# Difference between revisions of "PDE Geometric Analysis seminar"

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Abstract: We study state-constraint static Hamilton-Jacobi equations in a sequence of domains $\{\Omega_k\}$ in $\mathbb R^n$ such that $\Omega_k \subset \Omega_{k+1}$ for all $k \in \mathbb N$. We obtain rates of convergence of $u_k$, the solution to the state-constraint problem in $\Omega_k$, to $u$, the solution to the corresponding problem in $\Omega=\bigcup_k \Omega_k$. In many cases, the rates obtained are proven to be optimal (it's a joint work with Yeoneung Kim and Hung V. Tran). | Abstract: We study state-constraint static Hamilton-Jacobi equations in a sequence of domains $\{\Omega_k\}$ in $\mathbb R^n$ such that $\Omega_k \subset \Omega_{k+1}$ for all $k \in \mathbb N$. We obtain rates of convergence of $u_k$, the solution to the state-constraint problem in $\Omega_k$, to $u$, the solution to the corresponding problem in $\Omega=\bigcup_k \Omega_k$. In many cases, the rates obtained are proven to be optimal (it's a joint work with Yeoneung Kim and Hung V. Tran). | ||

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+ | ===Jin Woo Jang=== | ||

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+ | Title: On a Cauchy problem for the Landau-Boltzmann equation | ||

+ | |||

+ | Abstract: In this talk, I will introduce a recent development in the global well-posedness of the Landau equation (1936) in a general smooth bounded domain, which has been a long-outstanding open problem. This work proves the global stability of the Landau equation in an $L^\infty_{x,v}$ framework with the Coulombic potential in a general smooth bounded domain with the specular reflection boundary condition for initial perturbations of the Maxwellian equilibrium states. Our methods consist of the generalization of the well-posedness theory for the kinetic Fokker-Planck equation (HJV-2014, HJJ-2018) and the extension of the boundary value problem to a whole space problem, as well as the use of a recent extension of De Giorgi-Nash-Moser theory for the kinetic Fokker-Planck equations (GIMV-2016) and the Morrey estimates (BCM-1996) to further control the velocity derivatives, which ensures the uniqueness. This is a joint work with Y. Guo, H. J. Hwang, and Z. Ouyang. |

## Revision as of 00:52, 2 October 2019

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 2020-Spring 2021

## PDE GA Seminar Schedule Fall 2019-Spring 2020

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

Sep 9 | Scott Smith (UW Madison) | Recent progress on singular, quasi-linear stochastic PDE | Kim and Tran |

Sep 14-15 | AMS Fall Central Sectional Meeting https://www.ams.org/meetings/sectional/2267_program.html | ||

Sep 23 | Son Tu (UW Madison) | State-Constraint static Hamilton-Jacobi equations in nested domains | Kim and Tran |

Sep 28-29, VV901 | https://www.ki-net.umd.edu/content/conf?event_id=993 | Recent progress in analytical aspects of kinetic equations and related fluid models | |

Oct 7 | Jin Woo Jang (Postech) | On a Cauchy problem for the Landau-Boltzmann equation | Kim |

Oct 14 | Stefania Patrizi (UT Austin) | TBA | Tran |

Oct 21 | Claude Bardos (Université Paris Denis Diderot, France) | From d'Alembert paradox to 1984 Kato criteria via 1941 1/3 Kolmogorov law and 1949 Onsager conjecture | Li |

Oct 28 | Albert Ai (UW Madison) | TBA | Ifrim |

Nov 4 | Yunbai Cao (UW Madison) | TBA | Kim and Tran |

Nov 11 | Speaker (Institute) | TBA | Host |

Nov 18 | Speaker (Institute) | TBA | Host |

Nov 25 | Mathew Langford (UT Knoxville) | TBA | Angenent |

Feb 17 | Yannick Sire (JHU) | TBA | Tran |

Feb 24 | Speaker (Institute) | TBA | Host |

March 2 | Theodora Bourni (UT Knoxville) | TBA | Angenent |

March 9 | Ian Tice (CMU) | TBA | Kim |

March 16 | No seminar (spring break) | TBA | Host |

March 23 | Jared Speck (Vanderbilt) | TBA | SCHRECKER |

March 30 | Speaker (Institute) | TBA | Host |

April 6 | Speaker (Institute) | TBA | Host |

April 13 | Speaker (Institute) | TBA | Host |

April 20 | Hyunju Kwon (IAS) | TBA | Kim |

April 27 | Speaker (Institute) | TBA | Host |

## Abstracts

### Scott Smith

Title: Recent progress on singular, quasi-linear stochastic PDE

Abstract: This talk with focus on quasi-linear parabolic equations with an irregular forcing . These equations are ill-posed in the traditional sense of distribution theory. They require flexibility in the notion of solution as well as new a priori bounds. Drawing on the philosophy of rough paths and regularity structures, we develop the analytic part of a small data solution theory. This is joint work with Felix Otto, Hendrik Weber, and Jonas Sauer.

### Son Tu

Title: State-Constraint static Hamilton-Jacobi equations in nested domains

Abstract: We study state-constraint static Hamilton-Jacobi equations in a sequence of domains $\{\Omega_k\}$ in $\mathbb R^n$ such that $\Omega_k \subset \Omega_{k+1}$ for all $k \in \mathbb N$. We obtain rates of convergence of $u_k$, the solution to the state-constraint problem in $\Omega_k$, to $u$, the solution to the corresponding problem in $\Omega=\bigcup_k \Omega_k$. In many cases, the rates obtained are proven to be optimal (it's a joint work with Yeoneung Kim and Hung V. Tran).

### Jin Woo Jang

Title: On a Cauchy problem for the Landau-Boltzmann equation

Abstract: In this talk, I will introduce a recent development in the global well-posedness of the Landau equation (1936) in a general smooth bounded domain, which has been a long-outstanding open problem. This work proves the global stability of the Landau equation in an $L^\infty_{x,v}$ framework with the Coulombic potential in a general smooth bounded domain with the specular reflection boundary condition for initial perturbations of the Maxwellian equilibrium states. Our methods consist of the generalization of the well-posedness theory for the kinetic Fokker-Planck equation (HJV-2014, HJJ-2018) and the extension of the boundary value problem to a whole space problem, as well as the use of a recent extension of De Giorgi-Nash-Moser theory for the kinetic Fokker-Planck equations (GIMV-2016) and the Morrey estimates (BCM-1996) to further control the velocity derivatives, which ensures the uniqueness. This is a joint work with Y. Guo, H. J. Hwang, and Z. Ouyang.