# Difference between revisions of "PDE Geometric Analysis seminar"

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===Serguei Denissov=== | ===Serguei Denissov=== | ||

We consider the patch evolution under the 2D Euler dynamics and study how the geometry of the boundary can deteriorate in time. | We consider the patch evolution under the 2D Euler dynamics and study how the geometry of the boundary can deteriorate in time. | ||

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+ | ===Bing Wang=== | ||

+ | We show that the mean curvature blows up at the first finite singular time for a closed smooth embedded mean curvature flow in R3. This is a joint work with H.Z. Li. | ||

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===Andrei Tarfulea=== | ===Andrei Tarfulea=== | ||

We consider a model for three-dimensional fluid flow on the torus that also keeps track of the local temperature. The momentum equation is the same as for Navier-Stokes, however the kinematic viscosity grows as a function of the local temperature. The temperature is, in turn, fed by the local dissipation of kinetic energy. Intuitively, this leads to a mechanism whereby turbulent regions increase their local viscosity and | We consider a model for three-dimensional fluid flow on the torus that also keeps track of the local temperature. The momentum equation is the same as for Navier-Stokes, however the kinematic viscosity grows as a function of the local temperature. The temperature is, in turn, fed by the local dissipation of kinetic energy. Intuitively, this leads to a mechanism whereby turbulent regions increase their local viscosity and | ||

dissipate faster. We prove a strong a priori bound (that would fall within the Ladyzhenskaya-Prodi-Serrin criterion for ordinary Navier-Stokes) on the thermally weighted enstrophy for classical solutions to the coupled system. | dissipate faster. We prove a strong a priori bound (that would fall within the Ladyzhenskaya-Prodi-Serrin criterion for ordinary Navier-Stokes) on the thermally weighted enstrophy for classical solutions to the coupled system. | ||

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## Revision as of 19:36, 11 February 2017

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 Fall 2017

# PDE GA Seminar Schedule Spring 2017

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

January 23 Special time and location: 3-3:50pm, B325 Van Vleck |
Sigurd Angenent (UW) | Ancient convex solutions to Mean Curvature Flow | Kim & Tran |

January 30 | Serguei Denissov (UW) | Instability in 2D Euler equation of incompressible inviscid fluid | Kim & Tran |

February 6 | Benoit Perthame (University of Paris VI) | Wasow lecture | |

February 13 | Bing Wang (UW) | The extension problem of the mean curvature flow | Kim & Tran |

February 20 | Eric Baer (UW) | ||

February 27 | Ben Seeger (University of Chicago) | Tran | |

March 7 - Applied math/PDE/Analysis seminar | Roger Temam (Indiana University) | Mathematics Department Distinguished Lecture | |

March 8 - Applied math/PDE/Analysis seminar | Roger Temam (Indiana University) | Mathematics Department Distinguished Lecture | |

March 13 | Sona Akopian (UT-Austin) | Kim | |

March 27 - Analysis/PDE seminar | Sylvia Serfaty (Courant) | Tran | |

March 29 | Sylvia Serfaty (Courant) | Wasow lecture | |

April 3 | Zhenfu Wang (Maryland) | Kim | |

April 10 | Andrei Tarfulea (Chicago) | Improved estimates for thermal fluid equations | Baer |

April 24 | Chris Henderson (Chicago) | TBA | Lin |

May 1st | Jeffrey Streets (UC-Irvine) | Bing Wang |

# Abstracts

### Sigurd Angenent

The Huisken-Hamilton-Gage theorem on compact convex solutions to MCF shows that in forward time all solutions do the same thing, namely, they shrink to a point and become round as they do so. Even though MCF is ill-posed in backward time there do exist solutions that are defined for all t<0 , and one can try to classify all such “Ancient Solutions.” In doing so one finds that there is interesting dynamics associated to ancient solutions. I will discuss what is currently known about these solutions. Some of the talk is based on joint work with Sesum and Daskalopoulos.

### Serguei Denissov

We consider the patch evolution under the 2D Euler dynamics and study how the geometry of the boundary can deteriorate in time.

### Bing Wang

We show that the mean curvature blows up at the first finite singular time for a closed smooth embedded mean curvature flow in R3. This is a joint work with H.Z. Li.

### Andrei Tarfulea

We consider a model for three-dimensional fluid flow on the torus that also keeps track of the local temperature. The momentum equation is the same as for Navier-Stokes, however the kinematic viscosity grows as a function of the local temperature. The temperature is, in turn, fed by the local dissipation of kinetic energy. Intuitively, this leads to a mechanism whereby turbulent regions increase their local viscosity and dissipate faster. We prove a strong a priori bound (that would fall within the Ladyzhenskaya-Prodi-Serrin criterion for ordinary Navier-Stokes) on the thermally weighted enstrophy for classical solutions to the coupled system.