arXiv subjects

Wendong Wang

Publications and source records attributed to Wendong Wang.

At least 73 records · Page 4Linked to original sources

arXiv2021

Interior and Boundary Regularity Criteria for the 6D steady Navier-Stokes Equations

It is shown in this paper that suitable weak solutions to the 6D steady incompressible Navier-Stokes are Hölder continuous at $0$ provided that $\int_{B_1}|u(x)|^3dx+\int_{B_1}|f(x)|^qdx$ or $\int_{B_1}|\nabla u(x)|^2dx$+$\int_{B_1}|\nabla u(x)|^2dx\left(\int_{B_1}|u(x)|dx\right)^2+\int_{B_1}|f(x)|^qdx$ with $q>3$ is sufficiently small, which implies that the 2D Hausdorff measure of the set of singular points is zero. For the boundary case, we obtain that $0$ is regular provided that $\int_{B_1^+} |u(x)|^3 dx + \int_{B_1^+} |f(x)|^3 dx$ or $\int_{B_1^+} |\nabla u(x)|^2 dx + \int_{B_1^+} |f(x)|^3 dx$ is sufficiently small. These results improve previous regularity theorems by Dong-Strain (\cite{DS}, Indiana Univ. Math. J., 2012), Dong-Gu (\cite{DG2}, J. Funct. Anal., 2014), and Liu-Wang (\cite{LW}, J. Differential Equations, 2018), where either the smallness of the pressure or the smallness on all balls is necessary.

math.AP↗

arXiv2021

An attempt at axiomatization of extending mechanism of solutions to the fluid dynamical systems

Note that some classic fluid dynamical systems such as the Navier-Stokes equations, Magnetohydrodynamics (MHD), Boussinesq equations, and etc are observably different from each other but obey some energy inequalities of the similar type. In this paper, we attempt to axiomatize the extending mechanism of solutions to these systems, merely starting from several basic axiomatized conditions such as the local existence, joint property of solutions and some energy inequalities. The results established have nothing to do with the concrete forms of the systems and, thus, give the extending mechanisms in a unified way to all systems obeying the axiomatized conditions. The key tools are several new multiplicative interpolation inequalities of Besov type, which have their own interests.

math.AP↗

arXiv2021

Classification of solutions of the 2D steady Navier-Stokes equations with separated variables in cone-like domains

We investigate the problem of classification of solutions for the steady Navier-Stokes equations in any cone-like domains. In the form of separated variables, $$u(x,y)=\left( \begin{array}{c} φ_1(r)v_1(θ) φ_2(r)v_2(θ) \end{array} \right) ,$$ where $x=r\cosθ$ and $y=r\sinθ$ in polar coordinates, we obtain the expressions of all smooth solutions with $C^0$ Dirichlet boundary condition. In particular, it shows that (i) some solutions are found, which are Hölder continuous on the boundary, but their gradients blow up at the corner; (ii) all solutions in the entire plane of $\mathbb{R}^2$ like harmonic functions or Stokes equations, are polynomial expressions.

math.AP↗

arXiv2021

Interfacial ferroelectricity in marginally twisted 2D semiconductors

Twisted heterostructures of two-dimensional crystals offer almost unlimited scope for the design of novel metamaterials. Here we demonstrate a room-temperature ferroelectric semiconductor that is assembled using mono- or few- layer MoS2. These van der Waals heterostructures feature broken inversion symmetry, which, together with the asymmetry of atomic arrangement at the interface of two 2D crystals, enables ferroelectric domains with alternating out-of-plane polarisation arranged into a twist-controlled network. The latter can be moved by applying out-of-plane electrical fields, as visualized in situ using channelling contrast electron microscopy. The interfacial charge transfer for the observed ferroelectric domains is quantified using Kelvin probe force microscopy and agrees well with theoretical calculations. The movement of domain walls and their bending rigidity also agrees well with our modelling results. Furthermore, we demonstrate proof-of-principle field-effect transistors, where the channel resistance exhibits a pronounced hysteresis governed by pinning of ferroelectric domain walls. Our results show a potential venue towards room temperature electronic and optoelectronic semiconductor devices with built-in ferroelectric memory functions.

