# 2021-22

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

**Time & place:** B3.02 Tuesdays 12-1 (term 1&2) and Thursdays (term 3) in week 2,4,6, 8 and 10

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 ;-)

**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.

**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.

**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.

**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.

**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 to capture nontrivial hydrodynamic behavior. In this (nearly elastic) regime, we prove a result of convergence of the inelastic Boltzmann equation toward 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.

**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.

**Abstract: **Quasi-variational inequalities (QVIs) can be thought of as generalisations of variational inequalities where the constraint set in which the solution is sought depends on the unknown solution itself. In this talk, I'll discuss various aspects of elliptic quasi-variational inequalities of obstacle type including existence results, sensitivity analysis of the source-to-solution map as well as optimal control problems with QVI constraints and associated stationarity systems. The talk will be based on recent joint work with Michael Hintermüller (Berlin) and Carlos N. Rautenberg (Virginia).

**Abstract: **The classical Erdős-Turán inequality on the distribution of roots for complex polynomials can be equivalently stated in a potential theoretic formulation, that is, if the logarithmic potential generated by a probability measure on the unit circle is close to 0, then this probability measure is close to the uniform distribution. We generalize this classical inequality from $d=1$ to higher dimensions $d>1$ with the class of Riesz potentials which includes the logarithmic potential as a special case. In order to quantify how close a probability measure is to the uniform distribution in a general space, we use Wasserstein-infinity distance as a canonical extension of the concept of discrepancy. Then we give a compact description of this distance. Then for every dimension $d$, we prove inequalities bounding the Wasserstein-infinity distance between a probability measure $\rho$ and the uniform distribution by the $L^p$-norm of the Riesz potentials generated by $\rho$. Our inequalities are proven to be sharp up to the constants for singular Riesz potentials. Our results indicate that the phenomenon discovered by Erdős and Turán about polynomials is much more universal than it seems. Finally we apply these inequalities to prove stability theorems for energy minimizers, which provides a complementary perspective on the recent construction of energy minimizers with clustering behavior.

In this talk I will give an overview on the stochastic approach to the study of the evolution by mean curvature flow in the Heisenberg group. I will focus in particular on the asymptotic behaviour of the optimal controls for the p-regularizing problem vs the optimal controls of the limit problem (i.e. the horizontal evolution in the Heisenberg group). Join work with Nicolas Dirr and Raffaele Grande.

**Abstract : **Inverse problems arise whenever a physical quantity has to be reconstructed from indirect measurements. Whenever physics plays a crucial role in the description of the inverse problem, the model is based on a partial differential equation. As such, the model is infinite dimensional, so that, in theory, infinitely many measurements are required to recover the unknown. However, in all practical applications, only finite physical measures may be acquired. In this talk, I will discuss methods based on sampling, compressed sensing, and machine learning that allow us to obtain theoretical results for inverse problems in PDE with finite measures.

**Abstract:**

*Two kinetic models for non-instantaneous binary alignment collisions*, arXiv:2203.15711

**Abstract:** Semilinear wave equations in three spatial dimensions with wave--wave nonlinearities exhibit interesting and well-studied phenomena: from John's famous blow-up examples, to the null condition of Christodoulou and Klainerman, and more recently to the weak null condition of Lindblad and Rodnianski. The study of coupled semilinear wave and Klein-Gordon equations is less well-developed, and interesting problems occur across the possible spectrum of wave--wave, wave--KG and KG--KG interactions. In this talk I will discuss some recent results, in collaboration with Shijie Dong (SUSTech), on such mixed systems. This includes a recent proof of asymptotic stability for a Dirac--Klein-Gordon system in two spatial dimensions.

**Abstract:** We study the Keller-Segel system in the plane with an initial condition with sufficient decay and critical mass 8 pi. We find a function u0 with mass 8 pi such that for any initial condition sufficiently close to u0 and mass 8 pi, the solution is globally defined and blows up in infinite time. This proves the non-radial stability of the infinite-time blow up for some initial conditions, answering a question by Ghoul and Masmoudi (2018). This is joint work with Manuel del Pino (U. of Bath), Jean Dolbeault (U. Paris Dauphine), Monica Musso (U. of Bath) and Juncheng Wei (UBC).

**Abstract:** We use entropy methods to show that the heat equation with Dirichlet boundary conditions on the complement of a compact set in R^d shows a self-similar behaviour much like the usual heat equation on R^d, once we account for the loss of mass due to the boundary. Giving good lower bounds for the fundamental solution on these sets is surprisingly a relatively recent result, and we find some improvements using some advances in logarithmic Sobolev inequalities. This is a work in collaboration with Alejandro Gárriz and Fernando Quirós.