Searcharxiv⌕ Search

arXiv subjects

Dapeng Du

Publications and source records attributed to Dapeng Du.

10 recordsLinked to original sources
arXiv2025

On self-similar singular solutions to a vorticity stretching equation

We consider the following model equation: \begin{equation} \omega_{t} = Z_{11}\omega\,\omega , \end{equation} where \begin{equation} Z_{11} = \partial_{11}\Delta^{-1} \end{equation} is a Calderon-Zygmond operator. We get the existence of self-similar singular solutions with a special form. The main difficulty is the degeneracy of the operator $Z_{11}$ that is overcome by the spectral uncertainty principle. We also show that the solution to this model blows up in finite time if the initial datum is compactly supported and has a positive integral.

math.AP↗
arXiv2023

Iterative projection method for unsteady Navier-Stokes equations with high Reynolds numbers

A new iterative projection method is proposed to solve the unsteady Navier-Stokes equations with high Reynolds numbers. The convectional projection method attempts to project the intermediate velocity to the divergence free space only once per time step. However, such a velocity is not genuinely divergence free in general practice, which can yield large errors when the Reynolds number is high. The new method has several important features: the BDF2 time discretization, the skew-symmetric convection in a semi-implicit form, two modulating parameters, and the iterative projections in each time step. A major difficulty in the proof of iteration convergence is the nonlinear convection. We solve this problem by first analyzing the non-convective scheme with a focus on the spectral properties of the iterative matrix, and then employing a delicate perturbation analysis for the convective scheme. The work achieves the weakly divergence free velocity (strongly divergence free for divergence free finite element spaces), and the rigorous stability and error analysis when the iterations converge. The three dimensional numerical tests confirm that this new method can effectively treat high Reynolds numbers with only a few iterations per time step, where the convectional projection method and the iterative projection method with the explicit convection would fail.

math.NA↗
arXiv2022

Global solutions to the Nernst-Planck-Euler system on bounded domain

We show that the Nernst-Planck-Euler system, which models ionic electrodiffusion in fluids, has global strong solutions for arbitrarily large data in the two dimensional bounded domains. The assumption on species is either there are two species or the diffusivities and the absolute values of ionic valences are the same if the species are arbitrarily many. In particular, the boundary conditions for the ions are allowed to be inhomogeneous. The proof is based on the energy estimates, integration along the characteristic line and the regularity theory of elliptic and parabolic equations.

math.AP↗
arXiv2019

On some model equations of Euler and Navier-Stokes equations

We propose a two-dimensional generalization of Constantin-Lax-Majda model [2]. Some results about singular solutions are given. This model might be the first step toward the singular solutions of the Euler equations. Along the same line (vorticity formulation), we present some further model equations. They possibly models various aspects of difficulties related with the singular solutions of the Euler and Navier-Stokes equations. We also make some discussions on the possible connection between turbulence and the singular solutions of the Navier-Stokes equations.

math.AP↗
arXiv2019

Translate-to-Recognize Networks for RGB-D Scene Recognition

Cross-modal transfer is helpful to enhance modality-specific discriminative power for scene recognition. To this end, this paper presents a unified framework to integrate the tasks of cross-modal translation and modality-specific recognition, termed as Translate-to-Recognize Network (TRecgNet). Specifically, both translation and recognition tasks share the same encoder network, which allows to explicitly regularize the training of recognition task with the help of translation, and thus improve its final generalization ability. For translation task, we place a decoder module on top of the encoder network and it is optimized with a new layer-wise semantic loss, while for recognition task, we use a linear classifier based on the feature embedding from encoder and its training is guided by the standard cross-entropy loss. In addition, our TRecgNet allows to exploit large numbers of unlabeled RGB-D data to train the translation task and thus improve the representation power of encoder network. Empirically, we verify that this new semi-supervised setting is able to further enhance the performance of recognition network. We perform experiments on two RGB-D scene recognition benchmarks: NYU Depth v2 and SUN RGB-D, demonstrating that TRecgNet achieves superior performance to the existing state-of-the-art methods, especially for recognition solely based on a single modality.

cs.CV↗
arXiv2013

On Blow-up criterion for the Nonlinear Schrödinger Equation

The blowup is studied for the nonlinear Schrödinger equation $iu_{t}+Δu+ |u|^{p-1}u=0$ with $p$ is odd and $p\ge 1+\frac 4{N-2}$ (the energy-critical or energy-supercritical case). It is shown that the solution with negative energy $E(u_0)<0$ blows up in finite or infinite time. A new proof is also presented for the previous result in \cite{HoRo2}, in which a similar result but more general in a case of energy-subcritical was shown.

math.AP↗
arXiv2009

The Navier-Stokes equations in the critical Lebesgue space

We study regularity criteria for the $d$-dimensional incompressible Navier-Stokes equations. We prove in this paper that if $u\in L_\infty^tL_{d}^x((0,T)\times {\mathbb R}^d)$ is a Leray-Hopf weak solution, then $u$ is smooth and unique in $(0,T)\times \bR^d$. This generalizes a result by Escauriaza, Seregin and Šverák. Additionally, we show that if $T=\infty$ then $u$ goes to zero as $t$ goes to infinity.

math.AP↗
arXiv2005

On the local Smoothness of Solutions of the Navier-Stokes Equations

We consider the Cauchy problem for incompressible Navier-Stokes equations $u_t+u\nabla_xu-Δu+\nabla p=0, div u=0 in R^d \times R^+$ with initial data $a\in L^d(R^d)$, and study in some detail the smoothing effect of the equation. We prove that for $T<\infty$ and for any positive integers $n$ and $m$ we have $t^{m+n/2}D^m_tD^{n}_x u\in L^{d+2}(R^d\times (0,T))$, as long as the $\|u\|_{L^{d+2}_{x,t}(R^d\times (0,T))}$ stays finite.

math.AP↗