cond-mat.mes-hall↗

arXiv2021

Remarks on Liouville type theorems for the steady MHD and Hall-MHD equations

In this note we investigate Liouville type theorems for the steady three dimensional MHD and Hall-MHD equations, and show that the velocity field $u$ and the magnetic field $B$ are vanishing provided that $B\in L^{6,\infty}(\mathbb{R}^3)$ and $u\in BMO^{-1}(\mathbb{R}^3)$, which state that the velocity field plays an important role. Moreover, the similar result holds in the case of partial viscosity or diffusivity for the three dimensional MHD equations.

math.AP↗

arXiv2021

Low-rank matrix recovery via regularized nuclear norm minimization

In this paper, we theoretically investigate the low-rank matrix recovery problem in the context of the unconstrained regularized nuclear norm minimization (RNNM) framework. Our theoretical findings show that, the RNNM method is able to provide a robust recovery of any matrix $X$ (not necessary to be exactly low-rank) from its few noisy measurements $\textbf{b}=\mathcal{A}(X)+\textbf{n}$ with a bounded constraint $\|\textbf{n}\|_{2}\leqε$, provided that the $tk$-order restricted isometry constant (RIC) of $\mathcal{A}$ satisfies a certain constraint related to $t>0$. Specifically, the obtained recovery condition in the case of $t>4/3$ is found to be same with the sharp condition established previously by Cai and Zhang (2014) to guarantee the exact recovery of any rank-$k$ matrix via the constrained nuclear norm minimization method. More importantly, to the best of our knowledge, we are the first to establish the $tk$-order RIC based coefficient estimate of the robust null space property in the case of $0<t\leq1$.

math.NA↗

arXiv2020

Atomic Resolution Imaging of CrBr$_{3}$ using Adhesion-Enhanced Grids

Suspended specimens of 2D crystals and their heterostructures are required for a range of studies including transmission electron microscopy (TEM), optical transmission experiments and nanomechanical testing. However, investigating the properties of laterally small 2D crystal specimens, including twisted bilayers and air sensitive materials, has been held back by the difficulty of fabricating the necessary clean suspended samples. Here we present a scalable solution which allows clean free-standing specimens to be realized with 100% yield by dry-stamping atomically thin 2D stacks onto a specially developed adhesion-enhanced support grid. Using this new capability, we demonstrate atomic resolution imaging of defect structures in atomically thin CrBr3, a novel magnetic material which degrades in ambient conditions.

cond-mat.mtrl-sci↗

arXiv2020

Apparent Liquid Permeability in Mixed-Wet Shale Permeable Media

Apparent liquid permeability (ALP) in ultra-confined permeable media is primarily governed by the pore confinement and fluid-rock interactions. A new ALP model is required to predict the interactive effect of the above two on the flow in mixed-wet, heterogeneous nanoporous media. This study derives an ALP model and integrates the compiled results from molecular dynamics (MD) simulations, scanning electron microscopy, atomic force microscopy, and mercury injection capillary pressure. The ALP model assumes viscous forces, capillary forces, and liquid slippage in tortuous, rough pore throats. Predictions of the slippage of water and octane are validated against MD data reported in the literature. In up-scaling the proposed liquid transport model to the representative-elementary-volume scale, we integrate the geological fractals of the shale rock samples including their pore size distribution, pore throat tortuosity, and pore-surface roughness. Sensitivity results for the ALP indicate that when the pore size is below 100 nm pore confinement allows oil to slip in both hydrophobic and hydrophilic pores, yet it also restricts the ALP due to the restricted intrinsic permeability. The ALP reduces to the well-established Carman-Kozeny equation for no-slip viscous flow in a bundle of capillaries, which reveals a distinguishable liquid flow behavior in shales versus conventional rocks. Compared to the Klinkenberg equation, the proposed ALP model reveals an important insight into the similarities and differences between liquid versus gas flow in shales.

