Homepage of Frédéric Hérau

 
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Research interests

Partial differential equations, Kinetic theory and Mathematical physics

Semiclassical analysis, pseudospectral estimates, spectrum, tunneling effect, Fokker-Planck operators, Witten Laplacian, magnetic tunneling effect, magnetic Schroedinger operator

Hypocoercivity, hypoellipticity, trend to the equilibrium for kinetic equations, linearized kinetic equations, Fokker-Planck, Vlasov-Poisson-Fokker-Planck, linear relaxation, Boltzmann, Landau, ...

Transparent conditions, Dirichlet to Neumann operators, Ventcel's type conditions appearing in mechanics.

Semi-bounded pseudo-differential operators, Gaarding inequalities, Fefferman-Phong inequality, SAK principle, paradifferential operators


Prepublications

  1. Semiclassical asymptotics of the Bloch-Torrey operator in two dimensions, with David Kreijcirik and Nicolas Raymond
    preprint 30 p. (February 2024). Pdf


  2. Spectral asymptotics of the Neumann Laplacian with variable magnetic fields on a smooth bounded domain in three dimension, with Maha AAfarani, Khaled Abou Alfa and Nicolas Raymond
    preprint 21 p. (June 2023). Pdf


  3. Quantitative Geometric Control in Linear Kinetic Theory, with Helge Dietert, Harsha Hutridurga, Clément Mouhot
    preprint 34 p. (September 2022). Pdf



Publications

  1. Trajectorial hypocoercivity and application to control theory, with Helge Dietert, Harsha Hutridurga, Clément Mouhot
    Seminaire Ihés, exposé no VIII, EDP appl. 10 p. (2022). Pdf


  2. Special macroscopic modes and hypocoercivity, with Kleber Carrapatoso, Jean Dolbeault, Stéphane Mischler, Clément Mouhot and Christian Schmeiser
    To appear in JEMS 65 p. (2024). Pdf


  3. Semiclassical spectral gaps of the 3D Neumann Laplacian with constant magnetic field, with Nicolas Raymond
    to appear in Annales de l'Institut Fourier 51 p. (2022). Pdf


  4. Weighted Korn and Poincaré-Korn inequalities in the Euclidean space and associated operators, with Kleber Carrapatoso, Jean Dolbeault, Stéphane Mischler and Clément Mouhot
    Archive for Rational Mechanics and Analysis volume 243, 1565-1596, (2022). Pdf


  5. Purely magnetic tunneling effect in two dimensions, with V. Bonnaillie-Noël and N. Raymond
    Inventionnes Matematicae, 227, no 2, 745-793 (2022). Pdf


  6. Regularization estimates and Cauchy theory for inhomogeneous Boltzmann equation for hard potentials without cut-off, with D. Tonon and I. Tristani
    Comm. in Math. Physics, 377 (1), 697-771, (2020). Pdf


  7. Coercivity, hypocoercivity, exponential time decay and simulations for discrete fokker-Planck equations, with G. Dujardin and P. Lafitte
    Numerische Mathematik, volume 144, 615-697, (2020). Pdf


  8. Global hypoelliptic and symbolic estimates for the linearized Boltzmann operator without angular cutoff, with R. Alexandre and W.-X. Li
    Journal de Mathématiques Pures et Appliquées Volume 126, 1-71, (2019). Pdf


  9. Holomorphic extensions of the De Gennes function, with V. Bonnaillie-Noël and N. Raymond
    Annales Mathématiques Blaise Pascal 24, no 2, 225-234, (2018). Pdf


  10. Introduction to hypocoercive methods and applications for simple linear inhomogeneous kinetic models,
    Lectures on the Analysis of Nonlinear Partial Differential Equations Vol. 5, MLM5, 119-147 (2017). Pdf


  11. Curvature induced magnetic bound states: towards the tunneling effect for the ellipse, with V. Bonnaillie-Noël and N. Raymond
    Procedings of "journées EDP" at Roscoff 15 p. (2017). Pdf


  12. On global existence and trend to the equilibrium for the Vlasov-Poisson-Fokker-Planck system with exterior confining potential, with L. Thomann
    J. Funct. Anal 271, no.5 pp 1301-1340 (2016). Pdf


  13. Semiclassical tunneling and magnetic flux effects on the circle, with V. Bonnaillie-Noël and N. Raymond
    Journal of Spectral Theory EMS, 7 (3), pp.771-796 (2017). Pdf


  14. Magnetic WKB constructions, with V. Bonnaillie-Noël and N. Raymond
    Arch. Ration. Mech. Anal. 221, no.2 817-891 (2016). Pdf


