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The optimal large time behavior of 3D quasilinear hyperbolic equations with nonlinear damping 具有非线性阻尼的三维准线性双曲方程的最佳大时间行为
IF 1 4区 数学 Q1 MATHEMATICS Pub Date : 2024-02-14 DOI: 10.1007/s10473-024-0317-6
Han Wang, Yinghui Zhang

We are concerned with the large-time behavior of 3D quasilinear hyperbolic equations with nonlinear damping. The main novelty of this paper is two-fold. First, we prove the optimal decay rates of the second and third order spatial derivatives of the solution, which are the same as those of the heat equation, and in particular, are faster than ones of previous related works. Second, for well-chosen initial data, we also show that the lower optimal L2 convergence rate of the k (∈ [0, 3])-order spatial derivatives of the solution is ({(1 + t)^{ - {{3 + 2k} over 4}}}). Therefore, our decay rates are optimal in this sense. The proofs are based on the Fourier splitting method, low-frequency and high-frequency decomposition, and delicate energy estimates.

我们关注的是具有非线性阻尼的三维准线性双曲方程的大时间行为。本文的主要新颖之处有两方面。首先,我们证明了解的二阶和三阶空间导数的最佳衰减率,其衰减率与热方程的衰减率相同,特别是比以往相关研究的衰减率更快。其次,对于精心选择的初始数据,我们还证明了解的 k (∈ [0, 3])阶空间导数的较低最优 L2 收敛率是({(1 + t)^{ - {{3 + 2k}over 4}}})。因此,我们的衰减率在这个意义上是最优的。证明基于傅立叶分裂法、低频和高频分解以及微妙的能量估计。
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引用次数: 0
On a universal inequality for approximate phase isometries 关于近似相等距的普遍不等式
IF 1 4区 数学 Q1 MATHEMATICS Pub Date : 2024-02-14 DOI: 10.1007/s10473-024-0303-z
Duanxu Dai, Haixin Que, Longfa Sun, Bentuo Zheng

Let X and Y be two normed spaces. Let ({cal U}) be a non-principal ultrafilter on ℕ. Let g: XY be a standard ε-phase isometry for some ε ≥ 0, i.e., g(0) = 0, and for all u, v ϵ X,

$$|,,|,||g(u) + g(v)|| pm ||g(u) - g(v)||,| - |,||u + v|| pm ||u - v||,|,,|, le varepsilon .$$

The mapping g is said to be a phase isometry provided that ε = 0. In this paper, we show the following universal inequality of g: for each ({u^ * } in {w^ * } - exp ,,||{u^ * }||{B_{{X^ * }}}), there exist a phase function ({sigma _{{u^ * }}}:X to { - 1,1} ) and φ ϵ Y* with (||varphi || = ||{u^ * }|| equiv alpha ) satisfying that

$$|leftlangle {{u^ * },u} rightrangle - {sigma _{{u^ * }}}(u)leftlangle {varphi ,g(u)} rightrangle | le {5 over 2}varepsilon alpha ,,,,{rm{for}},{rm{all}},u in X.$$

In particular, let X be a smooth Banach space. Then we show the following: (1) the universal inequality holds for all u* ∈ X*; (2) the constant ({5 over 2}) can be reduced to ({3 over 2}) provided that Y* is strictly convex; (3) the existence of such a g implies the existence of a phase isometry Θ: XY such that (Theta (u) = mathop {lim }limits_{n,{cal U}} {{g(nu)} over n}) provided that Y** has the w*-Kadec-Klee property (for example, Y is both reflexive and locally uniformly convex).

