Research journey to Lyon

Thanks to his LMS Scheme 3 grant worth £1200, Baptiste enjoyed a two weeks stay in Lyon from November 10 to 25. Baptiste worked with his collaborator Francesco Fanelli on a new project that aims to explore the question of the inviscid limit for some hydrodynamical equations – as the Euler equation with variable density – in the wake of the important work of Jacob Bedrossian and Nader Masmoudi on Euler equations. This project lies mathematically at the intersection of Baptiste’s line of research on Gevrey regularity and Francesco’s knowledge on hydrodynamical systems.

Baptiste was also invited to the Analysis Seminar of Besançon and gave a talk on weakly hyperbolic systems. The audience was very interested by this subject and raised many interesting questions. It may be the sign that Baptiste will go back to Besançon in the near future!

New team member: Frank Rösler

On 1 October 2018 we were joined by a new team member, Dr Frank Rösler. Frank completed his PhD at Durham University and worked as a Research Assistant in Freiburg (Germany).

He is interested in the spectral theory of non-selfadjoint operators and other operator-theoretic questions in PDE theory. His past projects involved pseudospectra of non-normal Schrödinger Operators and more general resolvent norm estimates of partial differential operators. More recently, he studied problems in Asymptotic Analysis and Homogenisation from an operator-theoretic perspective.

Welcome Frank!

Marie Skłodowska-Curie Fellowship Success

Jonathan Ben-Artzi and Junyong Zhang have been awarded a Marie Skłodowska-Curie Fellowship which will commence on 1 July 2018 for a period of two years. Their project, entitled “Geometric Analysis of Dilute Plasmas” (GRANDPA), will focus on studying regularity theory and long-time behavior of plasmas governed by the Vlasov-Maxwell system. The abstract reads:

“The ultimate goal of this Fellowship is to understand the long time behaviour of plasmas governed by the relativistic Vlasov- Maxwell system (RVM). The main difficulty is the hyperbolic nature of Maxwell’s equations (the electromagnetic fields propagate at the speed of light): particles that travel close to the speed of light nearly interact with their own fields. It is not currently known whether particles can be accelerated to such speeds, and, if so, whether this necessarily leads to development of singularities. This is a major open problem.”

 

The combined expertise of Jonathan and Junyong in kinetic theory and in dispersive equations played a central role in the success of this application. The total value of the award is €195,455.

New team member: Junyong Zhang

This month we welcomed Dr Junyong Zhang as a research associate. He joins us from the Beijing Institute of Technology, where he maintains his affiliation. Junyong is interested in harmonic analysis, spectral analysis and PDEs. Specifically, he studies problems related to the long-time behaviour of nonlinear dispersive equations, as well as Strichartz and restriction estimates. An added complication is that he considers such problems on nontrivial underlying manifolds.

He obtained his PhD in 2011 at the Institute of Applied Physics and Computational Mathematics in Beijing and has since then also spent a year at both the Australian National University and Stanford University.

Welcome Junyong!

New paper: Arbitrarily Large Solutions of the Vlasov-Poisson System

Jonathan Ben-Artzi recently uploaded a new paper entitled Arbitrarily large solutions of the Vlasov-Poisson system, co-authored with Stephen Pankavich and Simone Calogero.

The Vlasov-Poisson system, which models the statistical behavior of many-particle systems, is known to have global-in-time classical solutions (in three dimensions). However, the underlying particle systems (of attractive or repulsive particles) may have singularities appearing in finite time. For instance, attractive particles (stars) can collapse to a single point in finite time. It is therefore interesting to ask how close to a singularity can the Vlasov-Poisson system get?

This has recently been done by Rein & Taegert in the attractive case, however the repulsive case remained open. The main result states that for any constants C_1,C_2>0 there exists initial data with density whose L^\infty norm is initially bounded by C_1 but that at some later time T>0 is greater than C_2 . The main tool is obtaining a priori estimates for particle trajectories and choosing initial data carefully. This data is chosen to be supported on a spherical shell about the origin, with initial velocities pointing inwards.

New team member: Baptiste Morisse

Baptiste Morisse, who recently graduated from Université Paris Diderot (Paris 7), joined the project on 1 September 2017. Baptiste, who was a student of Benjamin Texier, brings expertise in well-posedness theory for first-order quasilinear operators, in particular in the context of Gevrey spaces. Baptiste will be in Cardiff for the next three years.

Welcome Baptiste!