  15. Tunnel effect for semiclassical random walk, with J.-F. Bony and L. Michel
    Anal. PDE 8, no. 2 pp 289-332 (2015). Pdf


  16. Artifical conditions for the linear elasticity equations, with V. Bonnaillie-Noël, M.Dambrine and G. Vial
    Math of Comp. 84 pp.1599-1632 (2015). Pdf


  17. Global hypoelliptic estimates for Landau-type operators with external potential, with W.-X. Li
    Kyoto J. Math. 53, no. 3 pp. 533-565. (2013). Pdf


  18. Supersymmetric structures for second order differential operators, with J. Sjöstrand and M. Hitrik
    Algebra i analiz vol25 n2 pp 125-154 (2013). Pdf


  19. Tunnel effect and symmetries for Kramers Fokker-Planck type operators, with J. Sjöstrand and M. Hitrik
    Journal of the Inst. of Math. Jussieu 10(3) pp. 567--634 (2011). Pdf


  20. Multiscale expansion and numerical approximation for surface defects, with V. Bonnaillie-Noël, D. Brancherie, M. Dambrine, S. Tordeux and G. Vial
    proceedings of CANUM 2010 pp. 22-35 (2011). Pdf


  21. Hypoelliptic estimates for some linear diffusive kinetic equations,
    Proceedings of the Journees EADP, Port d'albret 13 p. (2010). Pdf


  22. Anisotropic hypoelliptic estimates for Landau-type operators, with K. Pravda-Starov
    Journal de mathématiques pures et appliquées 95, pp. 513-552 (2011). Pdf


  23. On generalized Ventcel's type boundary conditions for Laplace operator in a bounded domain, with V. Bonnaillie-Noël, M.Dambrine and G. Vial
    SIAM J. Math. Anal. 42 (2) pp.931-945. (2010). Pdf


  24. Tunnel effect for the Kramers-Fokker-Planck type operators: return to equilibrium and applications, with J. Sjöstrand and M. Hitrik
    Int. Math. Res. Not. 2008, Article ID rnn057 48 p. (2008). Pdf


  25. Tunnel effect for the Kramers-Fokker-Planck type operators, with J. Sjöstrand et M. Hitrik
    Ann. Henri Poincaré 9, 2 pp. 209--274 (2008). Pdf


  26. Hypocoercivity and exponential time decay for the linear inhomogeneous relaxation Boltzmann equation,
    Asymptot. Anal. 46, 3-4 pp. 349--359 (2006). Pdf


  27. Short and long time behavior of the Fokker-Planck equation in a confining potential and applications,
    J. Funct. Anal., 244, 1 pp. 95--118 (2007). Pdf


  28. Semiclassical analysis for the Kramers-Fokker-Planck equation, with J.Sjostrand et C. Stolk
    Comm. Partial Differential Equations 30, 4-6 pp. 689--760 (2005). Pdf


  29. Isotropic hypoellipticity and trend to the equilibrium for the Fokker-Planck equation with high degree potential, with F. Nier
    Arch. Rat. Mech. An. 171, 2 pp. 151-218 (2004). Pdf


  30. Isotropic hypoellipticity and trend to the equilibrium for the Fokker-Planck equation with high degree potential (announced),
    Actes colloque Forges-les-Eaux 13 p. (2002). Pdf


  31. Melin inequality for paradifferential operators and applications,
    Comm. Partial Differential Equations 39, 7-8 pp. 1659-1680 (2002). Pdf


  32. Melin-Hörmander inequality in a Wiener type pseudo-differential algebra,
    Ark. För Mat. 39, 2 pp. 311-338 (2001). Pdf


  33. Fefferman's SAK principle in one dimension,
    Annales de l'Institut Fourier 50, 4 pp. 1229-1264 (2000). Pdf


  34. Fonction maximale associée à des symboles d'opérateurs pseudodifférentiels semi-bornés,
    C. R. Acad. Sci. 330, I pp. 449-454 (2000). Pdf


  35. Une inégalité de Garding à bord,
    Proceedings of the EADP conference in La Chapelle sur Erdre Exp. 5 12 p., Nantes (2000). Pdf





Other Works

  1. Short time diffusion properties of inhomogeneous kinetic equations with fractional collision kernel, with D. Tonon and I. Tristani
    Unpublished, 49 p. (April 2018). Pdf


  2. Methodes microlocales pour les equations cinetiques,
    Habilitation a diriger des recherches, Reims 89 p. (2007). Pdf


  3. Opérateurs pseudo-différentiels semi-bornés,
    Thèse de l'Universite de Rennes 1 137 p. (1999). Pdf