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Oana Ivanovici

Publications and source records attributed to Oana Ivanovici.

17 recordsLinked to original sources
arXiv2025

Strichartz and dispersive estimates for quantum bouncing ball model: exponential sums and Van der Corput methods in 1d semi-classical Schrödinger equations

We analyze the one-dimensional semi-classical Schrödinger equation on the half-line with a linear potential and Dirichlet boundary conditions. Our main focus is on establishing improved dispersive and Strichartz estimates for this model, which govern the space-time behavior of solutions. We prove refined Strichartz bounds using Van der Corput-type derivative tests, beating previous known results where Strichartz estimates incur 1/4 losses. Moreover, assuming sharp bounds for certain exponential sums, our results indicate the possibility to reduce these losses further to $1/6 + ε$ for all $ε>0$, which would be sharp. We further expect that analogous Strichartz bounds should hold within the Friedlander model domain in higher dimensions.

math.AP↗
arXiv2024

Dispersion for the wave and Schrödinger equations outside a ball and counterexamples

We consider the wave equation with Dirichlet boundary conditions in the exterior of the unit ball $B_{d}(0,1)$ of $\mathbb{R}^d$. For $d=3$, we obtain a global in time parametrix and derive sharp dispersive estimates, matching the $\mathbb{R}^{3}$ case, for all frequencies (low and high). For $d\geq 4$, we provide an explicit solution at large frequency $1/h$, $h\in (0,1)$, with a smoothed Dirac data at a point at distance $h^{-1/3}$ from the origin in $\mathbb{R}^d$ 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 Schr{ö}dinger flow. Moreover, we generalize these counterexamples, first announced in \cite{ildispext}, to the case of the wave and Schr{ö}dinger equations outside cylindrical domains of the form $B\_{d\_1}(0,1)\times \mathbb{R}^{d\_2}$ in $\mathbb{R}^d$ with $d=d\_1+d\_2$ and $d\_1\geq 4$, for which we construct solutions, as done \cite{IaIv23} for $d\_1=2$, $d\_2=1$, whose decay rates exhibit a $h^{-(d\_1-3)/3}$ loss with respect to the boundary less case (at observation points around the mirror image of the source with respect to the origin).

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arXiv2023

Dispersion for the wave equation inside strictly convex domains II: the general case

We consider the wave equation on a manifold $(Ω,g)$ of dimension $d\geq 2$ with smooth strictly convex boundary $\partialΩ\neq\emptyset$, 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. We precisely describe where and when these losses occur and relate them to swallowtail type singularities in the wave front set, proving that our decay is optimal. Moreover, we derive better than expected Strichartz estimates, balancing lossy long time estimates at a given incidence with short time ones with no loss: for $d=3$, it heuristically means that, on average the decay loss is only $t^{1/6}$.

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arXiv2021

Strichartz estimates for the wave equation on a 2d model convex domain

We prove better Strichartz type estimates than expected from the (optimal) dispersion we obtained in our earlier work on a 2d convex model. This follows from taking full advantage of the space-time localization of caustics in the parametrix we obtain, despite their number increasing like the inverse square root of the distance from the source to the boundary. As a consequence, we improve known Strichartz estimates for the wave equation. Several improvements on our previous parametrix construction are obtained along the way and are of independent interest for further applications.

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arXiv2021

Dispersive estimates for the Schr{ö}dinger equation in a strictly convex domain and applications

We consider an anisotropic model case for a strictly convex domain of dimension $d\geq 2$ with smoothboundary and we describe dispersion forthe semi-classical Schr{ö}dinger equation with Dirichlet boundary condition. More specifically, we obtain the following fixed time decay rate for the linear semi-classical flow : a loss of $(\frac ht)^{1/4}$ 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 Schödinger equation on such a 3D model convex domain, hence matching known results on generic compact boundaryless manifolds.

math.AP↗
arXiv2020

Dispersive estimates for the wave and the Klein-Gordon equations in large time inside the Friedlander domain

We prove global in time dispersion 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.

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arXiv2017

Dispersion for the wave equation outside a ball and counterexamples

The purpose of this note is to 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 R^3. In higher dimensions d $\ge$ 4 we show that losses in dispersion do appear and this happens at the Poisson spot.

math.AP↗
arXiv2013

Dispersion for the wave equation inside strictly convex domains I: the Friedlander model case

We consider a model case for a strictly convex domain of dimension $d\geq 2$ with smooth 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 $t^{1/4}$ loss occurs with respect to the boundary less case, due to repeated occurrences of swallowtail type singularities in the wave front set.

math.AP↗
arXiv2010

Counter-examples to the Strichartz estimates for the wave equation in domains II

We consider a smooth and bounded domain of dimension d>1 and we construct solutions to the wave equation with Dirichlet boundary conditions which contradict the Strichartz estimates of the free space, at least for a subset of the usual range of indices. This is due to micro-local phenomena such as caustics generated in arbitrarily small time near the boundary.

math.AP↗
arXiv2009

On the Schrodinger equation outside strictly convex obstacles

We prove sharp Strichartz estimates for the semi-classical Schrodinger equation on a compact manifold with smooth, strictly geodesically concave boundary. We deduce sharp (classical) Strichartz estimates for the Schrodinger equation outside a strictly convex obstacle, local existence for the H^1-critical (quintic) Schrodinger equation and scattering for the sub-critical Schrodinger equation in 3D.

math.AP↗
arXiv2009

On the energy critical Schrodinger equation in 3D non-trapping domains

We prove that the quintic Schrodinger equation with Dirichlet boundary conditions is locally well posed for H^{1}_{0} data on any smooth, non-trapping domain of R^3. The key ingredient is a smoothing effect in L^{5}_{x}L^{2}_{t} for the linear equation. We also derive scattering results for the whole range of defocusing sub-quintic Schrodinger equations outside star-shaped domains.

math.AP↗
arXiv2009

Counter example to Strichartz estimates for the wave equation in domains

Let U be a bounded, regular, strictly convex domain of R^2 and consider the wave equation on U with Dirichlet boundary condition. We prove that in such a domain the Strichartz estimates for the wave equation suffer losses when compared to the case U=R^2, at least for a subset of the usual range of indices.

math.AP↗
arXiv2009

Square function and heat flow estimates on domains

The first purpose of this note is to provide a proof of the usual square function estimate on Lp (?). It turns out to follow directly from a generic Mikhlin multiplier theorem obtained by Alexopoulos, which mostly relies on Gaussian bounds on the heat kernel. We also provide a simple proof of a weaker version of the square function estimate, which is enough in most instances involving dispersive PDEs. Moreover, we obtain, by a relatively simple integration by parts, several useful Lp (?; H) bounds for the derivatives of the heat ?ow with values in a given Hilbert space H.

math.AP↗
arXiv2009

Precise smoothing effect in the exterior of balls

We are interested in this article in investigating the smoothing effect properties of the solutions of the Schrodinger equation. We deduce global well-posedness results for the cubic Schrodinger equation in the exterior of several convex obstacles of R^{3}.

math.AP↗