让 X 和 Y 是两个规范空间。让 ({cal U}) 是ℕ上的一个非主超滤波器。对于某个 ε ≥ 0,让 g: X → Y 是一个标准的 ε 相等距,即g(0) = 0,并且对于所有 u、v ϵ X,$$||,,|,||g(u) + g(v)|| pm |g(u) - g(v)||,| - |,|u + v|| pm ||u - v||,|,,|,le varepsilon .$$只要 ε = 0,映射 g 就被称为相等几何。在本文中,我们证明了 g 的以下普遍不等式:对于每个 ({u^ * } in {w^ * } - exp ,,||{u^ * }||{B_{{X^ * }}}), 都存在一个相位函数 ({sigma _{u^ * }}}:X to { - 1,1} ) and φ ϵ Y* with (||varphi || = ||{u^ * }|| equiv alpha ) satisfying that $$|leftlangle {{u^ * }、u} - {sigma _{{u^ * }}}(u)leftlangle {varphi ,g(u)} rightrangle | le {5 over 2}varepsilon alpha ,,,{rm{for},{rm{all}},u in X.$$特别地,让 X 是一个光滑的巴拿赫空间。然后我们证明以下几点:(1) 对于所有u*∈X*,普遍不等式都成立;(2) 常量({5 over 2})可以简化为({3 over 2}),前提是Y*是严格凸的;(3) 这样一个g的存在意味着相等几何Θ的存在:X → Y such that (θ (u) = mathop {lim }limits_{n,{cal U}}{{g(nu)}overn}/),条件是 Y** 具有 w*-Kadec-Klee 属性(例如,Y 既是反折的,又是局部均匀凸的)。
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引用次数: 0
The global existence and analyticity of a mild solution to the 3D regularized MHD equations 三维正则化多流体力学方程温和解的全局存在性和解析性
IF 1 4区 数学 Q1 MATHEMATICS Pub Date : 2024-02-14 DOI: 10.1007/s10473-024-0311-z
Cuntao Xiao, Hua Qiu, Zheng-an Yao

In this paper, we study the three-dimensional regularized MHD equations with fractional Laplacians in the dissipative and diffusive terms. We establish the global existence of mild solutions to this system with small initial data. In addition, we also obtain the Gevrey class regularity and the temporal decay rate of the solution.

本文研究了耗散项和扩散项中带有分数拉普拉斯的三维正则化 MHD 方程。我们确定了该系统在较小初始数据下温和解的全局存在性。此外,我们还得到了解的 Gevrey 类正则性和时间衰减率。
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引用次数: 0
The limiting profile of solutions for semilinear elliptic systems with a shrinking self-focusing core 具有收缩自聚焦核心的半线性椭圆系统解的极限轮廓
IF 1 4区 数学 Q1 MATHEMATICS Pub Date : 2024-02-06 DOI: 10.1007/s10473-024-0212-1
Ke Jin, Ying Shi, Huafei Xie

In this paper, we consider the semilinear elliptic equation systems

$$left{ {matrix{{ - Delta u + u = alpha {Q_n}(x)|u{|^{alpha - 2}}|v{|^beta }u,,{rm{in}},{mathbb{R}^N},} hfill cr { - Delta v + v = beta Q(x)|u{|^alpha }|v{|^{beta - 2}}v,,,,{rm{in}},{mathbb{R}^N},} hfill cr } } right.$$

where (Ngeqslant 3,,,alpha ,,,beta > 1,,alpha + beta < {2^ * },,{2^ * } = {{2N} over {N - 2}}) and Qn are bounded given functions whose self-focusing cores {x ∈ ℍNQn(x) > 0} shrink to a set with finitely many points as n → ∞. Motivated by the work of Fang and Wang [13], we use variational methods to study the limiting profile of ground state solutions which are concentrated at one point of the set with finitely many points, and we build the localized concentrated bound state solutions for the above equation systems.

在本文中,我们考虑半线性椭圆方程系统 $$left{ {matrix{{ -Delta u + u = alpha {Q_n}(x)|u{|^{alpha - 2}}|v{|^beta }u、,{rm{in}},{mathbb{R}^N},} hfill cr { - Delta v + v = beta Q(x)|u{|^alpha }|v{|^{beta - 2}}v,,{rm{in}},{mathbb{R}^N},} hfill cr }}right.$$where (Ngeqslant 3,,,alpha ,,beta > 1,,alpha + beta < {2^ * },,{2^ * })= {{2N}over{N-2}})和 Qn 都是有界给定函数,当 n → ∞ 时,它们的自聚焦核心 {x∈ ℍNQn(x) > 0} 会收缩为具有有限个点的集合。受方和王[13]的研究启发,我们利用变分法研究了集中于有限多点集合中某一点的基态解的极限轮廓,并建立了上述方程系统的局部集中边界解。
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引用次数: 0
The extremes of dependent chi-processes attracted by the Brown-Resnick process 布朗-雷斯尼克过程所吸引的依存戚过程的极值
IF 1 4区 数学 Q1 MATHEMATICS Pub Date : 2024-02-06 DOI: 10.1007/s10473-024-0217-9
Junjie Sun, Zhongquan Tan

