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Critical system involving fractional Laplacian

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arXiv 2017年

作者： Zhen, Maoding He, Jinchun Xu, Haoyuan School of Mathematics and Statistics Huazhong University of Science and Technology Wuhan430074 China Hubei Key Laboratory of Engineering Modeling and Scientific Computing Huazhong University of Science and Technology Wuhan430074 China

In this paper, we study the following critical system with fractional Laplacian:(Formula Presented) , By using the Nehari manifold, under proper conditions, we establish the existence and nonexistence of positive leas... 详细信息

关键词： Laplace transforms

Carleman and observability estimates for stochastic beam equation

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arXiv 2017年

作者： Zhen, Maoding He, Jinchun Xu, Haoyuan Yang, Meihua School of Mathematics and Statistics Huazhong University of Science and Technology Wuhan430074 China Hubei Key Laboratory of Engineering Modeling and Scientific Computing Huazhong University of Science and Technology Wuhan430074 China

In this paper, we establish a weight identity for stochastic beam equation by means of the multiplier method. Based on this identity, we first establish the global Carleman estimate for the special system with zero initial value and end value, then a revised Carleman estimate for stochastic beam equation is established through a cutoff technique. Finally, we use the revised Carleman estimate to get the required boundary observability estimate. Copyright © 2017, The Authors. All rights reserved.

关键词： Stochastic systems

Spectral gap of sparse bistochastic matrices with exchangeable rows

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arXiv 2018年

作者： Bordenave, Charles Qiu, Yanqi Zhang, Yiwei Institut de Mathématiques de Marseille CNRS Aix-Marseille University 39 Rue Frédéric Joliot Curie Marseille13013 France Institute of Mathematics and Hua Loo-Keng Key Laboratory of Mathematics AMSS Chinese Academy of Sciences Beijing100190 China CNRS Institut de Mathématiques de Toulouse University of Toulouse III School of Mathematics and Statistics Center for Mathematical Sciences Hubei Key Laboratory of Engineering Modeling and Scientific Computing Huazhong University of Sciences and Technology Wuhan430074 China

We consider a random bistochastic matrix of size n of the form MQ where M is a uniformly distributed permutation matrix and Q is a given bistochastic matrix. Under sparsity and regularity assumptions on Q, we prove that the second largest eigenvalue of MQ is essentially bounded by the normalized Hilbert-Schmidt norm of Q when n grows large. We apply this result to random walks on random regular digraphs.60B20, 60C05, 05C80, 05C38 Copyright © 2018, The Authors. All rights reserved.

关键词： Eigenvalues and eigenfunctions

Lattice boltzmann modeling of wall-bounded ternary fluid flows

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arXiv 2017年

作者： Liang, Hong Xu, Jiangrong Chen, Jiangxing Chai, Zhenhua Shi, Baochang Department of Physics Hangzhou Dianzi University Hangzhou310018 China School of Mathematics and Statistics Huazhong University of Science and Technology Wuhan430074 China Hubei Key Laboratory of Engineering Modeling and Scientific Computing Huazhong University of Science and Technology Wuhan430074 China

In this paper, a wetting boundary scheme used to describe the interactions among ternary fluids and solid is proposed in the framework of the lattice Boltzmann method. This scheme for three-phase fluids can preserve the reduction consistency property with the diphasic situation such that it could give physically relevant results. Combining this wetting boundary scheme and the lattice Boltzmann (LB) ternary fluid model based on the multicomponent phase-field theory, we simulated several ternary fluid flow problems involving solid substrate, including the spreading of binary drops on the substrate, the spreading of a compound drop on the substrate, and the shear of a compound liquid drop on the substrate. The numerical results are found to be good agreement with the analytical solutions or some available results. Finally, as an application, we use the LB model coupled with the present wetting boundary scheme to numerically investigate the impact of a compound drop on a solid circular cylinder. It is found that the dynamics of a compound drop can be remarkably influenced by the wettability of the solid surface and the dimensionless Weber number. Copyright © 2017, The Authors. All rights reserved.

关键词： Circular cylinders

Phase-field-based lattice Boltzmann modeling of large-density-ratio two-phase flows

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arXiv 2017年

作者： Liang, Hong Xu, Jiangrong Chen, Jiangxing Wang, Huili Chai, Zhenhua Shi, Baochang Department of Physics Hangzhou Dianzi University Hangzhou310018 China School of Mathematics and Statistics Huazhong University of Science and Technology Wuhan430074 China Hubei Key Laboratory of Engineering Modeling and Scientific Computing Huazhong University of Science and Technology Wuhan430074 China

In this paper, we present a simple and accurate lattice Boltzmann (LB) model for immiscible two-phase flows, which is able to deal with large density contrasts. This model utilizes two LB equations, one of which is used to solve the conservative Allen-Cahn equation, and the other is adopted to solve the incompressible Navier-Stokes equations. A novel forcing distribution function is elaborately designed in the LB equation for the Navier-Stokes equations, which make it much simpler than the existing LB models. In addition, the proposed model can achieve superior numerical accuracy compared with previous Allen-Cahn type of LB models. Several benchmark two-phase problems, including static droplet, layered Poiseuille flow, and Spinodal decomposition are simulated to validate the present LB model. It is found that the present model can achieve relatively small spurious velocity in the LB community, and the obtained numerical results also show good agreement with the analytical solutions or some available results. At last, we use the present model to investigate the droplet impact on a thin liquid film with a large density ratio of 1000 and the Reynolds number ranging from 20 to 500. The fascinating phenomena of droplet splashing is successfully reproduced by the present model and the numerically predicted spreading radius exhibits to obey the power law reported in the literature. Copyright © 2017, The Authors. All rights reserved.

