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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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RANDOM PERTURBATIONS FOR THE CHEMOTAXIS-FLUID MODEL WITH FRACTIONAL DISSIPATION: GLOBAL PATHWISE WEAK SOLUTIONS

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arXiv 2023年

This paper considers a stochastically perturbed Keller-Segel-Navier-Stokes (KS-SNS) system arising from the biomathematics in two dimensions, where the diffusion of fluid is expressed by a fractional Laplacian with an exponent in [1/2, 1]. Our main result demonstrates that, under appropriate assumptions, the Cauchy problem of the KS-SNS system has a unique global probabilistically strong and analytically weak solution, which also confirms that the quadratic logistic source authentically contributes to the global existence of solutions. First, a three-layer approximate system is introduced, this system seems to be new in the studying of chemotaxis-fluid model and it enables one to construct approximate solutions in regular Hilbert spaces Hs(R2). Second, to accomplish the convergence progressively, a series of crucial entropy-energy inequalities for the approximations are derived, whose derivation strongly depends on the fine structure of the system and requires a novel strategy to adapt to the appearance of fractional dissipation and the unboundedness of domain. Third, based on these uniform bounds, we establish the existence of global martingale weak solutions by virtue of a stochastic compactness method. Finally, by applying the Littlewood-Paley theory, we show that the pathwise uniqueness holds for these martingale solution. As a by-product, a few interpolation inequalities in Besov spaces are obtained, which seems to be new and may have their own interest. Copyright © 2023, The Authors. All rights reserved.

关键词： Stochastic systems

WELL-POSEDNESS AND LARGE DEVIATIONS OF LÉVY-DRIVEN MARCUS STOCHASTIC LANDAU-LIFSHITZ-BARYAKHTAR EQUATION

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arXiv 2024年

作者： Xu, Fan Liu, Bin 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

This paper considers the stochastic Landau-Lifshitz-Baryakhtar (SLLBar) equation with pure jump noise in Marcus canonical form, which describes the dynamics of magnetic spin field in a ferromagnet at elevated temperatures with the effective field Heff influenced by external random noise. Under the natural assumption that the magnetic body O ⊂ Rd (d = 1, 2, 3) is bounded with smooth boundary, we shall prove that the initial-boundary value problem of SLLBar equation possesses a unique global probabilistically strong and analytically weak solution with initial data in the energy space H1(O). Then by employing the weak convergence method, we proceed to establish a Freidlin-Wentzell type large deviation principle for pathwise solutions to the SLLBar equation. Copyright © 2024, The Authors. All rights reserved.

关键词： Stochastic systems

A CHEMOTAXIS-FLUID MODEL DRIVEN BY LÉVY NOISE IN R2

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arXiv 2024年

In this paper, we investigate the existence and uniqueness of global solutions to the Cauchy problem for a coupled stochastic chemotaxis-Navier-Stokes system with multiplicative Lévy noises in R2. The existence of global martingale solutions is proved under a framework that is based on the Faedo-Galerkin approximation scheme and stochastic compactness method, where the verification of tightness depends crucially on a novel stochastic version of Lyapunov functional inequality and proper compactness criteria in Fréchet spaces. A pathwise uniqueness result is also established with suitable assumption on the jump noises, which indicates that the considered system admits a unique global strong solution. Copyright © 2024, The Authors. All rights reserved.

关键词： Lyapunov functions

GLOBAL MARTINGALE WEAK SOLUTIONS FOR THE THREE-DIMENSIONAL STOCHASTIC CHEMOTAXIS-NAVIER-STOKES SYSTEM WITH LÉVY PROCESSES

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arXiv 2022年

This paper considers the three-dimensional stochastic chemotaxis-Navier-Stokes (SCNS) system subjected to a Lévy-type random external force in a bounded domain. Until recently, the existed results concerning global solvability of SCNS system mainly concentrated on the case of two spatial dimensions, little is known about the SCNS system in dimension three. We prove in the present work that the initial-boundary value problem for the three-dimensional SCNS system has at least one global martingale solution under proper assumptions, which is weak in both the analytical and stochastic sense. A new stochastic analogue of entropy-energy inequality and an uniform boundedness estimate are derived, which enable us to construct global-in-time approximate solutions from a suitably regularized SCNS system via the Contraction Mapping Principle. The proof of the existence of martingale solution is based on a stochastic compactness method and an elaborate identification of the limits procedure, where the Jakubowski-Skorokhod Theorem is applied to deal with the phase spaces equipped with weak topology. Copyright © 2022, The Authors. All rights reserved.

关键词： Stochastic systems

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

Task Separation and Knowledge Sharing for Class Incremental Learning 5

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Task Separation and Knowledge Sharing for Class Incremental ...

5th International Conference on Intelligent computing and Human-Computer Interaction, ICHCI 2024

作者： Zhang, Jiali Qiao, Xiaoyan School of Computer Science and Technology Shandong Technology and Business University Yantai Key Laboratory of Big Data Modeling and Intelligent Computing Immersion Technology and Evaluation Shandong Engineering Research Center Shandong Yantai China School of Mathematics and Information Science Shandong Technology and Business University Yantai Key Laboratory of Big Data Modeling and Intelligent Computing Immersion Technology and Evaluation Shandong Engineering Research Center Shandong Yantai China

ISBN: (纸本)9798350368284

Methods based on dynamically expanding architectures can effectively mitigate catastrophic forgetting in class incremental learning (CIL), but they often overlook information sharing and integration between subnetworks, leading to performance degradation and increased parameter count and memory consumption as the network expands. To address these complex issues, we introduce a new method called Task Separation and Knowledge Sharing (TSKS), which transfers knowledge at both the logical and feature levels through Hybrid Knowledge Distillation (HKD). In logical distillation, we combine Global Knowledge Distillation (GKD) and Task Knowledge Distillation (TKD) to establish connections between tasks. To prevent the model from favoring new classes, we design an Attention- Based Feature Fusion (ABFF) mechanism to capture important features. Additionally, we introduce a Soft-Margin Auxiliary Loss (SMAL) to help the model learn new concepts. Extensive experiments show that TSKS outperforms existing models on CIFAR100 and ImageNet100, regardless of whether memory samples are used. Notably, TSKS demonstrates superior performance in memory-free settings, indicating a reduced reliance on rehearsal memory. © 2024 IEEE.

关键词： Adversarial machine learning

Lattice Boltzmann simulations of melting in a rectangular cavity heated locally from below at high Rayleigh number

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arXiv 2019年

作者： Zhao, Yong Wu, Yao Chai, Zhenhua Shi, Baochang Hubei Key Laboratory of Engineering Modeling and Scientific Computing School of Mathematics and Statistics Huazhong University of Science and Technology Wuhan430074 China

This work presented a block triple-relaxation-time (B-TriRT) lattice Boltzmann model for simulating melting in a rectangular cavity heated from below at high Rayleigh (Ra) number (Ra = 108). The test of benchmark problem show that present B-TriRT can dramatically reduce the numerical diffusion across the phase interface. In addition, the influences of location of the heated region are investigated. The results indicate that the location of heated region plays an essential role in melting rate and the full melting occur earliest when the heated region is located at the middle region. Copyright © 2019, The Authors. All rights reserved.

关键词： Melting