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 Spring 2012

date	speaker	title	host(s)
Feb 6	Yao Yao (UCLA)	Degenerate diffusion with nonlocal aggregation: behavior of solutions	Kiselev
March 12	Xuan Hien Nguyen (Iowa State)	Gluing constructions for solitons and self-shrinkers under mean curvature flow	Angenent
March 19	Nestor Guillen (UCLA)	The local geometry of maps with c-convex potentials	Feldman
March 26	Vlad Vicol (University of Chicago)	Shape dependent maximum principles and applications	Kiselev
April 9	Charles Smart (MIT)	TBA	Seeger
April 16	Jiahong Wu (Oklahoma)	The 2D Boussinesq equations with partial dissipation	Kiselev

Abstracts

Yao Yao (UCLA)

Degenerate diffusion with nonlocal aggregation: behavior of solutions

The Patlak-Keller-Segel (PKS) equation models the collective motion of cells which are attracted by a self-emitted chemical substance. While the global well-posedness and finite-time blow up criteria are well known, the asymptotic behaviors of solutions are not completely clear. In this talk I will present some results on the asymptotic behavior of solutions when there is global existence. The key tools used in the paper are maximum-principle type arguments as well as estimates on mass concentration of solutions. This is a joint work with Inwon Kim.

Xuan Hien Nguyen (Iowa State)

Gluing constructions for solitons and self-shrinkers under mean curvature flow

In the 1990s, Kapouleas and Traizet constructed new examples of minimal surfaces by desingularizing the intersection of existing ones with Scherk surfaces. Using this idea, one can find new examples of self-translating solutions for the mean curvature flow asymptotic at infinity to a finite family of grim reaper cylinders in general position. Recently, it has been shown that it is possible to desingularize the intersection of a sphere and a plane to obtain a family of self-shrinkers under mean curvature flow. I will discuss the main steps and difficulties for these gluing constructions, as well as open problems.

Nestor Guillen (UCLA)

We consider the Monge-Kantorovich problem, which consists in transporting a given measure into another "target" measure in a way that minimizes the total cost of moving each unit of mass to its new location. When the transport cost is given by the square of the distance between two points, the optimal map is given by a convex potential which solves the Monge-Ampère equation, in general, the solution is given by what is called a c-convex potential. In recent work with Jun Kitagawa, we prove local Holder estimates of optimal transport maps for more general cost functions satisfying a "synthetic" MTW condition, in particular, the proof does not really use the C^4 assumption made in all previous works. A similar result was recently obtained by Figalli, Kim and McCann using different methods and assuming strict convexity of the target.

Charles Smart (MIT)

TBA

Vlad Vicol (University of Chicago)

Title: Shape dependent maximum principles and applications

Abstract: We present a non-linear lower bound for the fractional Laplacian, when evaluated at extrema of a function. Applications to the global well-posedness of active scalar equations arising in fluid dynamics are discussed. This is joint work with P. Constantin.

Jiahong Wu (Oklahoma State)

"The 2D Boussinesq equations with partial dissipation"

The Boussinesq equations concerned here model geophysical flows such as atmospheric fronts and ocean circulations. Mathematically the 2D Boussinesq equations serve as a lower-dimensional model of the 3D hydrodynamics equations. In fact, the 2D Boussinesq equations retain some key features of the 3D Euler and the Navier-Stokes equations such as the vortex stretching mechanism. The global regularity problem on the 2D Boussinesq equations with partial dissipation has attracted considerable attention in the last few years. In this talk we will summarize recent results on various cases of partial dissipation, present the work of Cao and Wu on the 2D Boussinesq equations with vertical dissipation and vertical thermal diffusion, and explain the work of Chae and Wu on the critical Boussinesq equations with a logarithmically singular velocity.