Motivated by some recent works on the topic of the Brown-Resnick process, we study the functional limit theorem for normalized pointwise maxima of dependent chi-processes. It is proven that the properly normalized pointwise maxima of those processes are attracted by the Brown-Resnick process.

受最近一些关于布朗-雷斯尼克过程的著作的启发,我们研究了戚依过程的归一化点最大值的函数极限定理。研究证明,这些过程的适当归一化最大点受到布朗-雷斯尼克过程的吸引。
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引用次数: 0
The sparse representation related with fractional heat equations 与分数热方程有关的稀疏表示
IF 1 4区 数学 Q1 MATHEMATICS Pub Date : 2024-02-06 DOI: 10.1007/s10473-024-0211-2
Wei Qu, Tao Qian, Ieng Tak Leong, Pengtao Li

This study introduces a pre-orthogonal adaptive Fourier decomposition (POAFD) to obtain approximations and numerical solutions to the fractional Laplacian initial value problem and the extension problem of Caffarelli and Silvestre (generalized Poisson equation). As a first step, the method expands the initial data function into a sparse series of the fundamental solutions with fast convergence, and, as a second step, makes use of the semigroup or the reproducing kernel property of each of the expanding entries. Experiments show the effectiveness and efficiency of the proposed series solutions.

本研究介绍了一种预正交自适应傅里叶分解(POAFD)方法,用于获得分数拉普拉斯初值问题和 Caffarelli 与 Silvestre 扩展问题(广义泊松方程)的近似值和数值解。第一步,该方法将初始数据函数扩展为基本解的稀疏序列,并快速收敛;第二步,利用每个扩展项的半群或重现核属性。实验证明了所提出的数列解的有效性和效率。
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引用次数: 0
The long time behavior of the fractional Ornstein-Uhlenbeck process with linear self-repelling drift 具有线性自斥漂移的分数奥恩斯坦-乌伦贝克过程的长期行为
IF 1 4区 数学 Q1 MATHEMATICS Pub Date : 2024-02-06 DOI: 10.1007/s10473-024-0216-x
Xiaoyu Xia, Litan Yan, Qing Yang

Let BH be a fractional Brownian motion with Hurst index ({1 over 2} le H < 1). In this paper, we consider the equation (called the Ornstein-Uhlenbeck process with a linear self-repelling drift)

$${rm{d}}X_t^H = dB_t^H + sigma X_t^H{rm{d}}t + nu {rm{d}}t - theta left( {int_0^t {(X_{^t}^H - X_s^H){rm{d}}s} } right){rm{d}}t,$$

where θ < 0, σ, v ∈ ℝ. The process is an analogue of self-attracting diffusion (Cranston, Le Jan. Math Ann, 1995, 303: 87–93). Our main aim is to study the large time behaviors of the process. We show that the solution XH diverges to infinity as t tends to infinity, and obtain the speed at which the process XH diverges to infinity as t tends to infinity.

假设 BH 是一个分数布朗运动,其赫斯特指数为({1 over 2} le H < 1 )。在本文中,我们考虑的方程(称为具有线性自斥漂移的 Ornstein-Uhlenbeck 过程)为 $${rm{d}}X_t^H = dB_t^H + sigma X_t^H{rm{d}}t + nu {rm{d}}t - theta left( {int_0^t {(X_{^t}^H - X_s^H){rm{d}}s}}.}right){rm{d}}t,$$ 其中 θ < 0, σ, v∈ ℝ。这个过程类似于自吸引扩散(Cranston, Le Jan. Math Ann, 1995, 303: 87-93)。我们的主要目的是研究该过程的大时间行为。我们证明了解 XH 在 t 趋于无穷大时发散到无穷大,并得到了过程 XH 在 t 趋于无穷大时发散到无穷大的速度。
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引用次数: 0
A generalized scalar auxiliary variable method for the time-dependent Ginzburg-Landau equations 时变金兹堡-朗道方程的广义标量辅助变量法
IF 1 4区 数学 Q1 MATHEMATICS Pub Date : 2024-02-06 DOI: 10.1007/s10473-024-0215-y
Zhiyong Si