关键词： Navier Stokes equations

Jacobi polynomials on the Bernstein ellipse

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arXiv 2017年

作者： Wang, Haiyong Zhang, Lun School of Mathematics and Statistics Huazhong University of Science and Technology Wuhan430074 China Hubei Key Laboratory of Engineering Modeling and Scientific Computing Huazhong University of Science and Technology Wuhan430074 School of Mathematical Sciences Shanghai Key Laboratory for Contemporary Applied Mathe-matics Fudan University Shanghai200433

In this paper, we are concerned with Jacobi polynomials P(α,β)n(x) on the Bernstein ellipse with motivation mainly coming from recent studies of convergence rate of spectral interpolation. An explicit representation of P(α,β)n(x) is derived in the variable of parametrization. This formula further allows us to show that the maximum value of |P(α,β)n(z)| over the Bernstein ellipse is attained at one of the endpoints of the major axis if α+β ≥ -1. For the minimum value, we are able to show that for a large class of Gegenbauer polynomials (i.e., α = β), it is attained at two endpoints of the minor axis. These results particularly extend those previously known only for some special cases. Moreover, we obtain a more refined asymptotic estimate for Jacobi polynomials on the Bernstein ellipse.65N35, 65D05, 41A05, 41A25 Copyright © 2017, The Authors. All rights reserved.

关键词： Polynomials

Correction to: A Linearized Local Conservative Mixed Finite Element Method for Poisson–Nernst–Planck Equations

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Journal of scientific computing 2018年第2期77卷 818-818页

作者： Gao, Huadong Sun, Pengtao School of Mathematics and Statistics Huazhong University of Science and Technology Wuhan People’s Republic of China Hubei Key Laboratory of Engineering Modeling and Scientific Computing Huazhong University of Science and Technology Wuhan China Department of Mathematical Sciences University of Nevada Las Vegas Las Vegas USA

The original version of this article contained a mistake. There are error in line breaks in Eqs. 4.3 and 4.4 and the word “quad” was included inadvertently in Eq. 4.4.

The original version of this article contained a mistake. There are error in line breaks in Eqs. 4.3 and 4.4 and the word “quad” was included inadvertently in Eq. 4.4.

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A third-order discrete unified gas kinetic scheme for continuum and rarefied flows: Low-speed isothermal case

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arXiv 2017年

作者： Wu, Chen Shu, Chang Shi, Baochang Chen, Zhen State Key Laboratory of Coal Combustion Huazhong University of Science and Technology Wuhan430074 China Department of Mechanical Engineering National University of Singapore 10 Kent Ridge Crescent Singapore119260 Singapore School of Mathematics and Statistics Huazhong University of Science and Technology Wuhan430074 China Hubei Key Laboratory of Engineering Modeling and Scientific Computing Huazhong University of Science and Technology Wuhan430074 China

An efficient third-order discrete unified gas kinetic scheme (DUGKS) with efficiency is presented in this work for simulating continuum and rarefied flows. By employing two-stage time-stepping scheme and the high-order DUGKS flux reconstruction strategy, third-order of accuracy in both time and space can be achieved in the present method. It is also analytically proven that the second-order DUGKS is a special case of the present method. Compared with the high-order lattice Boltzmann equation LBE based methods, the present method is capable to deal with the rarefied flows by adopting the Newton-Cotes quadrature to approximate the integrals of moments. Instead of being constrained by the second-order (or lower-order) of accuracy in time splitting scheme as in the conventional high-order Runge-Kutta (RK) based kinetic methods, the present method solves the original BE, which overcomes the limitation in time accuracy. Typical benchmark tests are carried out for comprehensive evaluation of the present method. It is observed in the tests that the present method is advantageous over the original DUGKS in accuracy and capturing delicate flow structures. Moreover, the efficiency of the present third-order method is also shown in simulating rarefied flows. Copyright © 2017, The Authors. All rights reserved.

关键词： Boltzmann equation

Slow manifolds for stochastic systems with non-Gaussian stable Lévy noise

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arXiv 2017年

作者： Yuan, Shenglan Hu, Jianyu Liu, Xianming Duan, Jinqiao Center for Mathematical Sciences Huazhong University of Sciences and Technology Wuhan430074 China School of Mathematics and Statistics Huazhong University of Sciences and Technology Wuhan430074 China Hubei Key Laboratory of Engineering Modeling and Scientific Computing Huazhong University of Sciences and Technology Wuhan430074 China Department of Applied Mathematics Illinois Institute of Technology ChicagoIL60616 United States

This work is concerned with the dynamics of a class of slow-fast stochastic dynamical systems with non-Gaussian stable Lévy noise with a scale parameter. Slow manifolds with exponentially tracking property are constructed, eliminating the fast variables to reduce the dimension of these coupled dynamical systems. It is shown that as the scale parameter tends to zero, the slow manifolds converge to critical manifolds in distribution, which helps understand long time dynamics. The approximation of slow manifolds with error estimate in distribution are also considered. Copyright © 2017, The Authors. All rights reserved.

关键词： Differential equations