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 $\delta>0$, 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 $H^s$-Sobolev space for $s>\frac12$, to those of BO in the deep-water limit (as $\delta\to\infty$), and to those of KdV in the shallow-water limit (as $\delta\to 0$). This improves previous convergence results by Abdelouhab, Bona, Felland, and Saut (1989), which required $s>\frac32$ in the deep-water limit and $s\geq2$ in the shallow-water limit. Moreover, the convergence results also apply to the generalised ILW equation, i.e.~with nonlinearity $\partial_x (u^k)$ for $k\geq 2$. Furthermore, this work gives the first convergence results of generalised ILW solutions on the circle with regularity $s\geq \frac34$. 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 $\mathbb R^4$ 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 $\mathbb R^4$, 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 $\delta > 0$, in the deep-water limit ($\delta \to \infty$) and the shallow-water limit ($\delta \to 0$) 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, $0 < \delta < \infty$. 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 $\delta \to 0$, 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 $\mathbb{R}^{d}$ and $\mathbb{T}^{d}$ with $d \geq 3$. 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 $\mathbb{T}^d$, we prove global well-posedness with the stochastic forcing below the energy space.Comment: 26 page

Unconditional deep-water limit of the intermediate long wave equation in low-regularity

In this paper, we establish the unconditional deep-water limit of the intermediate long wave equation (ILW) to the Benjamin-Ono equation (BO) in low-regularity Sobolev spaces on both the real line and the circle. Our main tool is new unconditional uniqueness results for ILW in $H^s$ when $s_0<s\leq \frac 14$ on the line and $s_0<s< \frac 12$ on the circle, where $s_0 = 3-\sqrt{33/4}\approx 0.1277$. Here, we adapt the strategy of Mo\c{s}incat-Pilod (2023) for BO to the setting of ILW by viewing ILW as a perturbation of BO and making use of the smoothing property of the perturbation term.Comment: 27 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 L2L^2

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 $L^2$ well-posedness argument for BO by Molinet and the fourth author (2012), we prove global well-posedness of ILW in $L^2$ 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 $L^2$-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