# 2021-22-old

**Organisers:** Jo Evans, Marie-Therese Wolfram, Jose Rodrigo, Charlie Elliott

**Time & place:** Tuesdays 12-1 in week 2,4,6, 8 and 10 in B3.02

The PDEA seminar will be held in a hybrid format this year - speakers either come in person or join us via MS teams. You can do the same - come in person or watch it on teams ;-)

**Schedule for Term 1:**

- 12/10 Mahir Hadzic (UCL) - Speaker coming in person

Title: Dynamics of the Newtonian stars

Abstract: The starting point for this talk is the classical gravitational Euler-Pousson system describing isolated stars. After giving a brief review of what is mathematically known, I will focus on the question of stellar collapse and the intricately related scaling invariances of the system. I will the present recent works on the existence of self-similar imploding stars, obtained jointly with Yan Guo, Juhi Jang, and Matthew Schrecker.

- 26/10 Megan Griffin-Pickering (Durham) - Speaker coming in person

Title: Global well-posedness for the Vlasov-Poisson system with massless electrons in 3 dimensions

Abstract: The Vlasov-Poisson system is a PDE of kinetic type widely used in plasma physics. The precise structure of the model differs according to whether it describes the electrons or positively charged ions in the plasma, with the classical version of the system modelling the electrons. The Vlasov-Poisson system with massless electrons (VPME) describes instead the evolution of ions in a dilute plasma, interacting with thermalized electrons.

Compared to the electron case, the VPME system includes an additional exponential nonlinearity in the equation for the electrostatic potential, which creates several mathematical difficulties. In particular, while for the electron model the well-posedness theory in 3 dimensions is well established, the theory for ion models has been investigated more recently.

I will present results showing the global-in-time existence and uniqueness of classical solutions for the VPME system in 3 dimensions, generalising the known theory for the electron model to the ion case. This is based on joint work with Mikaela Iacobelli.

- 9/11 Andrew Kei-Fong Lam - Speaker online

- 16/11 Matthew Schrecker - Speaker coming in person

Title: Finite energy methods for compressible fluids with symmetry

Abstract: In this talk, I will survey recent work, partly joint with Gui-Qiang Chen, on the finite energy method for the isentropic Euler equations using the theory of compensated compactness. Developing this method has allowed us to prove the existence of global-in-time admissible solutions to the isentropic Euler equations under certain symmetry assumptions (e.g. spherical symmetry). The low regularity, finite energy framework means that our solutions continue (as weak solutions) even after shock formation or implosion phenomena. The methods used extend to a variety of other settings, such as the convergence of the vanishing viscosity limit from the Navier-Stokes equations (under symmetry) or to the Euler-Poisson equations for self-gravitating fluids.

- 23/11 Isabelle Tristani - Speaker online

Title : Hydrodynamic limit for the inelastic Boltzmann equationAbstract : In this talk, we are interested in the problem of rigorously deriving hydrodynamic equations from the Boltzmann equation for inelastic hard spheres with small inelasticity. One of the main difficulty is to identify the relation between the restitution coefficient (which quantifies the energy loss at the microscopic level) and the Knudsen number that allows us capture nontrivial hydrodynamic behavior. In this (nearly elastic) regime, we prove a result of convergence of the inelastic Boltzmann equation towards some hydrodynamic system which is an incompressible Navier-Stokes-Fourier system with self-consistent forcing terms. This is a joint work with Ricardo Alonso and Bertrand Lods.

- 7/12 Susana Gutierrez - speaker coming in person

**Title: **Self-similar solutions of the 1d Landau-Lifshitz-Gilbert equation.
**Abstract:** The Landau-Lifshitz-Gilbert equation (LLG) is a continuum model describing the dynamics for the spin in ferromagnetic materials. The main objective of this talk is to present an overview of the construction and study of the dynamical behaviour of self-similar solutions for this model in the one-dimensional case. We will consider both self-similar shrinker and expander solutions.