My publications are available also here : ORCID , HAL, arXiv.
| Dispersion estimates for the wave and the Schrödinger equations outside a ball and counterexamples, Oana Ivanovici |
| 2024 |
| Abstract : We consider the wave equation with Dirichlet boundary conditions in the exterior of the unit ball Bd(0, 1) of Rd. For d = 3, we obtain a global in time parametrix and derive sharp dispersive estimates, matching the R3 case, for all frequencies (low and high). For d ≥ 4, we provide an explicit solution at large frequency 1/h, h ∈ (0, 1), with a smoothed Dirac data at a point at distance h^{−1/3} from the origin in Rd whose decay rate exhibits h^{−(d−3)/3} loss with respect to the boundary less case, that occurs at observation points around the mirror image of the source with respect to the center of the ball (at the Poisson-Arago spot). Similar counterexample are obtained for the Schrödinger flow. Moreover, we generalize these counterexamples, first announced in a CRAS note in collaboration with G. Lebeau, to the case of the wave and Schrödinger equations outside cylindrical domains of the form Bd1 (0, 1) × Rd2 in Rd with d = d1 + d2 and d1 ≥ 4, for which we construct solutions whose decay rates exhibit a h^{−(d1−3)/3} loss with respect to the boundary less case (at observation points around the mirror image of the source with respect to the origin) |
| New counterexamples to Strichartz estimates for the wave equation on a 2D model convex domain, O. Ivanovici, G. Lebeau and F. Planchon |
| Journal de L’Ecole Polytechnique – Mathématiques, 8:1133-1157 (2021) |
| Abstract : We prove that the range of Strichartz estimates on a model 2D convex domain may be further restricted compared to the known counterexamples. Our new family of counterexamples is built on the parametrix construction from our recent paper “Dispersion for the wave equation inside strictly convex domains”. Interestingly enough, it is sharp in at least some regions of phase space. |
| Dispersive estimates for the wave equation outside a cylinder in R3, F.Iandoli and O. Ivanovici |
| J. Function. Analysis (2024), 286, no.9, 50 pages |
| Abstract : We consider the wave equation with Dirichlet boundary conditions in the exterior of a cylinder in R3 and we construct a sharp global in time parametrix to derive sharp dispersive estimates for all frequencies (low and high) and, as a corollary Strichartz estimates matching the R3 case. |
| Strichartz estimates for the wave equation on a 2D model convex domain, O. Ivanovici, G. Lebeau and F. Planchon |
| Journal of Differential Equations, Vol. 300 November (2021), 830-880 |
| Abstract : We prove sharper Strichartz estimates than expected from the optimal dispersion bounds. This follows from taking full advantage of the space-time localization of caustics. Several improvements on the parametrix construction from our previous works are obtained along theway and are of independent interest. |
| Dispersive estimates inside the Friedlander model domain for the Klein-Gordon equation and the wave equation in large time, Oana Ivanovici |
| Discrete and Continuous Dynamical Systems, December (2021), 41(12), 5707-5742 |
| Abstract : We prove global in time dispersion estimates for the wave and the Klein-Gordon equation inside the Friedlander domain by taking full advantage of the space-time localization of caustics and a precise estimate of the number of waves that may cross at a given, large time. Moreover, we uncover a significant difference between Klein-Gordon and the wave equation in the low frequency, large time regime, where Klein-Gordon exhibits a worse decay that the wave, unlike in the flat space. |
| Dispersive estimates for the Schrödinger equation in a strictly convex domain and applications, Oana Ivanovici |
| Annales de l’IHP Analyse Non-Linéaire, (2023) vol. 40, no. 4, pages 959-1008 |
| Abstract : We consider an anisotropic model case for a strictly convex domain of dimension d>1 with smooth, non-empty boundary and we describe dispersion for the semi-classical Schrödinger equation with Dirichlet boundary condition. More specifically, we obtain the following fixed time decay rate for the semi-classical Schrödinger flow : a loss of 1/4 occurs with respect to the boundary less case due to repeated swallowtail type singularities, and is proven optimal. Corresponding Strichartz estimates allow to solve the cubic nonlinear Schrödinger equation on such a 3D model convex domain, hence matching known results on generic compact boundary less manifolds. |
| Dispersion for the wave equation inside strictly convex domains II, Oana Ivanovici, Richard Lascar, Gilles Lebeau and Fabrice Planchon |
| Annals of PDE 9 (2023), no.2, no. 14, 117 pages |
| Abstract : In this paper, we consider the wave equation on a strictly convex domain Ω of dimension d ≥ 2 with smooth boundary and with Dirichlet boundary conditions. We construct a sharp local in time parametrix and then proceed to obtain dispersion estimates. |
