Oana Ivanovici




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Research interests and publications

  • Microlocal analysis, nonlinear PDE, hyperbolic equations on manifolds with/without boundary, dispersion, long time behavior of solutions, propagation et reflection of singularities, diffraction phenomenon.

  ORCID ID : 0000-0002-5474-887X


Preprints

    • preprint ; with Fabrice Planchon, Gilles Lebeau and Richard Lascar
    • In this very recent work (started in 2012), we consider the wave equation on a generic strictly convex domain of dimension at least two with smooth non empty boundary and with Dirichlet boundary conditions. We construct a sharp local in time parametrix and then proceed to obtain dispersion estimates: our fixed time decay rate for the Green function exhibits a t^{1/4} loss with respect to the boundary less case. Moreover, we precisely describe where and when these losses occur and relate them to swallowtail type singularities in the wave front set, proving that the resulting decay is optimal.
      The new methods in this paper (compared to the Friedlander model case) provide a much deeper understanding of the complex propagation pattern near the boundary and extends the parametrix construction to the largest possible phase space region. This may have far reaching consequences, beyond pointwise bounds, as the parametrix will prove to be a powerful tool to prove sharp propagation of singularities results which were out of reach until now.

  • Dispersion estimates for wave and Schrödinger equations outside a strictly convex obstacles and counterexamples
    • preprint ; in collaboration with Gilles Lebeau
     
    • Abstract: We consider the linear wave equation and the linear Schrödinger equation
      outside a compact, strictly convex obstacle in R^d with smooth boundary. In dimension
      d = 3 we show that the linear flow satisfies the dispersive estimates as in R3. For
      d >3 and if the obstacle is a ball, we show that there exists points (near the Poisson-Arago
      spot) where the dispersive estimates fail for both wave and Schrödinger quations.

    • The question about whether or not dispersion did hold outside general strictly convex obstacles was raised more than 20 years ago : in this work we give sharp answers which highlight the importance of diffractive effects, especially in higher dimensions where we provide unexpected counterexamples. 
  • Strichartz estimates for the wave equation in strictly convex domains : d=2
    • preprint in collaboration with Gilles Lebeau and Fabrice Planchon
    • Abstract: We prove sharper Strichartz estimates than expected from the
      optimal dispersion bounds. This follows from reducing the analysis to gliding
      wave packets and taking full advantage of the space-time localization of caustics. Several improvements on the parametrix construction are obtained along the way and are of independent interest. Moreover, we extend the range of known counterexamples by propagating carefully constructed Gaussian beams, proving that our Strichartz estimates are sharp, up to logarithmic losses, in some regions of phase space.
  • Strichartz estimates for the wave equation in strictly convex domains : d=3,4
    • in collaboration with Gilles Lebeau and Fabrice Planchon

Publications

    • in collaboration with Gilles Lebeau
    • Abstract: In this paper we prove dispersive estimates for the wave equation outside a ball in R^d. If d = 3, we show that the linear  flow satisfies the dispersive estimates as in the free case. In higher dimensions d >3, we show that losses in dispersion do appear and this happens at the Poisson spot.
    • to appear in Communications and PDE (2017)
    • Abstract: The first purpose of this note is to provide a proof of the usual square function estimate on L^p(M) (which follows from a generic Mikhlin multiplier theorem). We also relate such bounds to a weaker version of the square function estimate which is enough in most instances involving dispersive PDEs and relies on Gaussian bounds on the heat kernel. Moreover, we obtain several useful L^p(M;H) bounds for (the derivatives of) the heat flow with values in a Hilbert space H.
  • Estimations de Strichartz pour l'équation des ondes dans un domaine strictement convexe : le cas général


    • in collaboration with Fabrice Planchon et Gilles Lebeau

    • Publications de la SMF, Séminaires et Congrès, 30 (2017) pages 69-79

    • Abstract: In this note we establish Strichartz estimates with 1/6 loss for the wave equation inside a generic stritcly convex domain of R^3. To do that, we need to focus on the space-time localization of caustics and show that they appear at exceptional times so that, averaging by integration in time, can produce better Strichartz estimates (recall that we have obtained optimal dispersive estimates  with a loss of 1/4).
    • Abstract: In this paper we show that inside a strictly convex model domain there is a loss compared to the dispersive estimates in the free case and this is due to microlocal phenomena such as caustics generated in arbitrarily small time near the boundary. This result is optimal.

    • Journal of the European Math.Soc. (JEMS), volume 14, issue 5 (2012), pages 1357-1388
    • a detailed version  (59 pages) is available  here.
    • Mathematische Annalen vol. 347 issue 3 (2010), pages 627-672
    • Abstract: In these two papers (dealing first with a model case, then with a general convex domain) I showed that the (local in time) Strichartz estimates for the wave equation suffer losses
      when compared to the usual case R^d, at least for a subset of the usual range of admissible
      indices (q; r). This result was striking since the whispering gallery modes, which seemed
      to have the maximum amount of concentration and easily rule out the spectral projectors
      estimate with a bound like in the flat case, do NOT rule out the Strichartz estimates : so
      people felt that the latter should hold for all admissible exponents.
    • Annales de l'IHP Analyse Non-Linéaire vol.27. no.5 (2010)
    • Abstract: In this work we prove a local well-posedness theory for the solution to the quintic nonlinear Schrödinger equation outside non-trapping obstacles, bypassing the absence of suitable
      known Strichartz estimates (back then).

    • Analysis and PDE vol.3 no.3 (2010), pages 261-293
    • Abstract: Outside a strictly convex obstacle, I obtained the full set of Strichartz estimates (except for
      the end-points) for solutions to the Schrödinger equation, an open question since Smith and
      Sogge had obtained similar results for the wave in 1995.
    • Asymptotic Analysis vol. 53 no. 4 (2007), pages 189-208
    • Abstract: We investigate the smoothing effect properties of the solutions of the Schrödinger equation outside one or more balls in R^3.


Proceedings

    • Séminaire Laurent Schwartz (XEDP) (2013-2014)
    • Oberwolfach Reports (2013)

PhD Thesis 

    • Thèse de Doctorat, Université Paris Sud, Faculté des Sciences d'Orsay (2009)
    • The final thesis report is available here.