# Minisymposium on PDE and Mathematical Physics @ ETH

## Tuesday, April 23, 2024, in room HG G 19.2.

## Schedule

Time | Speaker | Title | Abstract |
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9:00-9:30 | Stefano Rossi | On the asymptotic behavior of solutions to the screened Vlasov-Poisson system in the whole space | I will discuss some results regarding the asymptotic behavior of solutions to the screened Vlasov-Poisson system with small initial data in the Euclidean space $\mathbb{R}^d$. The focus will be mainly on dispersive techniques used to obtain sharp decay estimates for the electric field, and how these depend on the dimension. This is part of an ongoing project with M. Iacobelli and K. Widmayer. |

9:30-10:00 | Hyunju Kwon | A glimpse of ideal turbulence | I will discuss solutions to the 3D incompressible Euler equations that reflect phenomena in ideal turbulence, based on recent joint works with Matthew Novack and Vikram Giri. |

10:00-10:30 | Christoph Kehle | The black hole formation threshold | Solutions to the Einstein equations with small initial data disperse to Minkowski space while sufficiently focused initial data can lead to the formation of a black hole. It is a fundamental problem in general relativity to understand how these different classes—dispersing and collapsing—fit together in the moduli space of solutions. I will present recent work on this question which characterizes parts of the interface between collapse and dispersion known as the black hole formation threshold. |

10:30-11:00 | Coffee break |
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11:00-11:30 | Antoine Gagnebin | Landau damping for Vlasov-type systems on the torus | I will talk about Vlasov-type equations on the torus $\mathbb{T}^d$, PDEs that model confined plasma physics when you neglect collisions between particles and external magnetic effects. I will talk about a class of equilibria for such equations and look at the asymptotic stability in time for this kind of equilibrium. I will present the work of Landau (1946) concerning linear evolution and then talk about the asymptotic stability of the dynamic of ions in plasma. |

11:30-12:00 | Ata Deniz Aydin | Quantization of measures on manifolds via covering growth estimates | I will discuss recent advances on the quantization of measures on Riemannian manifolds, joint with M. Iacobelli. The quantization problem concerns the approximation of measures by finitely supported ones with respect to the Wasserstein distance. We characterize the asymptotics of the quantization error in terms of a metric notion for the growth of the manifold, which we call covering growth, and also give estimates on covering growth in terms of other conditions such as lower bounds on Ricci curvature. |

12:00-12:30 | Thomas Stucker | Quasinormal modes for the Kerr black hole | Quasinormal modes (QNMs) or resonances are the complex frequencies describing the characteristic modes of vibration of a dissipative system. We will discuss the importance of QNMs in the study of wave equations and provide a rigorous definition of QNMs for the scalar wave equation on Kerr spacetime. They are obtained as the poles of a certain meromorphic family of operators. If time permits, we will mention two results relating to the distribution of the Kerr QNMs in the complex plane. |

12:45-13:45 | Lunch break |
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14:00-14:30 | Mitchell Taylor | The dynamics of the free boundary Euler equations | Free boundary problems are equations in fluid mechanics where the evolution of the fluid boundary is strongly coupled to that of the flow. Such equations govern the dynamics of water waves, fluid droplets and gaseous stars. In this talk, we will discuss several recent developments in the study of such problems, including how to obtain an optimal local well-posedness theory, a sharp breakdown criterion, and a complete classification of the physical parameters allowing for solitary waves. The material in this talk is based on joint works with Mihaela Ifrim, Ben Pineau and Daniel Tataru. |

14:30-15:00 | Alexandre Rege | On the Bernstein-Landau paradox | We investigate the existence of undamped waves in magnetized collisionless plasmas, a phenomenon first discovered by Bernstein and now known as the Bernstein-Landau paradox. We consider this problem from a novel viewpoint, by reformulating the magnetized Vlasov-Poisson system as a Schrödinger equation with a self-adjoint operator. We show that the spectrum of this operator changes sharply when the magnetic field vanishes. This explains the Bernstein-Landau paradox. |

15:00-15:30 | Coffee break |
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15:30-16:00 | Therese Mörschell | Non-uniqueness of Parabolic solutions for the advection-diffusion equation | For divergence free velocity fields that are square-integrable in time and space, uniqueness of parabolic solutions to the advection diffusion equation (ADE) is guaranteed by the DiPerna-Lions theory. But is this integrability condition sharp? We show that if we lower either integrability in time or in space to some $p<2$ arbitrary but fixed, we obtain non-uniqueness of parabolic solutions for the ADE on the two-dimensional torus. We construct two explicit examples, using a stochastic-Lagrangian approach. This is joint work with Massimo Sorella (EPFL). |