Math. Sci. Bldg.)

Future seminars

### 2017/11/21

10:30-11:30   Room #056 (Graduate School of Math. Sci. Bldg.)
Felix Schulze (University College London)
Optimal isoperimetric inequalities for surfaces in any codimension
[ Abstract ]
Let $(M^n,g)$ be simply connected, complete, with non-positive sectional
curvatures, and $\Sigma$ a 2-dimensional surface in $M^n$. Let $S$ be an area
minimising 3-current such that $\partial S = \Sigma$. We use a weak mean
curvature flow, obtained via elliptic regularisation, starting from
$\Sigma$, to show that $S$ satisfies the optimal Euclidean isoperimetric
inequality: $|S| \leq 1/(6\sqrt{\pi}) |\Sigma|^{3/2}$. We also obtain the
optimal estimate in case the sectional curvatures of $M$ are bounded from
above by $\kappa < 0$ and characterise the case of equality. The proof
follows from an almost monotonicity of a suitable isoperimetric
difference along the approximating flows in one dimension higher.

### 2017/12/12

10:30-11:30   Room #056 (Graduate School of Math. Sci. Bldg.)
Alex Mahalov (Arizona State University)
Stochastic Three-Dimensional Navier-Stokes Equations + Waves: Averaging, Convergence, Regularity and Nonlinear Dynamics (English)
[ Abstract ]
We establish multi-scale stochastic averaging, convergence and regularity theorems in a general framework by bootstrapping from global regularity of the averaged stochastic resonant equations. The averaged covariance operator couples stochastic and wave effects. We also present theoretical results for 3D nonlinear dynamics.