# Difference between revisions of "PDE Geometric Analysis seminar"

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Abstract: In this talk, I will present some recent results concerning the 1D isentropic Euler equations using the theory of compensated compactness in the framework of finite energy solutions. In particular, I will discuss the convergence of the vanishing viscosity limit of the compressible Navier-Stokes equations to the Euler equations in one space dimension. I will also discuss how the techniques developed for this problem can be applied to the existence theory for the spherically symmetric Euler equations and the transonic nozzle problem. One feature of these three problems is the lack of a priori estimates in the space $L^\infty$, which prevent the application of the standard theory for the 1D Euler equations. | Abstract: In this talk, I will present some recent results concerning the 1D isentropic Euler equations using the theory of compensated compactness in the framework of finite energy solutions. In particular, I will discuss the convergence of the vanishing viscosity limit of the compressible Navier-Stokes equations to the Euler equations in one space dimension. I will also discuss how the techniques developed for this problem can be applied to the existence theory for the spherically symmetric Euler equations and the transonic nozzle problem. One feature of these three problems is the lack of a priori estimates in the space $L^\infty$, which prevent the application of the standard theory for the 1D Euler equations. | ||

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+ | ===Analaura Stingo=== | ||

+ | |||

+ | Title: Global existence of small solutions to a model wave-Klein-Gordon system in 2D | ||

+ | |||

+ | Abstract: This talk deals with the problem of global existence of solutions to a quadratic coupled wave-Klein-Gordon system in space dimension 2, when initial data are small, smooth and mildly decaying at infinity. Some physical models, especially related to general relativity, have shown the importance of studying such systems. At present, most of the existing results concern the 3-dimensional case or that of compactly supported initial data.We content ourselves here with studying the case of a model quadratic quasi-linear non-linearity, that expresses in terms of « null forms » . | ||

+ | Our aim is to obtain some energy estimates on the solution when some Klainerman vector fields are acting on it, and sharp uniform estimates. The former ones are recovered making systematically use of normal forms’ arguments for quasi-linear equations, in their para-differential version. We derive the latter ones by deducing a system of ordinary differential equations from the starting partial differential system, this strategy maying leading us in the future to treat the case of the most general non-linearities. | ||

+ | |||

===Lei Wu=== | ===Lei Wu=== |

## Revision as of 22:38, 22 September 2018

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 2019-Spring 2020

## PDE GA Seminar Schedule Fall 2018-Spring 2019

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

August 31 (FRIDAY), | Julian Lopez-Gomez (Complutense University of Madrid) | The theorem of characterization of the Strong Maximum Principle | Rabinowitz |

September 10, | Hiroyoshi Mitake (University of Tokyo) | On approximation of time-fractional fully nonlinear equations | Tran |

September 12 and September 14, | Gunther Uhlmann (UWash) | TBA | Li |

September 17, | Changyou Wang (Purdue) | Some recent results on mathematical analysis of Ericksen-Leslie System | Tran |

Sep 28, Colloquium | Gautam Iyer (CMU) | Stirring and Mixing | Thiffeault |

October 1, | Matthew Schrecker (UW) | Finite energy methods for the 1D isentropic Euler equations | Kim and Tran |

October 8, | Anna Mazzucato (PSU) | TBA | Li and Kim |

October 15, | Lei Wu (Lehigh) | Hydrodynamic Limits in Kinetic Equations with Boundary Layer Effects | Kim |

October 22, | Annalaura Stingo (UCD) | Global existence of small solutions to a model wave-Klein-Gordon system in 2D | Mihaela Ifrim |

