New paper: Weak Poincaré inequalities in absence of spectral gaps

Jonathan Ben-Artzi recently uploaded a new paper entitled Weak Poincaré inequalities in absence of spectral gaps, co-authored with Amit Einav.

For Markov semigroups it is well-known that the following are equivalent:

  • The generator has a spectral gap,
  • The generator satisfies a Poincaré inequality,
  • Solutions decay exponentially

In this paper, they study semigroups which lack a spectral gap (such as the heat semigroup in \mathbb{R}^d ) and try to see how much of the above theorem remains true. They prove that an estimate on the density of the spectrum near 0 leads to a weak Poincaré inequality, which in turn leads to an algebraic decay rate.

This is applied to the heat semigroup, where the optimal decay rate t^{-d/2} is recovered. In this case, the weak Poincaré inequality is no more than the Nash inequality. This is done for the fractional Laplacian as well, with similar results.

 

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!

Postdoc position advertised

We are in the process of hiring another postdoc to join our team (see current members). We are looking for someone with research interests in analysis, and, more precisely, someone who is interested in how to approximate infinite dimensional objects in finite dimensions. A typical example is approximating operators in Hilbert space. If you think this sounds interesting please apply. Of course, you will also be free to continue your own research.

The advert is here.

This will involve joint work with our colleagues Marco Marletta (Cardiff) and Anders Hansen (Cambridge).

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.