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Unconditionally optimal convergence of an energy-conserving and linearly implicit scheme for nonlinear wave equations

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Science China Mathematics 2022年第8期65卷 1731-1748页

作者： Waixiang Cao Dongfang Li Zhimin Zhang School of Mathematical Sciences Beijing Normal UniversityBeijing 100875China School of Mathematics and Statistics Huazhong University of Science and TechnologyWuhan 430074China Hubei Key Laboratory of Engineering Modeling and Scientific Computing Huazhong University of Science and TechnologyWuhan 430074China Beijing Computational Science Research Center Beijing 100193China Department of Mathematics Wayne State UniversityDetroitMI 48202USA

In this paper,we present and analyze an energy-conserving and linearly implicit scheme for solving the nonlinear wave *** error estimates in time and superconvergent error estimates in space are established without certain time-step *** key is to estimate directly the solution bounds in the H-norm for both the nonlinear wave equation and the corresponding fully discrete scheme,while the previous investigations rely on the temporal-spatial error splitting *** examples are presented to confirm energy-conserving properties,unconditional convergence and optimal error estimates,respectively,of the proposed fully discrete schemes.

关键词： scalar auxiliary variable wave equations stability error estimate superconvergence

Analysis of L1-Galerkin FEMs for Time-Fractional Nonlinear Parabolic Problems

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Communications in Computational Physics 2018年第6期24卷 86-103页

作者： Dongfang Li Hong-Lin Liao Weiwei Sun Jilu Wang Jiwei Zhang School of Mathematics and Statistics Huazhong University of Science and TechnologyWuhan430074China Hubei Key Laboratory of Engineering Modeling and Scientific Computing Huazhong University of Science and TechnologyWuhan 430074China Department of Mathematics City University of Hong KongKowloonHong Kong Department of Mathematics Nanjing University of Aeronautics and AstronauticsNanjing211106China Department of Scientific Computing Florida StateUniversityTallahasseeFL 32306USA Beijing Computational Science Research Center Beijing 100094China.

This paper is concerned with numerical solutions of time-fractional nonlinear parabolic problems by a class of L1-Galerkin finite element *** analysis of L1 methods for time-fractional nonlinear problems is limited mainly due to the lack of a fundamental Gronwall type *** this paper,we establish such a fundamental inequality for the L1 approximation to the Caputo fractional *** terms of the Gronwall type inequality,we provide optimal error estimates of several fully discrete linearized Galerkin finite element methods for nonlinear *** theoretical results are illustrated by applying our proposed methods to the time fractional nonlinear Huxley equation and time fractional Fisher equation.

关键词： Time-fractional nonlinear parabolic problems L1-Galerkin FEMs Error estimates discrete fractional Gronwall type inequality Linearized schemes

Boundary stabilization of coupled wave system with spatially-varying coefficients and internal anti-damping

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Boundary stabilization of coupled wave system with spatially...

第三十九届中国控制会议

作者： Xiaodan Feng Zhifei Zhang Huazhong University of Science and Technology Hubei Key Laboratory of Engineering Modeling and Scientific Computing Huazhong University of Science and Technology

In this paper, we consider the exponential stabilization of coupled wave equations with spatially-varying coefficients and internal anti-damping. In contrast to the previous work in the literature, the kernel equations of this system are much more complicated. Using the backstepping method, we verify the kernel equations, which is a system of coupled linear wave equations with spatially-varying coefficients. Then by designing a new characteristic line, the well-posedness of the kernel equations is obtained. Finally we use a Lyapunov function to get the exponential stabilization. A numerical example is presented to illustrate the result.

关键词： backstepping boundary control coupled wave system spatially-varying coefficients

EFFECT OF RANDOM NOISES ON PATHWISE SOLUTIONS TO THE HIGH-DIMENSIONAL MODIFIED EULER-POINCARÉ SYSTEM

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arXiv 2021年

作者： Zhang, Lei School of Mathematics and Statistics Hubei Key Laboratory of Engineering Modeling and Scientific Computing Huazhong University of Science and Technology Hubei Wuhan430074 China

In this paper, we study the Cauchy problem for the stochastically perturbed high-dimensional modified Euler-Poincaré system (MEP2) on the torus Td, d ≥ 1. We first establish a local well-posedness framework in the sense of Hadamard for the MEP2 driven by general nonlinear multiplicative noises. Then two kinds of global existence and uniqueness results are demonstrated: One indicates that the MEP2 perturbed by nonlocal-type random noises with proper intensity admits a unique large global strong solution;The other one infers that, if the initial data is sufficiently small, then the MEP2 perturbed by linear multiplicative noise has a unique global solution with high probability. In the case of one dimension, we find that the stochastic MEP2 will break down in finite time when the initial data meets appropriate shape condition. Copyright © 2021, The Authors. All rights reserved.

关键词： Stochastic systems

GLOBAL STRONG SOLUTION FOR THE STOCHASTIC TAMED CHEMOTAXIS-NAVIER-STOKES SYSTEM IN R3

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arXiv 2024年

作者： Xu, Fan Zhang, Lei Liu, Bin School of Mathematics and Statistics Hubei Key Laboratory of Engineering Modeling and Scientific Computing Huazhong University of Science and Technology Hubei Wuhan430074 China

In this work, we consider the 3D Cauchy problem for a coupled system arising from the biomathematics, which consists of a chemotaxis model with cubic logistic source and the stochastic tamed Navier-Stokes equations (STCNS, for short). Our main goal is to establish the existence and uniqueness of global strong solution (strong in both the probabilistic and PDE senses) for the 3D STCNS system with large initial data. To achieve this, we first introduce a triple approximation scheme by virtue of the Friedrichs mollifier, frequency truncation operators and cut-off functions, which makes it possible for constructing sufficiently smooth approximation solutions, and facilitating the effective use of stochastic entropy-energy estimates. Then, building on the entropy-energy estimates, we further derive some higher-order energy estimates, which together with the stochastic compactness method permits us to construct the strong solution for the STCNS system. Copyright © 2024, The Authors. All rights reserved.

