# Nonlinear Partial Differential Equations

## Fall 2023

Date: | September 26, 2023 |

Time: | 3:00pm |

Location: | BLOC 302 |

Speaker: | William M Golding, University of Texas at Austin |

Title: | Strong solutions for the homogeneous Landau equation |

Abstract: | The Landau equation is one of the fundamental models of statistical physics for describing the evolution of plasmas. Nevertheless, the existence of global-in-time smooth solutions remains widely open, even for the homogeneous equation. I will discuss some recent progress in constructing local-in-time strong solutions for large initial data under minimal integrability assumptions. Our method relies crucially on a priori short time smoothing estimates obtained via the De Giorgi method. While for Landau these smoothing estimates require some integrability assumptions on the initial data, these assumptions may be removed for critical perturbations of the Landau equation. Consequently, arbitrarily small critical perturbations of the Landau equation admit global-in-time smooth solutions for generic, possibly large initial data. |

Date: | October 10, 2023 |

Time: | 3:00pm |

Location: | Zoom |

Speaker: | Lucas Huysmans, University of Cambridge |

Title: | Non-uniqueness and inadmissibility of the vanishing viscosity limit of the passive scalar transport equation |

Abstract: | We study the vanishing viscosity/diffusivity limit of the passive scalar transport equation along a given bounded divergence-free vector field on the two-dimensional torus. We construct two such velocity vector fields each exhibiting peculiar behaviour. For the first we demonstrate that along different subsequences of viscosities the limit of transport may converge strongly to different solutions to the inviscid transport. Both of these limits are renormalised solutions to the transport equation, and so equally physically admissible. For the second velocity vector field we prove the uniqueness of the vanishing viscosity limit of viscous transport, however, for any initial data, this unique vanishing viscosity solution is mixed to its spatial average and after a short delay perfectly unmixes to its original state. Therefore the vanishing viscosity limit exhibits a dissipation of energy/entropy and later a reverse of this dissipation. |

Date: | October 17, 2023 |

Time: | 3:00pm |

Location: | Zoom/See link be |

Speaker: | Hussain Ibdah, University of Maryland |

Title: | Bypassing Holder super-criticality barriers in viscous, incompressible fluids |

Abstract: | We will go over the main ideas used in showing that as long as super-critical Holder semi-norms of the classical solution to the incompressible Navier-Stokes system (in any dimension) are under control, the solution remains smooth. The key idea is exploiting an enhanced regularity effect coming from the transport term at the level of a simple one-dimensional drift-diffusion equation, allowing us to break the criticality barrier. We then employ ideas introduced by Kiselev, Nazarov, Volberg, and Shterenberg to propagate this to abstract drift-diffusion equations, providing to our knowledge the very first reasonable extension of the celebrated parabolic regularity result of Nash to an equation that is not in divergence form. Such an approach coupled with subtle pressure estimates due to Silvestre allows us to rigorously treat the incompressible Navier-Stokes as a perturbation of drift-diffusion, obtaining, to our knowledge, the very first genuinely super-critical regularity criterion for this system of equations. Zoom link: https://tamu.zoom.us/j/93413906718?pwd=VDlXYS8zbm9uQ2dMWGowcGVOeGRPdz09 |

Date: | October 24, 2023 |

Time: | 3:00pm |

Location: | BLOC 302 |

Speaker: | Collin Victor, Texas A&M University |

Title: | Leveraging Observational Data to Enhance Continuous Data Assimilation |

Abstract: | In this talk, I will discuss my investigation of a computationally efficient algorithm for data assimilation in recovering solutions to turbulent fluids, specifically the Azouani-Olson-Titi continuous data assimilation algorithm. I will present modifications aimed at enhancing the algorithm's performance while maintaining its relevance to physically realistic use cases. Furthermore, I will demonstrate the application of the algorithm to a range of dissipative partial differential equations, from the 1D Allen-Cahn equation to the 2D and 3D Navier-Stokes equations, and a high-resolution realistic ocean model. A key focus of my research is the role of observational data in convergence rates, examining optimal strategies for gathering observations, and effectively assimilating sparse observational data. Through my analysis, I provide insights into improving turbulent fluid recovery by leveraging the algorithm's potential in various scenarios and identifying the crucial factors that contribute to its success. |

Date: | October 31, 2023 |

Time: | 3:00pm |

Location: | BLOC 302 |

Speaker: | Yongming Li, Texas A&M University |

Title: | Dispersive estimates for 1D matrix Schrödinger operators with threshold resonance |

Abstract: | In this talk, we will discuss dispersive and local decay estimates for a class of matrix Schrödinger operators that naturally arise from the linearization of focusing nonlinear Schrödinger equations around a solitary wave. We review the spectral properties of these linearized operators, and discuss how threshold resonances may appear in their spectrum. In the presence of threshold resonances, it will be shown that the slowdown of the local decay rate can be pinned down to a finite rank operator corresponding to the threshold resonances. Some applications for the linearized equation for the 1D focusing cubic Schrödinger equation will be discussed. |

Date: | December 5, 2023 |

Time: | 3:00pm |

Location: | BLOC 302 |

Speaker: | Maxime van de Mortel, Rutgers University |

Title: | Asymptotic behavior of the Klein-Gordon equation on a Schwarzschild black hole |

Abstract: | It has long been conjectured that the Klein-Gordon equation on a Schwarzschild black hole behaves very differently from the wave equation at late-times, and only decays at a slow t^{-5/6} rate. Despite its apparent simplicity, this conjecture had remained open. We discuss its resolution and our recent result establishing late-time tails at the rate t^{-5/6} for each angular mode. Joint work with Federico Pasqualotto and Yakov Shlapentokh-Rothman. |