218 research outputs found
Deep-water and shallow-water limits of the intermediate long wave equation
In this paper, we study the low regularity convergence problem for the
intermediate long wave equation (ILW), with respect to the depth parameter
, on the real line and the circle. As a natural bridge between the
Korteweg-de Vries (KdV) and the Benjamin-Ono (BO) equations, the ILW equation
is of physical interest. We prove that the solutions of ILW converge in the
-Sobolev space for , to those of BO in the deep-water limit (as
), and to those of KdV in the shallow-water limit (as
). This improves previous convergence results by Abdelouhab, Bona,
Felland, and Saut (1989), which required in the deep-water limit
and in the shallow-water limit. Moreover, the convergence results also
apply to the generalised ILW equation, i.e.~with nonlinearity for . Furthermore, this work gives the first convergence
results of generalised ILW solutions on the circle with regularity . Overall, this study provides mathematical insights for the behaviour
of the ILW equation and its solutions in different water depths, and has
implications for predicting and modelling wave behaviour in various
environments.Comment: 41 page
Global well-posedness of the 4-d energy-critical stochastic nonlinear Schr\"{o}dinger equations with non-vanishing boundary condition
We consider the energy-critical stochastic cubic nonlinear Schr\"odinger
equation on with additive noise, and with the non-vanishing
boundary conditions at spatial infinity. By viewing this equation as a
perturbation to the energy-critical cubic nonlinear Schr\"odinger equation on
, we prove global well-posedness in the energy space. Moreover, we
establish unconditional uniqueness of solutions in the energy space.Comment: 18 pages. arXiv admin note: text overlap with arXiv:1904.06793,
arXiv:1112.1354 by other author
On the deep-water and shallow-water limits of the intermediate long wave equation from a statistical viewpoint
(Due to the limit on the number of characters for an abstract set by arXiv,
the full abstract can not be displayed here. See the abstract in the paper.) We
study convergence problems for the intermediate long wave equation (ILW), with
the depth parameter , in the deep-water limit ()
and the shallow-water limit () from a statistical point of view.
In particular, we establish convergence of invariant Gibbs dynamics for ILW in
both the deep-water and shallow-water limits. For this purpose, we first
construct the Gibbs measures for ILW, . As they are
supported on distributions, a renormalization is required. With the Wick
renormalization, we carry out the construction of the Gibbs measures for ILW.
We then prove that the Gibbs measures for ILW converge in total variation to
that for the Benjamin-Ono equation (BO) in the deep-water limit. In the
shallow-water regime, after applying a scaling transformation, we prove that,
as , the Gibbs measures for the scaled ILW converge weakly to
that for the Korteweg-de Vries equation (KdV). We point out that this second
result is of particular interest since the Gibbs measures for the scaled ILW
and KdV are mutually singular (whereas the Gibbs measures for ILW and BO are
equivalent).
We also discuss convergence of the associated dynamical problem.
Lastly, we point out that our results also apply to the generalized ILW
equation in the defocusing case, converging to the generalized BO in the
deep-water limit and to the generalized KdV in the shallow-water limit. In the
non-defocusing case, however, our results can not be extended to a nonlinearity
with a higher power due to the non-normalizability of the corresponding Gibbs
measures.Comment: 75 page
Global well-posedness of the energy-critical stochastic nonlinear wave equations
We consider the Cauchy problem for the defocusing energy-critical stochastic
nonlinear wave equations (SNLW) with an additive stochastic forcing on
and with . By adapting the
probabilistic perturbation argument employed in the context of the random data
Cauchy theory by B\'enyi-Oh-Pocovnicu (2015) and Pocovnicu (2017) and in the
context of stochastic PDEs by Oh-Okamoto (2020), we prove global well-posedness
of the defocusing energy-critical SNLW. In particular, on , we
prove global well-posedness with the stochastic forcing below the energy space.Comment: 26 page
Optimal divergence rate of the focusing Gibbs measure
We study the focusing Gibbs measure with critical/supercritical potentials.
