48 research outputs found
Quadratic Nonlinear Derivative Schr\"odinger Equations - Part 2
In this paper we consider the local well-posedness theory for the quadratic
nonlinear Schr\"odinger equation with low regularity initial data in the case
when the nonlinearity contains derivatives. We work in 2+1 dimensions and prove
a local well-posedness result close to scaling for small initial data.Comment: a typo in the statement of the main theorem has been correcte
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
The optimal trilinear restriction estimate for a class of hypersurfaces with curvature
Bennett, Carbery and Tao established nearly optimal trilinear
restriction estimates in under transversality assumptions
only. In this paper we show that the curvature improves the range of exponents,
by establishing estimates, for any in the
case of double-conic surfaces. The exponent is shown to
be the universal threshold for the trilinear estimate.Comment: arXiv admin note: text overlap with arXiv:1601.0100
On Schr\"odinger Maps
We study the local well-posedness theory for the Schr\"odinger Maps equation.
We work in dimensions, for , and prove a local well-posedness
for small initial data in H^{\frac{n}{2}+\e}
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.Comment: added references and typos correcte
Optimal multilinear restriction estimates for a class of surfaces with curvature
Bennett, Carbery and Tao considered the -linear restriction estimate in
and established the near optimal 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 . 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 .Comment: Notation, setup and general reductions are adopted from our works
arXiv:1601.01000 and arXiv:1603.02965, where we address similar questions in
the bilinear and trilinear setup. The analysis is significantly more involve
Near soliton evolution for equivariant Schroedinger Maps in two spatial dimensions
We consider the Schr\"odinger Map equation in dimensions, with values
into . This admits a lowest energy steady state , namely the
stereographic projection, which extends to a two dimensional family of steady
states by scaling and rotation. We prove that is unstable in the energy
space . However, in the process of proving this we also show that
within the equivariant class is stable in a stronger topology
On global well-posedness and scattering for the massive Dirac-Klein-Gordon system
We prove global well-posedness and scattering for the massive
Dirac-Klein-Gordon system with small initial data of subcritical regularity in
dimension three. To achieve this, we impose a non-resonance condition on the
masses.Comment: 24 page
Global wellposedness in the energy space for the Maxwell-Schr\"odinger system
We prove that the Maxwell-Schr\"odinger system in is globally
well-posed in the energy space. The key element of the proof is to obtain a
short time wave packet parametrix for the magnetic Schr\"odinger equation,
which leads to linear, bilinear and trilinear estimates. These, in turn, are
extended to larger time scales via a bootstrap argument
The almost optimal multilinear restriction estimate for hypersurfaces with curvature: the case of hypersurfaces in R^n
In this paper we establish the optimal multilinear restriction estimate for
n-1 hypersurfaces with some curvature, where is the dimension of the
underlying space. The result is sharp up to the endpoint and the role of
curvature is made precise in terms of the shape operator