# 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.

### Previous PDE/GA seminars

## 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 21(Wednesday!), Room 901 Van Vleck | Nestor Guillen (UCLA) | Feldman | |

March 26 | Vlad Vicol (University of Chicago) |
Shape dependent maximum principles and applications |
Kiselev |

April 9 | Charles Smart (MIT) | Seeger | |

April 16 | Jiahong Wu (Oklahoma) |
The 2D Boussinesq equations with partial dissipation |
Kiselev |

April 23 | Joana Oliveira dos Santos Amorim (Universite Paris Dauphine) |
A geometric look on Aubry-Mather theory and a theorem of Birkhoff |
Bolotin |

April 27 (Colloquium. Friday at 4pm, in Van Vleck B239) | Gui-Qiang Chen (Oxford) |
Nonlinear Partial Differential Equations of Mixed Type |
Feldman |

May 14 | Jacob Glenn-Levin (UT Austin) |
Incompressible Boussinesq equations in borderline Besov spaces |
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)

*PDE methods for the Abelian sandpile*

Abstract: The Abelian sandpile growth model is a deterministic diffusion process for chips placed on the $d$-dimensional integer lattice. One of the most striking features of the sandpile is that it appears to produce terminal configurations converging to a peculiar lattice. One of the most striking features of the sandpile is that it appears to produce terminal configurations converging to a peculiar fractal limit when begun from increasingly large stacks of chips at the origin. This behavior defied explanation for many years until viscosity solution theory offered a new perspective. This is joint work with Lionel Levine and Wesley Pegden.

### 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.

### Joana Oliveira dos Santos Amorim (Universit\'e Paris Dauphine)

"A geometric look on Aubry-Mather theory and a theorem of Birkhoff"

Given a Tonelli Hamiltonian $H:T^*M \lto \Rm$ in the cotangent bundle of a compact manifold $M$, we can study its dynamic using the Aubry and Ma\~n\'e sets defined by Mather. In this talk we will explain their importance and give a new geometric definition which allows us to understand their property of symplectic invariance. Moreover, using this geometric definition, we will show that an exact Lipchitz Lagrangian manifold isotopic to a graph which is invariant by the flow of a Tonelli Hamiltonian is itself a graph. This result, in its smooth form, was a conjecture of Birkhoff.

### Gui-Qiang Chen (Oxford)

"Nonlinear Partial Differential Equations of Mixed Type"

Many nonlinear partial differential equations arising in mechanics and geometry naturally are of mixed hyperbolic-elliptic type. The solution of some longstanding fundamental problems in these areas greatly requires a deep understanding of such nonlinear partial differential equations of mixed type. Important examples include shock reflection-diffraction problems in fluid mechanics (the Euler equations), isometric embedding problems in in differential geometry (the Gauss-Codazzi-Ricci equations), among many others. In this talk we will present natural connections of nonlinear partial differential equations with these longstanding problems and will discuss some recent developments in the analysis of these nonlinear equations through the examples with emphasis on identifying/developing mathematical approaches, ideas, and techniques to deal with the mixed-type problems. Further trends, perspectives, and open problems in this direction will also be addressed. This talk will be based mainly on the joint work correspondingly with Mikhail Feldman, Marshall Slemrod, as well as Myoungjean Bae and Dehua Wang.

### Jacob Glenn-Levin (UT Austin)

We consider the Boussinesq equations, which may be thought of as inhomogeneous, incompressible Euler equations, where the inhomogeneous term is a scalar quantity, typically density or temperature, governed by a convection-diffusion equation. I will discuss local- and global-in-time well-posedness results for the incompressible 2D Boussinesq equations, assuming the density equation has nonzero diffusion and that the initial data belongs in a Besov-type space.