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A CHEMOTAXIS-FLUID MODEL DRIVEN BY LÉVY NOISE IN R2

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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 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

WELL-POSEDNESS AND INVARIANT MEASURES FOR THE STOCHASTICALLY PERTURBED LANDAU-LIFSHITZ-BARYAKHTAR EQUATION

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arXiv 2024年

In this paper, we study the initial-boundary value problem for the stochastic Landau-Lifshitz-Baryakhtar (SLLBar) equation with Stratonovich-type noise in bounded domains O ⊂ Rd, d = 1, 2, 3. Our main results can be briefly described as follows: (1) for d = 1, 2, 3 and any u0 ∈ H1, the SLLBar equation admits a unique local-in-time pathwise weak solution;(2) for d = 1 and small-data u0 ∈ H1, the SLLBar equation has a unique global-in-time pathwise weak solution and at least one invariant measure;(3) for d = 1, 2 and small-data u0 ∈ L2, the SLLBar equation possesses a unique global-in-time pathwise very weak solution and at least one invariant measure, while for d = 3 only the existence of martingale solution is obtained due to the loss of pathwise uniqueness. Copyright © 2024, The Authors. All rights reserved.

关键词： Stochastic systems

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

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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 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

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

Multiple-relaxation-time finite-difference lattice Boltzmann model for the nonlinear convection-diffusion equation

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Physical Review E 2021年第3期104卷 035308-035308页

作者： Xinmeng Chen Zhenhua Chai Jinlong Shang Baochang Shi School of Mathematics and Statistics Huazhong University of Science and Technology Wuhan 430074 China Hubei Key Laboratory of Engineering Modeling and Scientific Computing Huazhong University of Science and Technology Wuhan 430074 China

In this paper, a multiple-relaxation-time finite-difference lattice Boltzmann method (MRT-FDLBM) is developed for the nonlinear convection-diffusion equation (NCDE). Through designing the equilibrium distribution function and the source term properly, the NCDE can be recovered exactly from MRT-FDLBM. We also conduct the von Neumann stability analysis on the present MRT-FDLBM and its special case, i.e., single-relaxation-time finite-difference lattice Boltzmann method (SRT-FDLBM). Then, a simplified version of MRT-FDLBM (SMRT-FDLBM) is also proposed, which can save about 15% computational cost. In addition, a series of real and complex-value NCDEs, including the isotropic convection-diffusion equation, Burgers-Fisher equation, sine-Gordon equation, heat-conduction equation, and Schrödinger equation, are used to test the performance of MRT-FDLBM. The results show that both MRT-FDLBM and SMRT-FDLBM have second-order convergence rates in space and time. Finally, the stability and accuracy of five different models are compared, including the MRT-FDLBM, SMRT-FDLBM, SRT-FDLBM, the previous finite-difference lattice Boltzmann method [H. Wang, B. Shi et al., Appl. Math. Comput. 309, 334 (2017)], and the lattice Boltzmann method (LBM). The stability tests show that the sequence of stability from high to low is as follows: MRT-FDLBM, SMRT-FDLBM, SRT-FDLBM, the previous finite-difference lattice Boltzmann method, and LBM. In most of the precision test results, it is found that the order from high to low of precision is MRT-FDLBM, SMRT-FDLBM, SRT-FDLBM, and the previous finite-difference lattice Boltzmann method.

关键词： Lattice-Boltzmann methods

Multiple-distribution-function lattice Boltzmann method for convection-diffusion-system-based incompressible Navier-Stokes equations

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Physical Review E 2022年第5期106卷 055305-055305页

作者： Zhenhua Chai Baochang Shi Chengjie Zhan School of Mathematics and Statistics Huazhong University of Science and Technology Wuhan 430074 China Hubei Key Laboratory of Engineering Modeling and Scientific Computing Huazhong University of Science and Technology Wuhan 430074 China

In this paper, a multiple-distribution-function lattice Boltzmann method (MDF-LBM) with a multiple-relaxation-time model is proposed for incompressible Navier-Stokes equations which are considered as coupled convection-diffusion equations. Through direct Taylor expansion analysis, we show that the Navier-Stokes equations can be recovered correctly from the present MDF-LBM, and additionally, it is also found that the velocity and pressure can be directly computed through the zero and first-order moments of the distribution function. Then in the framework of the present MDF-LBM, we develop a locally computational scheme for the velocity gradient in which the first-order moment of the nonequilibrium distribution is used; this scheme is also extended to calculate the velocity divergence, strain rate tensor, shear stress, and vorticity. Finally, we also conduct some simulations to test the MDF-LBM and find that the numerical results not only agree with some available analytical and numerical solutions but also have a second-order convergence rate in space.

关键词： Lattice-Boltzmann methods

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

General propagation lattice Boltzmann model for nonlinear advection-diffusion equations

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Physical Review E 2018年第4期97卷 043310-043310页

作者： Xiuya Guo Baochang Shi Zhenhua Chai School of Mathematics and Statistics Huazhong University of Science and Technology Wuhan 430074 China Hubei Key Laboratory of Engineering Modeling and Scientific Computing Huazhong University of Science and Technology Wuhan 430074 China

In this paper, a general propagation lattice Boltzmann model is proposed for nonlinear advection-diffusion equations (NADEs), and the Chapman-Enskog analysis shows that the NADEs with variable coefficients can be recovered correctly from the present model. We also perform some simulations of the linear advection-diffusion equation, nonlinear heat conduction equation, NADEs with anisotropic diffusion, and variable coefficients to test the present model, and find that the numerical results agree well with the corresponding analytical solutions. Moreover, it is also shown that by properly adjusting the two free parameters introduced into the propagation step, the present model could be more stable and more accurate than the standard lattice Bhatnagar-Gross-Krook model.

关键词： Lattice-Boltzmann methods

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