| Dispersion for the wave flow outside a ball in Rd, with Gilles Lebeau |
| Comptes Rendus – Mathématique, 355 (2017), pages 774-779 |
| Abstract : In this paper we prove dispersive estimates for the wave equation outside a ball in Rd. 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. |
| Square function and heat flow estimates on domains, with Fabrice Planchon |
| Communications in PDE (2017), 42(9): 1447-1466 |
| Abstract: The first purpose of this note is to provide a proof of the usual square function estimate on Lp(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 Lp(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, with F. Planchon and G. Lebeau |
| Publications de la SMF, 30 (2017), pages 69-79 |
| Abstract : In this paper we establish Strichartz estimates with 1/6 loss for the wave equation inside a generic strictly convex domain of R3. To do that, we need to focus on the space-time localisation of caustics and show that they appear at exceptional times so that, averaging by integration in time, can produce better Strichartz estimates (recall that dispersion holds with 1/4 loss). |
| Dispersion for the wave equation inside strictly convex domains I : the Friedlander model case, with Gilles Lebeau and Fabrice Planchon |
| Annals of Mathematics, vol.180 issue 1 (2014), pages 323-380 |
| Abstract : We consider a model case for a strictly convex domain Ω ⊂ Rd of dimension d>1 with smooth, nonempty boundary and we describe dispersion for the wave equation with Dirichlet boundary conditions. More specifically, we obtain the optimal fixed time decay rate for the smoothed out Green function: a t1/4 loss occurs with respect to the boundary less case, due to repeated occurrences of swallowtail type singularities in the wave front set. |
| Counter-examples to the Strichartz estimates for the wave equation in general domains with boundary, Oana Ivanovici |
| Journal of the European Math. Soc. (JEMS), vol. 14, issue 5 (2012), pages 1357-1388 |
| Abstract : In this paper we prove shown that the (local in time) Strichartz estimates for the wave equation with Dirichlet boundary condition inside a general convex domain suffer losses when compared to the usual flat case R^d, at least for a subset of the usual range of admissible indices (q,r). |
| Counter-examples to the Strichartz estimates for the wave equation in domains, Oana Ivanovici |
| Mathematische Annalen vol 347 issue 3 (2010), pages 627-672 |
| Abstract : We consider the wave equation with Dirichlet boundary condition inside the Friedlander model domain in dimension d>1 and we prove that the (local in time) Strichartz estimates suffer losses when compared to the usual flat case Rd, at least for a subset of the usual range of admissible indices (q,r). This result was striking as the whispering gallery modes, which seemed to have the maximum amount of concentration and easily rule out the spectral projectors estimates with a bound like in the flat case, do NOT rule out the Strichartz estimates : so one could think that the latter should hold for all admissible exponents. |
| On the energy critical Schrödinger equation in non-trapping domains, Oana Ivanovici and Fabrice Planchon |
| 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. |
| On the Schrödinger equation outside strictly convex domains, Oana Ivanovici |
| Analysis and PDE, vol. 3, no.3 (2010), pages 261-293 |
| Abstract : Outside a strictly convex obstacle, we obtain the full set of Strichartz estimates (except for the end-points) for solutions to the Schrödinger equation, an open question since H.Smith and Ch.Sogge had obtained similar results for the wave equation ( in 1995 ). |
| Precise smoothing effect in the exterior of balls, Oana Ivanovici |
| Asymptotic Analysis vol.53 no. 4 (2007), pages 189-208 |
| Abstract : We investigate the smoothing properties of solutions to the Schrödinger equation outside one or more balls in R3. |
Proceedings
| Dispersive estimates for the wave equation outside a general strictly convex obstacle in R3 |
| Oberwolfach reports (2022) |
| Strichartz estimates for the 3D waves in strictly convex domains |
| Oberwolfach reports (2013) |
| Dispersive and Stricharz estimates for the wave equation in domains with boundary |
| Proceedings of the “Journées EDP” (2010) |
| Dispersive estimates for the wave equation in two dimensional convex domains |
| Oberwolfach reports (2010) |
HDR and PhD Thesis
| HDR |
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| Analyse des effets géométriques sur les équations dispersives |
| Sorbonne Université, 2021 |
| PhD |
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| Dispersive equations and boundary value problems |
| Université Paris XI, Orsay, 2009 |