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

Title: Structural optimization with anisotropy: Application to 3D printing.

Abstract: 3D printing is an umbrella term for a set of technologies that manufacture highly intricate and complex designs not feasible with traditional die-casting or injection molding methods. But despite their popularization in recent years, several limitations prevent further integration of 3D printing into existing production lines. One recurring issue relates to overhangs, which are regions of the constructed object that when placed in a certain orientation extend outwards without any underlying support. Some of these overhangs can deform under their own weight and, if not supported from below, present a risk in damaging the printed object.

Conventional wisdom from practitioners says that overhangs whose outer normal makes an angle greater than 135 degrees with the upwards vertical direction should be supported from below with scaffolding. These are then removed after a successful print, but increase the material and processing costs. Another remedy is to modify the design to be self-supporting as much as possible without compromising its intended functionality. In this talk we consider the latter within a structural topology optimization framework. Extending previous studies with a linear elasticity model, we realize an overhang angle constraint with the help of anisotropic perimeter functionals, and study the corresponding optimal control problem. Numerical examples are provided to demonstrate how we discourage designs that develop overhangs not respecting the angle constraint. It turns out that for our approach we have to work with non-differentiable functionals, and thus we turn to subdifferential calculus to derive the first order optimality conditions. This is a joint work with Harald Garcke (Regensburg), Robert Nurnberg (Trento) and Andrea Signori (Pavia).

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 equation

Abstract : 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.