PDE Geometric Analysis seminar

The seminar will be held in room B115 of Van Vleck Hall on Mondays from 3:30pm - 4:30pm, unless indicated otherwise.

Seminar Schedule Fall 2013

date	speaker	title	host(s)
September 9	Greg Drugan (U. of Washington)	Construction of immersed self-shrinkers	Angenent
October 7	Guo Luo (Caltech)	Potentially Singular Solutions of the 3D Incompressible Euler Equations.	Kiselev
November 18	Roman Shterenberg (UAB)	TBA.	Kiselev

Seminar Schedule Spring 2014

date	speaker	title	host(s)
March 3	Hongjie Dong (Brown)	TBA.	Kiselev
April 7	Zoran Grujic (University of Virginia)	TBA.	Kiselev

Abstracts

Greg Drugan (U. of Washington)

Construction of immersed self-shrinkers

Abstract: We describe a procedure for constructing immersed self-shrinking solutions to mean curvature flow. The self-shrinkers we construct have a rotational symmetry, and the construction involves a detailed study of geodesics in the upper-half plane with a conformal metric. This is a joint work with Stephen Kleene.

Guo Luo (Caltech)

Potentially Singular Solutions of the 3D Incompressible Euler Equations

Abstract: Whether the 3D incompressible Euler equations can develop a singularity in finite time from smooth initial data is one of the most challenging problems in mathematical fluid dynamics. This work attempts to provide an affirmative answer to this long-standing open question from a numerical point of view, by presenting a class of potentially singular solutions to the Euler equations computed in axisymmetric geometries. The solutions satisfy a periodic boundary condition along the axial direction and no-flow boundary condition on the solid wall. The equations are discretized in space using a hybrid 6th-order Galerkin and 6th-order finite difference method, on specially designed adaptive (moving) meshes that are dynamically adjusted to the evolving solutions. With a maximum effective resolution of over $(3 \times 10^{12})^{2}$ near the point of singularity, we are able to advance the solution up to $\tau_{2} = 0.003505$ and predict a singularity time of $t_{s} \approx 0.0035056$, while achieving a \emph{pointwise} relative error of $O(10^{-4})$ in the vorticity vector $\omega$ and observing a $(3 \times 10^{8})$-fold increase in the maximum vorticity $\norm{\omega}_{\infty}$. The numerical data is checked against all major blowup (non-blowup) criteria, including Beale-Kato-Majda, Constantin-Fefferman-Majda, and Deng-Hou-Yu, to confirm the validity of the singularity. A careful local analysis also suggests that the blowing-up solution develops a self-similar structure near the point of the singularity, as the singularity time is approached.