In particular, we prove asymptotic formulae for the frequency approximation of
the partition function, which captures the optimal divergence rate of the
partition function as the frequency truncation is removed.Comment: 15 page
Global well-posedness of one-dimensional cubic fractional nonlinear Schr\"odinger equations in negative Sobolev spaces
We study the Cauchy problem for the cubic fractional nonlinear Schr\"odinger
equation (fNLS) on the real line and on the circle. In particular, we prove
global well-posedness of the cubic fNLS with all orders of dispersion higher
than the usual Schr\"odinger equation in negative Sobolev spaces. On the real
line, our well-posedness result is sharp in the sense that a contraction
argument does not work below the threshold regularity. On the circle, due to
ill-posedness of the cubic fNLS in negative Sobolev spaces, we study the
renormalized cubic fNLS. In order to overcome the failure of local uniform
continuity of the solution map in negative Sobolev spaces, by applying a gauge
transform and partially iterating the Duhamel formulation, we study the
resulting equation with a cubic-quintic nonlinearity. In proving uniqueness, we
present full details justifying the use of the normal form reduction for rough
solutions, which seem to be missing from the existing literature. Our
well-posedness result on the circle extends those in Miyaji-Tsutsumi (2018) and
Oh-Wang (2018) to the endpoint regularity.Comment: 49 page
Deep-water limit of the intermediate long wave equation in
We study the well-posedness issue of the intermediate long wave equation
(ILW) on both the real line and the circle. By applying the gauge transform for
the Benjamin-Ono equation (BO) and adapting the well-posedness argument
for BO by Molinet and the fourth author (2012), we prove global well-posedness
of ILW in on both the real line and the circle. In the periodic setting,
this provides the first low regularity well-posedness of ILW. We then establish
convergence of the ILW dynamics to the BO dynamics in the deep-water limit at
the -level.Comment: 26 page
Towards Generic and Controllable Attacks Against Object Detection
Existing adversarial attacks against Object Detectors (ODs) suffer from two
inherent limitations. Firstly, ODs have complicated meta-structure designs,
hence most advanced attacks for ODs concentrate on attacking specific
detector-intrinsic structures, which makes it hard for them to work on other
detectors and motivates us to design a generic attack against ODs. Secondly,
most works against ODs make Adversarial Examples (AEs) by generalizing
image-level attacks from classification to detection, which brings redundant
computations and perturbations in semantically meaningless areas (e.g.,
backgrounds) and leads to an emergency for seeking controllable attacks for
ODs. To this end, we propose a generic white-box attack, LGP (local
perturbations with adaptively global attacks), to blind mainstream object
detectors with controllable perturbations. For a detector-agnostic attack, LGP
tracks high-quality proposals and optimizes three heterogeneous losses
simultaneously. In this way, we can fool the crucial components of ODs with a
part of their outputs without the limitations of specific structures. Regarding
controllability, we establish an object-wise constraint that exploits
foreground-background separation adaptively to induce the attachment of
perturbations to foregrounds. Experimentally, the proposed LGP successfully
attacked sixteen state-of-the-art object detectors on MS-COCO and DOTA
datasets, with promising imperceptibility and transferability obtained. Codes
are publicly released in https://github.com/liguopeng0923/LGP.gi
Distil the informative essence of loop detector data set: Is network-level traffic forecasting hungry for more data?
Network-level traffic condition forecasting has been intensively studied for
decades. Although prediction accuracy has been continuously improved with
emerging deep learning models and ever-expanding traffic data, traffic
forecasting still faces many challenges in practice. These challenges include
the robustness of data-driven models, the inherent unpredictability of traffic
dynamics, and whether further improvement of traffic forecasting requires more
sensor data. In this paper, we focus on this latter question and particularly
on data from loop detectors. To answer this, we propose an uncertainty-aware
traffic forecasting framework to explore how many samples of loop data are
truly effective for training forecasting models. Firstly, the model design
combines traffic flow theory with graph neural networks, ensuring the
robustness of prediction and uncertainty quantification. Secondly, evidential
learning is employed to quantify different sources of uncertainty in a single
pass. The estimated uncertainty is used to "distil" the essence of the dataset
that sufficiently covers the information content. Results from a case study of
a highway network around Amsterdam show that, from 2018 to 2021, more than 80\%
of the data during daytime can be removed. The remaining 20\% samples have
equal prediction power for training models. This result suggests that indeed
large traffic datasets can be subdivided into significantly smaller but equally
informative datasets. From these findings, we conclude that the proposed
methodology proves valuable in evaluating large traffic datasets' true
information content. Further extensions, such as extracting smaller, spatially
non-redundant datasets, are possible with this method.Comment: 13 pages, 5 figure
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