Searcharxiv⌕ Search

arXiv subjects

Ioan Bejenaru

Publications and source records attributed to Ioan Bejenaru.

At least 19 recordsLinked to original sources
arXiv2026

A new resolution space for nonlinear Schrödinger equations and applications

Resolution spaces play a central role in constructing solutions for nonlinear partial differential equations. One of the main goals in the area of nonlinear dispersive PDEs has been to construct effective resolution spaces which capture the known bilinear restrictions estimates for free solutions. In this paper we propose a new structure for the Schrödinger equation which effectively replicates the classical bilinear $L^2_{t,x}$ estimate. In addition, the new structure has the property that its "dual" is an effective candidate for a space for the forcing in the linear inhomogeneous Schrödinger equation, a feature that has been elusive so far in the literature. As an application, we show how these structures can recover the known global well-posedness results for derivative NLS with null structure, with Schrödinger Maps being one such model.

math.AP↗
arXiv2024

Near soliton evolution for $2$-equivariant Schrödinger Maps in two space dimensions

We consider equivariant solutions for the Schrödinger Map equation in $2+1$ dimensions, with values into $\mathbb{S}^2$. Within each equivariance class $m \in \mathbb{Z}$ this admits a lowest energy nontrivial steady state $Q^m$, which extends to a two dimensional family of steady states by scaling and rotation. If $|m| \geq 3$ then these ground states are known to be stable in the energy space $\dot H^1$, whereas instability and even finite time blow-up along the ground state family may occur if $|m| = 1$. In this article we consider the most delicate case $|m| = 2$. Our main result asserts that small $\dot H^1$ perturbations of the ground state $Q^2$ yield global in time solutions, which satisfy global dispersive bounds. Unlike the higher equivariance classes, here we expect solutions to move arbitrarily far along the soliton family; however, we are able to provide a time dependent bound on the growth of the scale modulation parameter. We also show that within the equivariant class the ground state is stable in a slightly stronger topology $X \subset \dot H^1$.

math.AP↗
arXiv2019

The multilinear restriction estimate: almost optimality and localization

The first result in this paper provides a very general $ε$-removal argument for the multilinear restriction estimate. The second result provides a refinement of the multilinear restriction estimate in the case when some terms have appropriate localization properties; this generalizes a prior result of the author.

math.CA↗
arXiv2016

Optimal multilinear restriction estimates for a class of surfaces with curvature

Bennett, Carbery and Tao considered the $k$-linear restriction estimate in $\mathbb{R}^{n+1}$ and established the near optimal $L^\frac2{k-1}$ estimate under transversality assumptions only. We have shown that the trilinear restriction estimate improves its range of exponents under some curvature assumptions. In this paper we establish almost sharp multilinear estimates for a class of hypersurfaces with curvature for $4 \leq k \leq n$. Together with previous results in the literature, this shows that curvature improves the range of exponents in the multilinear restriction estimate at all levels of lower multilinearity, that is when $k \leq n$.

math.CA↗
arXiv2016

The cubic Dirac equation: Small initial data in $H^{\frac12}(\mathbb{R}^2)$

Global well-posedness and scattering for the cubic Dirac equation with small initial data in the critical space $H^{\frac12}(\mathbb{R}^2)$ is established. The proof is based on a sharp endpoint Strichartz estimate for the Klein-Gordon equation in dimension $n=2$, which is captured by constructing an adapted systems of coordinate frames.

math.AP↗
arXiv2016

The optimal trilinear restriction estimate for a class of hypersurfaces with curvature

Bennett, Carbery and Tao established nearly optimal $L^1$ trilinear restriction estimates in $\mathbb{R}^{n+1}$ under transversality assumptions only. In this paper we show that the curvature improves the range of exponents, by establishing $L^p$ estimates, for any $p > \frac{2(n+4)}{3(n+2)}$ in the case of double-conic surfaces. The exponent $\frac{2(n+4)}{3(n+2)}$ is shown to be the universal threshold for the trilinear estimate.

math.CA↗
arXiv2016

Optimal bilinear restriction estimates for general hypersurfaces and the role of the shape operator

It is known that under some transversality and curvature assumptions on the hypersurfaces involved, the bilinear restriction estimate holds true with better exponents than what would trivially follow from the corresponding linear estimates. This subject was extensively studied for conic and parabolic surfaces with sharp results proved by Wolff and Tao, and with later generalizations by Lee. In this paper we provide a unified theory for general hypersurfaces and clarify the role of curvature in this problem, by making statements in terms of the shape operators of the hypersurfaces involved.

math.CA↗
arXiv2016

The multilinear restriction estimate: a short proof and a refinement

We provide an alternative and self contained proof of the main result of Bennett, Carbery, Tao regarding the multilinear restriction estimate. The approach is inspired by the recent result of Guth about the Kakeya version of multilinear restriction estimate. At lower levels of multilinearity we provide a refined estimate in the context of small support for one of the terms involved.

math.CA↗
arXiv2015

Well-posedness and scattering for the Zakharov system in four dimensions

The Cauchy problem for the Zakharov system in four dimensions is considered. Some new well-posedness results are obtained. For small initial data, global well-posedness and scattering results are proved, including the case of initial data in the energy space. None of these results is restricted to radially symmetric data.

math.AP↗
arXiv2014

The cubic Dirac equation: Small initial data in $H^1(\mathbb{R}^3)$

We establish global well-posedness and scattering for the cubic Dirac equation for small data in the critical space $H^1(\mathbb{R}^3)$. The main ingredient is obtaining a sharp end-point Strichartz estimate for the Klein-Gordon equation. In a classical sense this fails and it is related to the failure of the endpoint Strichartz estimate for the wave equation in space dimension three. In this paper, systems of coordinate frames are constructed in which endpoint Strichartz estimates are recovered and energy estimates are established.

math.AP↗
arXiv2010

Convolutions of singular measures and applications to the Zakharov system

Uniform L^2-estimates for the convolution of singular measures with respect to transversal submanifolds are proved in arbitrary space dimension. The results of Bennett-Bez are used to extend previous work of Bejenaru-Herr-Tataru. As an application, it is shown that the 3D Zakharov system is locally well-posed in the full subcritical regime.

math.AP↗
arXiv2010

Near soliton evolution for equivariant Schroedinger Maps in two spatial dimensions

We consider the Schrödinger Map equation in $2+1$ dimensions, with values into $§^2$. This admits a lowest energy steady state $Q$, namely the stereographic projection, which extends to a two dimensional family of steady states by scaling and rotation. We prove that $Q$ is unstable in the energy space $\dot H^1$. However, in the process of proving this we also show that within the equivariant class $Q$ is stable in a stronger topology $X \subset \dot H^1$.

math.AP↗