physics.flu-dyn↗

arXiv2020

The perturbation analysis of nonconvex low-rank matrix robust recovery

In this paper, we bring forward a completely perturbed nonconvex Schatten $p$-minimization to address a model of completely perturbed low-rank matrix recovery. The paper that based on the restricted isometry property generalizes the investigation to a complete perturbation model thinking over not only noise but also perturbation, gives the restricted isometry property condition that guarantees the recovery of low-rank matrix and the corresponding reconstruction error bound. In particular, the analysis of the result reveals that in the case that $p$ decreases $0$ and $a>1$ for the complete perturbation and low-rank matrix, the condition is the optimal sufficient condition $δ_{2r}<1$ \cite{Recht et al 2010}. The numerical experiments are conducted to show better performance, and provides outperformance of the nonconvex Schatten $p$-minimization method comparing with the convex nuclear norm minimization approach in the completely perturbed scenario.

cs.IT↗

arXiv2020

Scaling invariant Serrin criterion via one velocity component for the Navier-Stokes equations

In this paper, we prove that the Leray weak solution $u : \mathbb{R}^3\times (0, T)\rightarrow\mathbb{R}^3 $ of the Navier-Stokes equations is regular in $\mathbb{R}^3\times (0,T)$ under the scaling invariant Serrin condition imposed on one component of the velocity $u_3\in L^{q,1}(0, T;L^p(\mathbb{R}^3))$ with \[ \frac{2}{q}+\frac{3}{p}\leq 1,\quad 3<p<+\infty. \] This result is an immediate consequence of a new local regularity criterion in terms of one velocity component for suitable weak solutions.

math.AP↗

arXiv2020

Selectively Controlled Magnetic Microrobots with Opposing Helices

Magnetic microrobots that swim through liquid media are of interest for minimally invasive medical procedures, bioengineering, and manufacturing. Many of the envisaged applications, such as micromanipulation and targeted cargo delivery, necessitate the use and adequate control of multiple microrobots, which will increase the velocity, robustness, and efficacy of a procedure. While various methods involving heterogeneous geometries, magnetic properties, and surface chemistries have been proposed to enhance independent control, the main challenge has been that the motion between all microwsimmers remains coupled through the global control signal of the magnetic field. Katsamba and Lauga proposed transchiral microrobots, a theoretical design with magnetized spirals of opposite handedness. The competition between the spirals can be tuned to give an intrinsic nonlinearity that each device can function only within a given band of frequencies. This allows individual microrobots to be selectively controlled by varying the frequency of the rotating magnetic field. Here we present the experimental realization and characterization of transchiral micromotors composed of independently driven magnetic helices. We show a swimming micromotor that yields negligible net motion until a critical frequency is reached and a micromotor that changes its translation direction as a function of the frequency of the rotating magnetic field. This work demonstrates a crucial step towards completely decoupled and addressable swimming magnetic microrobots.

cond-mat.soft↗

arXiv2020

An Optimal Condition of Robust Low-rank Matrices Recovery

In this paper we investigate the reconstruction conditions of nuclear norm minimization for low-rank matrix recovery. We obtain sufficient conditions $δ_{tr}<t/(4-t)$ with $0<t<4/3$ to guarantee the robust reconstruction $(z\neq0)$ or exact reconstruction $(z=0)$ of all rank $r$ matrices $X\in\mathbb{R}^{m\times n}$ from $b=\mathcal{A}(X)+z$ via nuclear norm minimization. Furthermore, we not only show that when $t=1$, the upper bound of $δ_r<1/3$ is the same as the result of Cai and Zhang \cite{Cai and Zhang}, but also demonstrate that the gained upper bounds concerning the recovery error are better. Moreover, we prove that the restricted isometry property condition is sharp. Besides, the numerical experiments are conducted to reveal the nuclear norm minimization method is stable and robust for the recovery of low-rank matrix.

cs.IT↗

arXiv2020

An analysis of noise folding for low-rank matrix recovery

Previous work regarding low-rank matrix recovery has concentrated on the scenarios in which the matrix is noise-free and the measurements are corrupted by noise. However, in practical application, the matrix itself is usually perturbed by random noise preceding to measurement. This paper concisely investigates this scenario and evidences that, for most measurement schemes utilized in compressed sensing, the two models are equivalent with the central distinctness that the noise associated with (\ref{eq.3}) is larger by a factor to $mn/M$, where $m,~n$ are the dimension of the matrix and $M$ is the number of measurements. Additionally, this paper discusses the reconstruction of low-rank matrices in the setting, presents sufficient conditions based on the associating null space property to guarantee the robust recovery and obtains the number of measurements. Furthermore, for the non-Gaussian noise scenario, we further explore it and give the corresponding result. The simulation experiments conducted, on the one hand show effect of noise variance on recovery performance, on the other hand demonstrate the verifiability of the proposed model.

