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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

Convergence analysis of one-point large deviations rate functions of numerical discretizations for stochastic wave equations with small noise

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arXiv 2022年

作者： Jin, Diancong Hong, Jialin Sheng, Derui 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 Academy of Mathematics and Systems Science Chinese Academy of Sciences Beijing100190 China School of Mathematical Sciences University of Chinese Academy of Sciences Beijing100049 China

In this work, we present the convergence analysis of one-point large deviations rate functions (LDRFs) of the spatial finite difference method (FDM) for stochastic wave equations with small noise, which is essentially about the asymptotical limit of minimization problems and not a trivial task for the nonlinear cases. In order to overcome the difficulty that objective functions for the original equation and the spatial FDM have different effective domains, we propose a new technical route for analyzing the pointwise convergence of the one-point LDRFs of the spatial FDM, based on the Γ -convergence of objective functions. Based on the new technical route, the intractable convergence analysis of one-point LDRFs boils down to the qualitative analysis of skeleton equations of the original equation and its numerical discretizations. Copyright © 2022, The Authors. All rights reserved.

关键词： Finite difference method

Numerical Investigations of a Class of Biological Models on Unbounded Domain

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Numerical Mathematics(Theory,Methods and Applications) 2019年第1期12卷 169-186页

作者： Qiumei Huang Dongfang Li Jiwei Zhang Beijing Institute for Scientific and Engineering Computing Beijing University of TechnologyBeijing 100124China School of Mathematics and Statistics Huazhong University of Science and TechnologyWuhan 430074China Hubei Key Laboratoryof Engineering Modeling and Scientific Computing Huazhong University of Science and TechnologyWuhan 430074China Beijing Computational Science Research Center Beijing1000193China

This paper is concerned with numerical computations of a class of biologi-cal models on unbounded spatial *** overcome the unboundedness of spatial domain,we first construct efficient local absorbing boundary conditions(labCs)to re-formulate the Cauchy problem into an initial-boundary value(IBV)*** that,we construct a linearized finite difference scheme for the reduced IVB problem,and provide the corresponding error estimates and stability *** delay-dependent dynamical properties on the Nicholson’s blowflies equation and the Mackey-Glass equa-tion are numerically ***,numerical examples are given to demonstrate the efficiency of our labCs and theoretical results of the numerical scheme.

关键词： Biological models artificial boundary method absorbing boundary conditions finite difference method stability,convergence

Multiple-relaxation-time lattice Boltzmann method for the Navier-Stokes and nonlinear convection-diffusion equations: modeling, analysis, and elements

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Physical Review E 2020年第2期102卷 023306-023306页

作者： Zhenhua Chai (柴振华) Baochang Shi (施保昌) School of Mathematics and Statistics Huazhong University of Science and Technology Wuhan 430074 China and Hubei Key Laboratory of Engineering Modeling and Scientific Computing Huazhong University of Science and Technology Wuhan 430074 China

In this paper, we first present a unified framework of multiple-relaxation-time lattice Boltzmann (MRT-LB) method for the Navier-Stokes and nonlinear convection-diffusion equations where a block-lower-triangular-relaxation matrix and an auxiliary source distribution function are introduced. We then conduct a comparison of the four popular analysis methods (Chapman-Enskog analysis, Maxwell iteration, direct Taylor expansion, and recurrence equations approaches) that have been used to obtain the macroscopic Navier-Stokes and nonlinear convection-diffusion equations from the MRT-LB method and show that from mathematical point of view, these four analysis methods can give the same equations at the second-order of expansion parameters. Finally, we give some elements that are needed in the implementation of the MRT-LB method and also find that some available LB models can be obtained from this MRT-LB method.

关键词： Lattice-Boltzmann methods

Spiking hierarchy in an adaptive exponential integrate-and-fire network synchronization

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arXiv 2021年

作者： Wu, Xiaoyue Lin, Congping Zhang, Yiwei School of Mathematics and Statistics Huazhong University of Sciences and Technology Wuhan China Center for Mathematical Sciences Huazhong University of Science and Technology Wuhan China Hubei Key Laboratory of Engineering Modeling and Scientific Computing Huazhong University of Science and Technology Wuhan China

Neuronal network synchronization has received wide interests. Network connection structure is known to play a key role in its synchronization. In the present manuscript, we study the influence of initial membrane potentials together with network topology on bursting synchronization, in particular the sequential spiking order in stabilized inter bursts. We find a hierarchical phenomenon on spiking order. We grade neurons into different layers: Primary neurons are those initiate the spike and lower-layer neurons are then determined via network connections successively. This constructs a directed graph to indicate spiking propagation among different layers of neurons. Neurons in upper layers spike earlier than those in lower layers. More interestingly, we find that among the same layer, neuron spiking order is mainly associated with stimuli they receive from upper layer via network connections;more stimuli leads to earlier spiking. Furthermore, we find that with effectively the same stimuli from upper layer, neurons with less connections spike earlier. © 2021, CC BY.

