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arXiv 2020年

作者： Hou, Xiaobo Tian, Xueting Zhang, Yiwei School of Mathematical Sciences Fudan University Shanghai200433 China 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

Pfister and Sullivan proved that if a topological dynamical system (X, T) satisfies almost product property and uniform separation property, then for each nonempty compact subset K of invariant measures, the entropy of saturated set GK satisfies (0.1) hBtop(T, GK) = inf{h(T, µ): µ ∈ K}, where hBtop(T, GK) is Bowen's topological entropy of T on GK, and h(T, µ) is the Kolmogorov-Sinai entropy of µ. In this paper, we investigate topological complexity of GK by replacing Bowen's topological entropy with upper capacity entropy and packing entropy in (0.1) and obtain the following formulas: hUCtop (T, GK) = htop(T, X) and hPtop(T, GK) = sup{h(T, µ): µ ∈ K}, where hUCtop (T, GK) is the upper capacity entropy of T on GK and hPtop(T, GK) is the packing entropy of T on GK. In the proof of these two formulas, uniform separation property is unnecessary. As applications, when (X, T) is a transitive Anosov diffeomorphism on a compact manifold, we show that (1) for any continuous function f, hUCtop (T, Sfop) = htop(T, X) where Sfop is the set of initial values of f-optimal orbits;(2) there exists a Baire generic subset F in the space of continuous functions and an open and dense subset G in the space of Hölder continuous or C1 smooth functions such that for any f ∈ F or f ∈ G, hBtop(T, Sfop) = hPtop(T, Sfop) = 0;(3) for any f ∈ G, hUCtop (T, SfMR) = 0 where SfMR is the set of initial values of measure-recurrent optimal orbits;(4) there exists a dense subset in the space of continuous functions such that for any f in the subset, hBtop(T, Sfop) = hPtop(T, Sfop) > 0. For equilibrium states, we prove that (1) for any continuous function f and equilibrium state µ of f, hUCtop (T, Gµ) = htop(T, X);(2) if f is Hölder continuous and is not cohomologous to a constant, then hBtop(T, Gµ) = hPtop(T, Gµ) fopMSC Codes 37C50, 37B20, 37B05, 37D45, 37C45 © 2020, CC BY.

关键词： Entropy

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GLOBAL-IN-TIME SOLVABILITY AND BLOW-UP FOR A NON-ISOSPECTRAL TWO-COMPONENT CUBIC CAMASSA-HOLM SYSTEM IN A CRITICAL BESOV SPACE

arXiv

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arXiv 2020年

作者： Zhang, Lei Qiao, Zhijun SCHOOL OF MATHEMATICS AND STATISTICS HUBEI KEY LABORATORY OF ENGINEERING MODELING AND SCIENTIFIC COMPUTING HUAZHONG UNIVERSITY OF SCIENCE AND TECHNOLOGY WUHANHUBEI430074 China SCHOOL OF MATHEMATICAL AND STATISTICAL SCIENCES THE UNIVERSITY OF TEXAS RIO GRANDE VALLEY EDINBURGTX78539 United States

In this paper, we prove the global Hadamard well-posedness of strong solutions to anon-isospectral two-component cubic Camassa-Holm system in the critical Besov space B22,1(T). 1 Our results shows that in comparison with the well-known work for classic Camassa-Holm-type equations, the existence of global solution only relies on the L1-integrability of the variable coefficients α(t) and γ(t), but nothing to do with the shape or smoothness of the initial data. The key ingredient of the proof hinges on the careful analysis of the mutual effect among two component forms, the uniform bound of approximate solutions, and several crucial estimates of cubic nonlinearities in low-regularity Besov spaces via the Littlewood-Paley decomposition theory. A reduced case in our results yields the global existence of solutions in a Besov space for two kinds of well-known isospectral peakon system with weakly dissipative terms. Moreover, we derive two kinds of precise blow-up criteria for a strong solution in both critical and non-critical Besov spaces, as well as providing specific characterization for the lower bound of the blow-up time, which implies the global existence with additional conditions on the time-dependent parameters α(t) an γ(t). Copyright © 2020, The Authors. All rights reserved.

关键词： Banach spaces

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

arXiv

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arXiv 2021年

作者： Cao, Waixiang Li, Dongfang Zhang, Zhimin School of Mathematical Sciences Beijing Normal University Beijing100875 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 Beijing Computational Science Research Center Beijing100193 China Department of Mathematics Wayne State University DetroitMI48202 United States

In this paper, we present and analyze an energy-conserving and linearly implicit scheme for solving the nonlinear wave equations. Optimal error estimates in time and superconvergent error estimates in space are established without time-step dependent on the spatial mesh size. The key is to estimate directly the solution bounds in the H2-norm for both the nonlinear wave equation and the corresponding fully discrete scheme, while the previous investigations rely on the temporal-spatial error splitting approach. Numerical examples are presented to confirm energy-conserving properties, unconditional convergence, and optimal error estimates, respectively, of the proposed fully discrete *** Codes 65M60, 65M12, 65N12 © 2021, CC0.