This paper develops a generalized scalar auxiliary variable (SAV) method for the time-dependent Ginzburg-Landau equations. The backward Euler method is used for discretizing the temporal derivative of the time-dependent Ginzburg-Landau equations. In this method, the system is decoupled and linearized to avoid solving the non-linear equation at each step. The theoretical analysis proves that the generalized SAV method can preserve the maximum bound principle and energy stability, and this is confirmed by the numerical result, and also shows that the numerical algorithm is stable.

本文针对时变金兹堡-朗道方程开发了一种广义标量辅助变量(SAV)方法。后向欧拉法用于对时变金兹堡-朗道方程的时变导数进行离散化。在该方法中,系统被解耦和线性化,以避免在每一步都求解非线性方程。理论分析证明广义 SAV 方法可以保持最大约束原理和能量稳定性,数值结果也证实了这一点,同时还表明数值算法是稳定的。
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引用次数: 0
The absence of singular continuous spectrum for perturbed Jacobi operators 扰动雅可比算子不存在奇异连续谱
IF 1 4区 数学 Q1 MATHEMATICS Pub Date : 2024-02-06 DOI: 10.1007/s10473-024-0208-x
Zhengqi Fu, Xiong Li

This paper is mainly about the spectral properties of a class of Jacobi operators

$$({H_{c,b}}u)(n) = {c_n}u(n + 1) + {c_{n - 1}}u(n - 1) + {b_n}u(n),$$

where ∣cn − 1∣ = O(n−α) and bn = O(n−1). We will show that, for α ≥ 1, the singular continuous spectrum of the operator is empty.

本文主要讨论一类雅可比算子$$({H_{c,b}}u)(n) = {c_n}u(n + 1) + {c_{n - 1}}u(n - 1) + {b_n}u(n)的谱性质,其中∣cn - 1∣ = O(n-α),bn = O(n-1)。我们将证明,对于 α ≥ 1,算子的奇异连续谱是空的。
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引用次数: 0
A flexible objective-constraint approach and a new algorithm for constructing the Pareto front of multiobjective optimization problems 构建多目标优化问题帕累托前沿的灵活目标约束方法和新算法
IF 1 4区 数学 Q1 MATHEMATICS Pub Date : 2024-02-06 DOI: 10.1007/s10473-024-0218-8
N. Hoseinpoor, M. Ghaznavi

In this article, a novel scalarization technique, called the improved objective-constraint approach, is introduced to find efficient solutions of a given multiobjective programming problem. The presented scalarized problem extends the objective-constraint problem. It is demonstrated that how adding variables to the scalarized problem, can lead to find conditions for (weakly, properly) Pareto optimal solutions. Applying the obtained necessary and sufficient conditions, two algorithms for generating the Pareto front approximation of bi-objective and three-objective programming problems are designed. These algorithms are easy to implement and can achieve an even approximation of (weakly, properly) Pareto optimal solutions. These algorithms can be generalized for optimization problems with more than three criterion functions, too. The effectiveness and capability of the algorithms are demonstrated in test problems.

本文介绍了一种新颖的标量化技术,即改进的目标-约束方法,用于寻找给定多目标程序设计问题的高效解决方案。提出的标量化问题扩展了目标-约束问题。它证明了如何在标量化问题中添加变量,从而找到(弱的、适当的)帕累托最优解的条件。应用所获得的必要条件和充分条件,设计了两种算法,用于生成双目标和三目标编程问题的帕累托前沿近似值。这些算法易于实现,并能实现对(弱、适当)帕累托最优解的均匀逼近。这些算法也可推广用于具有三个以上准则函数的优化问题。这些算法的有效性和能力在测试问题中得到了证明。
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引用次数: 0
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Acta Mathematica Scientia
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