关键词： Neurons

MAP-based Problem-Agnostic Diffusion Model for Inverse Problems

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arXiv 2025年

作者： Tao, Pingping Liu, Haixia Su, Jing Yang, Xiaochen Tan, Hongchen Shandong University Library No. 180 West Culture Road Shandong Weihai264209 China School of Mathematics and Statistics Hubei Key Laboratory of Engineering Modeling and Scientific Computing Institute of Interdisciplinary Research for Mathematics and Applied Science Huazhong University of Science and Technology No. 1037 Luoyu road Hubei Wuhan430074 China Dapartment of Basic Sciences Dalian University of Science and Technology No. 999-26 Harbor Road Liaoning Dalian116052 China School of Computer Science and Technology Harbin Institute of Technology No. 2 West Culture Road Shandong Weihai264209 China Institute of Future Technology Dalian University of Technology No. 2 Lingong Road Liaoning Dalian116081 China

Diffusion models have indeed shown great promise in solving inverse problems in image processing. In this paper, we propose a novel, problem-agnostic diffusion model called the maximum a posteriori (MAP)-based guided term estimation method for inverse problems. To leverage unconditionally pretrained diffusion models to address conditional generation tasks, we divide the conditional score function into two terms according to Bayes’ rule: an unconditional score function (approximated by a pretrained score network) and a guided term, which is estimated using a novel MAP-based method that incorporates a Gaussian-type prior of natural images. This innovation allows us to better capture the intrinsic properties of the data, leading to improved performance. Numerical results demonstrate that our method preserves contents more effectively compared to state-of-the-art methods—for example, maintaining the structure of glasses in super-resolution tasks and producing more coherent results in the neighborhood of masked regions during inpainting. Our numerical implementation is available at https://***/liuhaixias1/ MAP-DIFFUSION-IP. Copyright © 2025, The Authors. All rights reserved.

关键词： Inverse problems

A Phase-Field-Based Multiple-Relaxation-Time Lattice Boltzmann Method for Incompressible Multiphase Flows with Density and Viscosity Contrasts

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SSRN

SSRN 2022年

作者： Yuan, Xiaolei Wu, Yao Zhang, Chunhua Chai, Zhenhua Shi, Baochang College of Mathematics and Information Science Hebei University Baoding071002 China Institute of Interdisciplinary Research for Mathematics and Applied Science 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 Department of Mechanics and Aerospace Engineering Southern University of Science and Technology Guangdong Shenzhen518055 China

In this work, we present a unified framework of phase-field-based multiple-relaxation-time lattice Boltzmann (MRT-LB) method for incompressible multiphase flows with density and viscosity contrasts where a block-lower-triangular relaxation matrix is introduced. The present framework includes the classic MRT-LB model and central-moments-based LB model (CLBM, a particular version of the MRT-LB model). The governing equations of incompressible multiphase flows can be reproduced precisely through the direct Taylor expansion method at the second-order of expansion parameters. In this work, the pressure $p$ is re-estimated, and we prove that the present pressure expression is suitable for classic MRT model, CLBM and various lattice structures. In addition, theoretical analysis and modeling are both carried out in the velocity space. As a result, the present model has a similar structure to the traditional Bhatnagar-Gross-Krook (BGK) model and is very easy to analyze and implement. We then test the present model (mainly CLBM) by five classic physical problems. The results show that the present model is found to be accurate and has better numerical stability over the BGK model, and has the ability to simulate multiphase flow problems with large density ratio ($1000$). © 2022, The Authors. All rights reserved.

关键词： Multiphase flow

ON COMPUTABILITY OF EQUILIBRIUM STATES

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arXiv 2023年

作者： Binder, Ilia He, Qiandu Li, Zhiqiang Zhang, Yiwei Department of Mathematics University of Toronto Bahen Centre 40 St. George St. TorontoONM5S 2E4 Canada School of Mathematical Sciences Peking University Beijing100871 China School of Mathematical Sciences Beijing International Center for Mathematical Research Peking University Beijing100871 China School of Mathematics and Statistics Center for Mathematical Sciences Hubei Key Laboratory of Engineering Modeling and Scientific Computing Huazhong University of Science and Technology Wuhan430074 China

Equilibrium states are natural dynamical analogs of Gibbs states in thermodynamic formalism. This paper is devoted to the study of their computability in the sense of Computable Analysis. We show that the unique equilibrium state associated to a pair of a computable, topologically exact, distance-expanding, open map T: X → X and a computable Hölder continuous potential : X → R is always computable. Furthermore, the Hausdorff dimension of the Julia set and the equilibrium state for the geometric potential of a computable hyperbolic rational map are computable. On the other hand, we introduce a mechanism to establish non-uniqueness of equilibrium states. We also present some computable dynamical systems whose equilibrium states are all *** Codes Primary: 03D78, Secondary: 37D35, 37D20, 37F15 Copyright © 2023, The Authors. All rights reserved.

关键词： Dynamical systems

A diffuse-interface lattice Boltzmann method for the dendritic growth with thermosolutal convection

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arXiv 2022年

作者： Zhan, Chengjie Chai, Zhenhua Shi, Baochang Jiang, Ping Geng, Shaoning Sun, Dongke School of Mathematics and Statistics Huazhong University of Science and Technology Wuhan430074 China Institute of Interdisciplinary Research for Mathematics and Applied Science 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 The State Key Laboratory of Digital Manufacturing Equipment and Technology School of Mechanical Science and Engineering Huazhong University of Science and Technology Wuhan430074 China School of Mechanical Engineering Southeast University Nanjing211189 China

In this work, we proposed a diffuse interface model for the dendritic growth with thermosolutal convection. In this model, the sharp boundary between the fluid and solid dendrite is replaced by a thin but nonzero thickness diffuse interface, which is described by the order parameter governed by the phase-field equation for the dendritic growth. The governing equations for solute and heat transfer are modified such that the previous special treatments for source term can be avoided. To solve the model for the dendritic growth with thermosolutal convection, we also developed a diffuse-interface multi-relaxation-time lattice Boltzmann (LB) method. In this method, the order parameter in the phase-field equation is combined into the force caused by the fluid-solid interaction, and the treatment on the complex fluid-solid interface can be avoided. In addition, four LB models are developed for the phase-field equation, concentration equation, temperature equation and the Navier-Stokes equations in a unified framework. Finally, to test the present diffuse-interface LB method, we performed some simulations of the dendritic growth, and found that the numerical results are in good agreements with some previous works. © 2022, CC BY-NC-ND.

关键词： Navier Stokes equations