cs.IT↗

arXiv2020

Asymptotic behavior of the steady Navier-Stokes flow in the exterior domain

We consider an elliptic equation with unbounded drift in an exterior domain, and obtain quantitative uniqueness estimates at infinity, i.e. the non-trivial solution of $-\triangle u+W\cdot\nabla u=0$ decays in the form of $\exp(-C|x|\log^2|x|)$ at infinity provided $\|W\|_{L^\infty(\mathbb{R}^2\setminus B_1)}\lesssim 1$, which is sharp with the help of some counterexamples. These results also generalize the decay theorem by Kenig-Wang \cite{KW2015} in the whole space. As an application, the asymptotic behavior of an incompressible fluid around a bounded obstacle is also considered. Specially for the two-dimensional case, we can improve the decay rate in \cite{KL2019} to $\exp(-C|x|\log^2|x|)$, where the minimal decaying rate of $\exp(-C|x|^{\frac32+})$ is obtained by Kow-Lin in a recent paper \cite{KL2019} by using appropriate Carleman estimates.

math.AP↗

arXiv2019

Asymptotic properties of the plane shear thickening fluids with bounded energy integral

In this note we investigate the asymptotic behavior of plane shear thickening fluids around a bounded obstacle. Different from the Navier-Stokes case considered by Gilbarg-Weinberger in \cite{GW1978}, where the good structure of the vorticity can be exploited and weighted energy estimates can be applied, we have to overcome the nonlinear term of high order. The decay estimates of the velocity was obtained by combining Point-wise Behavior Theorem in \cite{Galdi} and Brezis-Gallouet inequality in \cite{BG1980} together, which is independent of interest.

math.AP↗

arXiv2019

RIP-based performance guarantee for low-tubal-rank tensor recovery

The essential task of tensor data analysis focuses on the tensor decomposition and the corresponding notion of rank. In this paper, by introducing the notion of tensor Singular Value Decomposition (t-SVD), we establish a Regularized Tensor Nuclear Norm Minimization (RTNNM) model for low-tubal-rank tensor recovery. As we know that many variants of the Restricted Isometry Property (RIP) have proven to be crucial analysis tools for sparse recovery. In the t-SVD framework, we initiatively define a novel tensor Restricted Isometry Property (t-RIP). Furthermore, we show that any third-order tensor $\boldsymbol{\mathcal{X}}$ can stably be recovered from few linear noise measurements under some certain t-RIP conditions via the RTNNM model. We note that, as far as the authors are aware, such kind of result has not previously been reported in the literature.

cs.IT↗

arXiv2019

Tensor Restricted Isometry Property Analysis For a Large Class of Random Measurement Ensembles

In previous work, theoretical analysis based on the tensor Restricted Isometry Property (t-RIP) established the robust recovery guarantees of a low-tubal-rank tensor. The obtained sufficient conditions depend strongly on the assumption that the linear measurement maps satisfy the t-RIP. In this paper, by exploiting the probabilistic arguments, we prove that such linear measurement maps exist under suitable conditions on the number of measurements in terms of the tubal rank r and the size of third-order tensor n1, n2, n3. And the obtained minimal possible number of linear measurements is nearly optimal compared with the degrees of freedom of a tensor with tubal rank r. Specially, we consider a random sub-Gaussian distribution that includes Gaussian, Bernoulli and all bounded distributions and construct a large class of linear maps that satisfy a t-RIP with high probability. Moreover, the validity of the required number of measurements is verified by numerical experiments.

stat.ML↗

arXiv2019

$C^α$ regularity of weak solutions of non-homogenous ultraparabolic equations with drift terms

Consider a class of non-homogenous ultraparabolic differential equations with drift terms or lower order terms arising from some physical models, and we prove that weak solutions are Hölder continuous, which also generalizes the classic results of parabolic equations of second order. The main ingredients are a type of weak Poincaré inequality satisfied by non-negative weak sub-solutions and Moser iteration.

math.AP↗