October 29, | Yeon-Eung Kim (UW) | TBA | Kim and Tran |

November 5, | Albert Ai (UC Berkeley) | TBA | Mihaela Ifrim |

December 3, | Trevor Leslie (UW) | TBA | Kim and Tran |

December 10, | ( ) | TBA | |

January 28, | ( ) | TBA | |

Time: TBD, | Jessica Lin (McGill University) | TBA | Tran |

March 4 | Vladimir Sverak (Minnesota) | TBA(Wasow lecture) | Kim |

March 18, | Spring recess (Mar 16-24, 2019) | ||

April 29, | ( ) | TBA |

## Abstracts

### Julian Lopez-Gomez

Title: The theorem of characterization of the Strong Maximum Principle

Abstract: The main goal of this talk is to discuss the classical (well known) versions of the strong maximum principle of Hopf and Oleinik, as well as the generalized maximum principle of Protter and Weinberger. These results serve as steps towards the theorem of characterization of the strong maximum principle of the speaker, Molina-Meyer and Amann, which substantially generalizes a popular result of Berestycki, Nirenberg and Varadhan.

### Hiroyoshi Mitake

Title: On approximation of time-fractional fully nonlinear equations

Abstract: Fractional calculus has been studied extensively these years in wide fields. In this talk, we consider time-fractional fully nonlinear equations. Giga-Namba (2017) recently has established the well-posedness (i.e., existence/uniqueness) of viscosity solutions to this equation. We introduce a natural approximation in terms of elliptic theory and prove the convergence. The talk is based on the joint work with Y. Giga (Univ. of Tokyo) and Q. Liu (Fukuoka Univ.)

### Changyou Wang

Title: Some recent results on mathematical analysis of Ericksen-Leslie System

Abstract: The Ericksen-Leslie system is the governing equation that describes the hydrodynamic evolution of nematic liquid crystal materials, first introduced by J. Ericksen and F. Leslie back in 1960's. It is a coupling system between the underlying fluid velocity field and the macroscopic average orientation field of the nematic liquid crystal molecules. Mathematically, this system couples the Navier-Stokes equation and the harmonic heat flow into the unit sphere. It is very challenging to analyze such a system by establishing the existence, uniqueness, and (partial) regularity of global (weak/large) solutions, with many basic questions to be further exploited. In this talk, I will report some results we obtained from the last few years.

### Matthew Schrecker

Title: Finite energy methods for the 1D isentropic Euler equations

Abstract: In this talk, I will present some recent results concerning the 1D isentropic Euler equations using the theory of compensated compactness in the framework of finite energy solutions. In particular, I will discuss the convergence of the vanishing viscosity limit of the compressible Navier-Stokes equations to the Euler equations in one space dimension. I will also discuss how the techniques developed for this problem can be applied to the existence theory for the spherically symmetric Euler equations and the transonic nozzle problem. One feature of these three problems is the lack of a priori estimates in the space $L^\infty$, which prevent the application of the standard theory for the 1D Euler equations.

### Analaura Stingo

Title: Global existence of small solutions to a model wave-Klein-Gordon system in 2D

Abstract: This talk deals with the problem of global existence of solutions to a quadratic coupled wave-Klein-Gordon system in space dimension 2, when initial data are small, smooth and mildly decaying at infinity. Some physical models, especially related to general relativity, have shown the importance of studying such systems. At present, most of the existing results concern the 3-dimensional case or that of compactly supported initial data.We content ourselves here with studying the case of a model quadratic quasi-linear non-linearity, that expresses in terms of « null forms » . Our aim is to obtain some energy estimates on the solution when some Klainerman vector fields are acting on it, and sharp uniform estimates. The former ones are recovered making systematically use of normal forms’ arguments for quasi-linear equations, in their para-differential version. We derive the latter ones by deducing a system of ordinary differential equations from the starting partial differential system, this strategy maying leading us in the future to treat the case of the most general non-linearities.

### Lei Wu

Title: Hydrodynamic Limits in Kinetic Equations with Boundary Layer Effects

Abstract: Hydrodynamic limits concern the rigorous derivation of fluid equations from kinetic theory. In bounded domains, kinetic boundary corrections (i.e. boundary layers) play a crucial role. In this talk, I will discuss a fresh formulation to characterize the boundary layer with geometric correction, and in particular, its applications in 2D smooth convex domains with in-flow or diffusive boundary conditions. We will focus on some newly developed techniques to justify the asymptotic expansion, e.g. weighted regularity in Milne problems and boundary layer decomposition.