# Difference between revisions of "PDE Geometric Analysis seminar"

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+ | A celebrated result in the theory of Sobolev spaces is Morrey's inequality, which establishes the continuous embedding of the continuous functions in certain Sobolev spaces. Interestingly enough the equality case of this inequality has not been thoroughly investigated (unless the underlying domain is R^n). We show that if the underlying domain is a bounded convex domain, then the extremal functions are determined up to a multiplicative factor. We will explain why the assertion is false if convexity is dropped and why convexity is not necessary for this result to hold. | ||

===Tau Shean Lim=== | ===Tau Shean Lim=== |

## Revision as of 22:15, 19 September 2016

The seminar will be held in room 901 of Van Vleck Hall on Mondays from 3:30pm - 4:30pm, unless indicated otherwise.

## Contents

### Previous PDE/GA seminars

### Tentative schedule for Spring 2017

# PDE GA Seminar Schedule Fall 2016

date | speaker | title | host(s) |
---|---|---|---|

September 12 | Daniel Spirn (U of Minnesota) | Dipole Trajectories in Bose-Einstein Condensates | Kim |

September 19 | Donghyun Lee (UW-Madison) | The Boltzmann equation with specular boundary condition in convex domains | Feldman |

September 26 | Kevin Zumbrun (Indiana) | Kim | |

October 3 | Will Feldman (UChicago ) | Lin & Tran | |

October 10 | Ryan Hynd (UPenn) | Extremal functions for Morrey’s inequality in convex domains | Feldman |

October 17 | Gung-Min Gie (Louisville) | Kim | |

October 24 | Tau Shean Lim (UW Madison) | Traveling Fronts of Reaction-Diffusion Equations with Ignition Media and Levy Operators | Kim & Tran |

October 31 | Tarek Elgindi ( Princeton) | Propagation of Singularities in Incompressible Fluids | Lee & Kim |

November 7 | Adrian Tudorascu (West Virginia) | Feldman | |

November 14 | Alexis Vasseur ( UT-Austin) | Feldman | |

November 21 | Minh-Binh Tran (UW Madison ) | Quantum Kinetic Problems | Hung Tran |

November 28 | ( ) | ||

December 5 | Brian Weber (University of Pennsylvania) | TBA | Bing Wang |

December 12 | David Kaspar (Brown) | Tran |

# Abstracts

### Daniel Spirn

Dipole Trajectories in Bose-Einstein Condensates

Bose-Einstein condensates (BEC) are a state of matter in which supercooled atoms condense into the lowest possible quantum state. One interesting important feature of BECs are the presence of vortices that form when the condensate is stirred with lasers. I will discuss the behavior of these vortices, which interact with both the confinement potential and other vortices. I will also discuss a related inverse problem in which the features of the confinement can be extracted by the propagation of vortex dipoles.

### Donghyun Lee

The Boltzmann equation with specular reflection boundary condition in convex domains

I will present a recent work (https://arxiv.org/abs/1604.04342) with Chanwoo Kim on the global-wellposedness and stability of the Boltzmann equation in general smooth convex domains.

### Ryan Hynd

Extremal functions for Morrey’s inequality in convex domains

A celebrated result in the theory of Sobolev spaces is Morrey's inequality, which establishes the continuous embedding of the continuous functions in certain Sobolev spaces. Interestingly enough the equality case of this inequality has not been thoroughly investigated (unless the underlying domain is R^n). We show that if the underlying domain is a bounded convex domain, then the extremal functions are determined up to a multiplicative factor. We will explain why the assertion is false if convexity is dropped and why convexity is not necessary for this result to hold.

### Tau Shean Lim

Traveling Fronts of Reaction-Diffusion Equations with Ignition Media and Levy Operators

We discuss traveling front solutions u(t,x) = U(x-ct) of reaction-diffusion equations u_t = Lu + f(u) with ignition media f and diffusion operators L generated by symmetric Levy processes X_t. Existence and uniqueness of fronts are well-known in the case of classical diffusion (i.e., Lu = Laplacian(u)) and non-local diffusion (Lu = J*u - u). Our work extends these results to general Levy operators. In particular, we show that a strong diffusivity in the underlying process (in the sense that the first moment of X_1 is infinite) prevents formation of fronts, while a weak diffusivity gives rise to a unique (up to translation) front U and speed c>0.