关键词： Navier Stokes equations

NEW GLOBAL CARLEMAN ESTIMATES AND NULL CONTROLLABILITY FOR FORWARD/BACKWARD SEMI-LINEAR PARABOLIC SPDES

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arXiv 2024年

作者： Zhang, Lei Xu, Fan Liu, Bin School of Mathematics and Statistics Hubei Key Laboratory of Engineering Modeling and Scientific Computing Huazhong University of Science and Technology Hubei Wuhan430074 China

In this paper, we study the null controllability for some linear and semi-linear parabolic SPDEs involving both the state and the gradient of the state. To start with, an improved global Carleman estimate for linear forward (resp. backward) parabolic SPDEs with general random coefficients and L2-valued source terms is derived. Based on this, we further develop a new global Carleman estimate for linear forward (resp. backward) parabolic SPDEs with H-1-valued source terms, which enables us to deal with the global null controllability for linear backward (resp. forward) parabolic SPDEs with gradient terms. As byproduct, a special energy-type estimate for the controlled system that explicitly depends on the parameters λ, μ and the weighted function θ is obtained. Furthermore, by employing a fixed-point argument, we extend the previous linear controllability results to some semi-linear backward (resp. forward) parabolic SPDEs. Copyright © 2024, The Authors. All rights reserved.

关键词： Controllability

ON THE KELLER-SEGEL MODELS INTERACTING WITH A STOCHASTICALLY FORCED INCOMPRESSIBLE VISCOUS FLOW IN R2

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arXiv 2022年

作者： Zhang, Lei Liu, Bin School of Mathematics and Statistics Hubei Key Laboratory of Engineering Modeling and Scientific Computing Huazhong University of Science and Technology Hubei Wuhan430074 China

This paper considers the Keller-Segel model coupled to stochastic Navier-Stokes equations (KS-SNS, for short), which describes the dynamics of oxygen and bacteria densities evolving within a stochastically forced 2D incompressible viscous flow. Our main goal is to investigate the existence and uniqueness of global solutions (strong in the probabilistic sense and weak in the PDE sense) to the KS-SNS system. A novel approximate KS-SNS system with proper regularization and cut-off operators in Hs(R2) is introduced, and the existence of approximate solution is proved by some a priori uniform bounds and a careful analysis on the approximation scheme. Under appropriate assumptions, two types of stochastic entropy-energy inequalities that seem to be new in their forms are derived, which together with the Prohorov theorem and Jakubowski-Skorokhod theorem enables us to show that the sequence of approximate solutions converges to a global martingale weak solution. In addition, when χ(·) ≡ const. > 0, we prove that the solution is pathwise unique, and hence by the Yamada-Wantanabe theorem that the KS-SNS system admits a unique global pathwise weak solution. Copyright © 2022, The Authors. All rights reserved.

关键词： Navier Stokes equations

GLOBAL WELL-POSEDNESS FOR THE LANDAU-LIFSHITZ-BARYAKHTAR EQUATION IN 3

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arXiv 2024年

作者： Xu, Fan Liu, Bin School of Mathematics and Statistics Hubei Key Laboratory of Engineering Modeling and Scientific Computing Huazhong University of Science and Technology Hubei Wuhan430074 China

This paper establishes the global well-posedness of the Landau-Lifshitz-Baryakhtar (LLBar) equation in the whole space 3. The study first demonstrates the existence and uniqueness of global strong solutions using the weak compactness approach. Furthermore, the existence and uniqueness of classical solutions, as well as arbitrary smooth solutions, are derived through a bootstrap argument. The proofs for the existence of these three types of global solutions are based on Friedrichs mollifier approximation and energy estimates, with the structure of the LLBar equation playing a crucial role in the derivation of the results. Copyright © 2024, The Authors. All rights reserved.

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Global well-posedness and blow-up for the damped perturbation of Camassa-Holm type equations

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arXiv 2020年

作者： Zhang, Lei SCHOOL OF MATHEMATICS AND STATISTICS HUBEI KEY LABORATORY OF ENGINEERING MODELING AND SCIENTIFIC COMPUTING HUAZHONG UNIVERSITY OF SCIENCE AND TECHNOLOGY HUBEI WUHAN430074 China

In this paper, we prove global well-posedness of strong solutions to a class of perturbed Camassa-Holm type equations in Besov spaces. It is shown that the existence of global solutions depends only on the L1-integrability of the time-dependent parameters, but not on the shape or smoothness conditions on initial data. As a by-product, we obtain a new global-in-time result for the weakly dissipative Camassa-Holm type equations in Besov spaces, which considerably improves the results in [11, 21, 22]. Moreover, we derive two kinds of precise blow-up criteria for a certain perturbed Camassa-Holm type equations in Sobolev space, which in some sense tell us how the time-dependent parameters affect the singularity formation of strong solutions. Copyright © 2020, The Authors. All rights reserved.

关键词： Sobolev spaces