# Nonlinear Partial Differential Equations

## Fall 2022

Date: | August 30, 2022 |

Time: | 09:00am |

Location: | Zoom |

Speaker: | Tak Kwong Wong, University of Hong Kong |

Title: | Regularity structure, global-in-time well-posedness and long-time behavior of energy conservative solutions to the Hunter-Saxton equation |

Abstract: | The Hunter-Saxton equation is an integrable equation in one spatial dimension, and can be used to study the nonlinear instability in the director field of a nematic liquid. In this talk, we will discuss the regularity structure, global-in-time well-posedness and long-time behavior of energy conservative solutions to the Hunter-Saxton equation. In particular, singularities for the energy measure may only appear at at most countably many times, and are completely determined by the absolutely continuous part of initial energy measure. The temporal and spatial locations of singularities are explicitly determined by the initial data as well. The long-time behavior of energy conservative solution is given by a kink-wave that is determined by the total energy of the system only. The analysis is based on using the method of characteristics in a generalized framework that consists of the evolutions of energy conservative solution and its energy measure. This is a joint work with Yu Gao and Hao Liu. |

Date: | September 13, 2022 |

Time: | 3:00pm |

Location: | BLOC 302 |

Speaker: | Edriss S. Titi, Texas A&M University and University of Cambridge |

Title: | Mathematical Analysis of Atmospheric and Oceanic Dynamics Models: Cloud Formation and Sea-ice Models |

Abstract: | In this talk we will present rigorous analytical results concerning global regularity, in the viscous case, and finite-time singularity, in the inviscid case, for oceanic and atmospheric dynamics models. Moreover, we will also provide a rigorous justification of the derivation of the Primitive Equations of planetary scale oceanic dynamics from the three-dimensional Navier-Stokes equations as the vanishing limit of the small aspect ratio of the depth to horizontal width. In addition, we will also show the global well-posedeness of the coupled three-dimensional viscous Primitive Equations with a micro-physics phase change moisture model for cloud formation. Eventually, we will also present short-time well-posedness of solutions to the Hibler’s sea-ice model. |

Date: | November 29, 2022 |

Time: | 3:00pm |

Location: | BLOC 302 |

Speaker: | Mary Vaughan, University of Texas at Austin |

Title: | Regularity theory for fractional elliptic equations in nondivergence form |

Abstract: | In this talk, we will define fractional powers of nondivergence form elliptic operators in bounded domains under minimal regularity assumptions and highlight several applications. We will characterize a Poisson problem driven by such operators with a degenerate/singular extension problem. We then develop the method of sliding paraboloids in the Monge–Ampère geometry to prove Harnack inequality and Hölder regularity for classical solutions to the extension equation. This in turn implies Harnack inequality and Hölder regularity for solutions to the fractional Poisson problem. This work is joint with Pablo Raúl Stinga (Iowa State University). |