One of my main topic of research is the issue of well-posedness for systems of first-order, quasilinear PDEs. The focus is on properties of the spectrum of the principal symbol of the system – reality of the spectrum, crossing of eigenvalues. In the cases of non-reality of the spectrum, I prove an Hadamard instability in Gevrey spaces, hence pointing out the strong, local ill-posedness of such systems. In some cases of crossing of real eigenvalues, I prove an energy estimate in Gevrey spaces.
In collaboration with Jonathan Ben-Artzi in the QUEST project, I work in kinetic theory. We are interested in many features of Vlasov-Poisson and Vlasov-Maxwell equations: Landau damping, time-decay of plasmas, micro- to mesoscopic limit…
I started recently a collaboration with Francesco Fanelli from Institut Camille Jordan (Lyon) on the wide and interesting topic of Gevrey well-posedness and inviscid limits in Gevrey regularity for different types of hydrodynamic systems, such as active scalar equations or inhomogeneous Euler equations.
You can find my (french) curriculum here.
- Averaging along degenerate flows on the annulus, with Jonathan Ben-Artzi (arxiv), 2019.
- Uniform convergence in von Neumann’s ergodic theorem in absence of a spectral gap, with Jonathan Ben-Arzti (arxiv), 2019.
- On hyperbolicity and Gevrey well-posedness. Part three: a class of weakly hyperbolic systems (arxiv – to appear in Indiana University Mathematics Journal ), 2018.
- On the action of pseudo-differential operators on Gevrey spaces, 2017 (pdf , arxiv).
- On hyperbolicity and Gevrey well-posedness. Part two: scalar or degenerate transitions, 2017 (pdf , arxiv – accepted in Journal of Differential Equations).
- On hyperbolicity and Gevrey well-posedness. Part one: the elliptic case, 2016 (pdf , arxiv).
- 2018 December 14th: SWAP at Bath.
- 2018 November 22nd: Séminaires Equations aux Dérivées Partielles at the Laboratoire de Mathématiques de Besançon
- 2018 April 24: Geometry and Analysis Seminar at QMUL, London.
- 2018 February 21st: Network on Hyperbolic Equations in Loughbourough.
- 2018 February 12th: GAPDE Seminar at Cambridge.
- 2017 November 28th: Seminaire Analyse-EDP de l’Institut Camille Jordan (Lyon, France).
- 2017 November 16th: Pure Analysis and PDEs Seminar at Imperial College (London, UK).
- 2017 October 2nd: Cardiff University Analysis Seminar.
- 2018 November: LMS Scheme 4 Grant “Research in Pairs” for a two weeks visit in Lyon for a collaboration with Francsco Fanelli.
Total amount: £1200.
- 2018 June: LMS Scheme 1 Grant to organize the Conference An Analyst, a Geometer and a Probabilist walk into a bar with Jonathan Ben-Artzi in Cardiff in June 2018.
Total amount: £6000.
- Le problème de Cauchy pour les systèmes quasi-linéaires faiblement hyperboliques ou non-hyperboliques en régularité Gevrey, 2017 (pdf).