关键词： Wave equations

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The influence of stochastic forcing on strong solutions to the Incompressible Slice Model in 2D bounded domain

arXiv

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arXiv 2020年

作者： Zhang, Lei Shi, Yu Hu, Chaozhu Wang, Weifeng 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 SCHOOL OF SCIENCE WUHAN UNIVERSITY OF TECHNOLOGY HUBEI WUHAN 430070 China SCHOOL OF SCIENCE HUBEI UNIVERSITY OF TECHNOLOGY HUBEI WUHAN 430068 China SCHOOL OF MATHEMATICS AND STATISTICS SOUTH-CENTRAL UNIVERSITY FOR NATIONALITIES WUHAN430074 China

The Cotter-Holm Slice Model (CHSM) was introduced to study the behavior of whether and specifically the formulation of atmospheric fronts, whose prediction is fundamental in meteorology. Considered herein is the influence of stochastic forcing on the Incompressible Slice Model (ISM) in a smooth 2D bounded domain, which can be derived by adapting the Lagrangian function in Hamilton’s principle for CHSM to the Euler-Boussinesq Eady incompressible case. First, we establish the existence and uniqueness of local pathwise solution (probability strong solution) to the ISM perturbed by nonlinear multiplicative stochastic forcing in Banach spaces Wk, p(D) with k > 1 + 1/ p and p ≥ 2. The solution is obtained by introducing suitable cut-off operators applied to the W1,∞-norm of the velocity and temperature fields, using the stochastic compactness method and the Yamada-Watanabe type argument based on the Gyöngy-Krylov characterization of convergence in probability. Then, when the ISM is perturbed by linear multiplicative stochastic forcing and the potential temperature does not vary linearly on the y-direction, we prove that the associated Cauchy problem admits a unique global-in-time pathwise solution with high probability, provided that the initial data is sufficiently small or the diffusion parameter is large enough. The results partially answer the problems left open in Alonso-Orán et al. (Physica D 392:99–118, 2019, pp. 117). Copyright © 2020, The Authors. All rights reserved.

关键词： Banach spaces

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The role of slow manifolds in parameter estimation for a multiscale stochastic system with α-stable Lévy noise

arXiv

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arXiv 2020年

作者： Chao, Ying Wei, Pingyuan Duan, Jinqiao School of Mathematics and Statistics Center for Mathematical Sciences Hubei Key Laboratory for Engineering Modeling and Scientific Computing Huazhong University of Science and Technology Wuhan430074 China Department of Applied Mathematics Illinois Institute of Technology ChicagoIL60616 United States

This work is about parameter estimation for a fast-slow stochastic system with non-Gaussian α-stable Lévy noise. When the observations are only available for slow components, a system parameter is estimated and the accuracy for this estimation is quantified by p-moment with p ∈ (1, α), with the help of a reduced system through random slow manifold approximation. This method provides an advantage in computational complexity and cost, due to the dimension reduction in stochastic systems. To numerically illustrate this method, and to corroborate that the parameter estimator based on the reduced slow system is a good approximation for the true parameter value of the original system, a prototypical example is present. Copyright © 2020, The Authors. All rights reserved.

关键词： Parameter estimation

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Cole-Hopf Transformation Based Lattice Boltzmann Model for One-dimensional Burgers' Equation

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Communications in Theoretical Physics 2018年第3期69卷 329-335页

作者： Xiao-Tong Qi Bao-Chang Shi Zhen-Hua Chai 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

In this paper,we present a Cole-Hopf transformation based lattice Boltzmann(LB)model for solving one-dimensional Burgers'equation,and compared to available LB models,the effect of nonlinear convection term can be *** Chapman-Enskog analysis,it can be found that the converted diffusion equation based on the Cole-Hopf transformation can be recovered correctly from present LB *** numerical tests are also performed to validate the present LB model,and the numerical results show that,similar to previous LB models,the present model also has a second-order convergence rate in space,but it is more accurate than the previous ones.

关键词： Burgers'equation Cole-Hopf transformation lattice Boltzmann model

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A multiple-relaxation-time lattice boltzmann model based four-level finite-difference scheme for one-dimensional diffusion equation

arXiv

引用

arXiv 2020年

作者： Lin, Yuxin Hong, Ning Shi, Baochang Chai, Zhenhua School of Mathematics and Statistics Huazhong University of Science and Technology Wuhan430074 China School of General Education Wuchang University of Technology Wuhan430223 China Hubei Key Laboratory of Engineering Modeling and Scientific Computing Huazhong University of Science and Technology Wuhan430074 China

In this paper, we first present a multiple-relaxation-time lattice Boltzmann (MRT-LB) model for one-dimensional diffusion equation where the D1Q3 (three discrete velocities in one-dimensional space) lattice structure is considered. Then through the theoretical analysis, we derive an explicit four-level finite-difference scheme from this MRT-LB model. The results show that the four-level finite-difference scheme is unconditionally stable, and through adjusting the weight coefficient ω0 and the relaxation parameters s1 and s2 corresponding to the first and second moments, it can also have a sixth-order accuracy in space. Finally, we also test the four-level finite-difference scheme through some numerical simulations, and find that the numerical results are consistent with our theoretical analysis. © 2020, CC BY.

关键词： Finite difference method

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OBSERVABLE SETS, POTENTIALS AND SCHRÖDINGER EQUATIONS

arXiv

引用

arXiv 2020年

作者： Huang, Shanlin Wang, Gengsheng Wang, Ming School of Mathematics and Statistics Hubei Key Laboratory of Engineering Modeling and Scientific Computing Huazhong University of Science and Technology Wuhan430074 China Center for Applied Mathematics Tianjin University Tianjin300072 China School of Mathematics and Physics China University of Geosciences Wuhan430074 China

We characterize observable sets for 1-dim Schrödinger equations in R: i∂tu = (−∂2x + x2m)u (with m ∈ N:= {0, 1, . . . }). More precisely, we obtain what follows: First, when m = 0, E ⊂ R is an observable set at some time if and only if it is thick, namely, there is γ > 0 and L > 0 so that E \ [x, x + L] ≥ γL for each x ∈ R;Second, when m = 1 (m ≥ 2 resp.), E is an observable set at some time (at any time resp.) if and only if it is weakly thick, namely lim |E T[−x, x]| > 0. x x→+∞ From these, we see how potentials x2m affect the observability (including the geometric structures of observable sets and the minimal observable time). Besides, we obtain several supplemental theorems for the above results, in particular, we find that a half line is an observable set at time T > 0 for the above equation with m = 1 if and only if T > π2 .MSC Codes 93B07, 35J10 Copyright © 2020, The Authors. All rights reserved.

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Characterizations of stabilizable sets for some parabolic equations in Rn

arXiv

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arXiv 2020年

We consider the parabolic type equation in Rn: (∂t + H)y(t, x) = 0, (t, x) ∈ (0, ∞) × Rn;y(0, x) ∈ L2(Rn), (0.1) where H can be one of the following operators: (i) a shifted fractional Laplacian;(ii) a shifted Hermite operator;(iii) the Schrödinger operator with some general potentials. We call a subset E ⊂ Rn as a stabilizable set for (0.1), if there is a linear bounded operator K on L2(Rn) so that the semigroup {e−t(H−χEK)}t≥0 is exponentially stable. (Here, χE denotes the characteristic function of E, which is treated as a linear operator on L2(Rn).) This paper presents different geometric characterizations of the stabilizable sets for (0.1) with different H. In particular, when H is a shifted fractional Laplacian, E ⊂ Rn is a stabilizable set for (0.1) if and only if E ⊂ Rn is a thick set, while when H is a shifted Hermite operator, E ⊂ Rn is a stabilizable set for (0.1) if and only if E ⊂ Rn is a set of positive measure. Our results, together with the results on the observable sets for (0.1) obtained in [1, 18, 24, 32], reveal such phenomena: for some H, the class of stabilizable sets contains strictly the class of observable sets, while for some other H, the classes of stabilizable sets and observable sets coincide. Besides, this paper gives some sufficient conditions on the stabilizable sets for (0.1) where H is the Schrödinger operator with some general potentials. Copyright © 2020, The Authors. All rights reserved.

关键词： Partial differential equations

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CONVERGENCE ANALYSIS FOR MINIMUM ACTION METHODS COUPLED WITH A FINITE DIFFERENCE METHOD

arXiv

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arXiv 2021年

作者： Hong, Jialin Jin, Diancong Sheng, Derui Academy of Mathematics and Systems Science Chinese Academy of Sciences Beijing100190 China School of Mathematical Sciences University of Chinese Academy of Sciences Beijing100049 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 Department of Applied Mathematics The Hong Kong Polytechnic University Hung Hom Kowloon Hong Kong

The minimum action method (MAM) is an effective approach to numerically solving minimums and minimizers of Freidlin–Wentzell (F-W) action functionals, which is used to study the most probable transition path and probability of the occurrence of transitions for stochastic differential equations (SDEs) with small noise. In this paper, we focus on MAMs based on a finite difference method, and present the convergence analysis of minimums and minimizers of the discrete F-W action functional. The main result shows that the convergence orders of the minimum of the discrete F-W action functional in the cases of multiplicative noises and additive noises are 1/2 and 1, respectively. Our main result also reveals the convergence of the stochastic θ-method for SDEs with small noise in terms of large deviations. Copyright © 2021, The Authors. All rights reserved.

关键词： Finite difference method

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