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Some Gradient Estimates for Nonlinear Heat-Type Equations on Smooth Metric Measure Spaces with Compact Boundary 具有紧凑边界的光滑度量空间上非线性热型方程的若干梯度估计
IF 0.7 4区 物理与天体物理 Q2 MATHEMATICS, APPLIED Pub Date : 2024-08-05 DOI: 10.1007/s44198-024-00220-1
Abimbola Abolarinwa

In this paper we prove some Hamilton type and Li–Yau type gradient estimates on positive solutions to generalized nonlinear parabolic equations on smooth metric measure space with compact boundary. The geometry of the space in terms of lower bounds on the weighted Bakry–Émery Ricci curvature tensor and weighted mean curvature of the boundary are key in proving generalized local and global gradient estimates. Various applications of these gradient estimates in terms of parabolic Harnack inequalities and Liouville type results are discussed. Further consequences to some specific models informed by the nature of the nonlinearities are highlighted.

本文证明了具有紧凑边界的光滑度量空间上广义非线性抛物方程正解的一些汉密尔顿型和李-尤型梯度估计。以加权 Bakry-Émery Ricci 曲率张量和边界加权平均曲率的下界表示的空间几何是证明广义局部和全局梯度估计的关键。讨论了这些梯度估计在抛物线哈纳克不等式和柳维尔类型结果方面的各种应用。此外,还强调了非线性性质对某些特定模型的影响。
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引用次数: 0
Asymptotic Stability of Two-Dimensional Magnetohydrodynamic System Near the Couette Flow in a Finite Channel 有限通道中库特流附近二维磁流体力学系统的渐近稳定性
IF 0.7 4区 物理与天体物理 Q2 MATHEMATICS, APPLIED Pub Date : 2024-07-29 DOI: 10.1007/s44198-024-00217-w
Fengjie Luo, Limei Li, Liangliang Ma

In this paper, we consider the asymptotic stability of the incompressible two-dimensional(2D) magnetohydrodynamic(MHD) system near the Couette flow at high Reynolds number and high magnetic Reynolds number in a finite channel (Omega =mathbb {T}times [-1,1]). We extend the results of the Navier–Stokes equations (for the previous results see[10]) to the MHD system. We prove that if the initial velocity (V_{in}) and the initial magnetic field (B_{in}) satisfy (Vert left( V_{in}-(y,0), B_{in}-(1,0)right) Vert _{H_{x,y}^{2}}le epsilon text {min}{nu ,mu }^frac{1}{2}) for some small (epsilon) independent of (nu ,mu), then the solution of the system remains within (mathcal{O}(text {min}{nu ,mu }^frac{1}{2})) of Couette flow, and close to Couette flow as (trightarrow infty); the magnetic field remains within (mathcal{O}(text {min}{nu ,mu }^frac{1}{2})) of the (1, 0), and close to (1, 0) as (trightarrow infty).

在本文中,我们考虑了不可压缩二维(2D)磁流体力学(MHD)系统在有限通道(Omega =mathbb {T}times [-1,1])中高雷诺数和高磁雷诺数下靠近库特流的渐近稳定性。我们将纳维-斯托克斯方程的结果(之前的结果见[10])扩展到 MHD 系统。我们证明,如果初始速度(V_{in})和初始磁场(B_{in})满足(Vert left( V_{in}-(y,0), B_{in}-(1,0)right) Vert _{H_{x,y}^{2}}le epsilon text {min}{nu 、对于某个独立于 (nu ,mu)的小 (epsilon),系统的解保持在 (mathcal{O}(text {min}{nu 、Couette flow)的范围内,并且接近于 Couette flow,即 (trightarrow infty);磁场保持在(1,0)的(mathcal{O}(text {min}{nu ,mu }^frac{1}{2})范围内,并接近(1,0)为(trightarrow infty)。
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引用次数: 0
Multi-Symplectic Method for the Two-Component Camassa–Holm (2CH) System 双分量卡马萨-霍尔姆(2CH)系统的多交映法
IF 0.7 4区 物理与天体物理 Q2 MATHEMATICS, APPLIED Pub Date : 2024-07-18 DOI: 10.1007/s44198-024-00216-x
Xiaojian Xi, Weipeng Hu, Bo Tang, Pingwei Deng, Zhijun Qiao

In this paper, the multi-symplectic formulations of the two-component Camassa–Holm system are presented. Both the multi-symplectic structure and two local conservation laws of the generalized two-component Camassa–Holm model are proposed for its first-order canonical form. Then, combining the Fourier pseudo-spectral method in the spatial domain with the midpoint method in the time dimension, the multi-symplectic Fourier pseudo-spectral scheme is constructed for the first-order canonical form. Meanwhile, the discrete scheme of the residuals of the multi-symplectic structure and two local conservation laws are also provided. By using the multi-symplectic Fourier pseudo-spectral scheme, the evolution of one- and two-soliton solutions for the generalized two-component Camassa–Holm model is regained. The structure-preserving properties and the reliability of the numerical scheme are illustrated by the tiny numerical residuals (less than 3.5 × 10−8) of the conservation laws as well as the tiny numerical variations (less than 1 × 10−9) of the amplitudes and the propagating velocities of the solitons.

本文提出了双分量卡马萨-霍姆系统的多交映公式。针对广义双分量 Camassa-Holm 模型的一阶典型形式,提出了该模型的多交映结构和两个局部守恒定律。然后,结合空间域的傅立叶伪谱法和时间维的中点法,构建了一阶典型形式的多交映傅立叶伪谱方案。同时,还提供了多交映结构残差的离散方案和两个局部守恒定律。通过使用多折射傅立叶伪谱方案,重新获得了广义双分量卡马萨-霍姆模型的单孑子和双孑子解的演化。守恒定律的微小数值残差(小于 3.5 × 10-8)以及孤子振幅和传播速度的微小数值变化(小于 1 × 10-9)说明了数值方案的结构保持特性和可靠性。
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引用次数: 0
Asymptotic Behavior and Numerical Simulations of a Conservative Phase-Field Model with Two Temperatures 双温保守相场模型的渐近行为和数值模拟
IF 0.7 4区 物理与天体物理 Q2 MATHEMATICS, APPLIED Pub Date : 2024-07-18 DOI: 10.1007/s44198-024-00209-w
Brice Landry Doumbe Bangola, Mohamed Ali Ipopa, Armel Andami Ovono

One of the types of problem that has attracted the attention of mathematicians in recent years is the phase field system. The field of application includes materials science, where phenomena such as phase separation in alloys, crystal formation and thermal welding are legion. Among these phase transition systems, the family of conservative systems is very popular with industry. Indeed, minimising losses in production systems is a major issue for their profitability. In this paper, we study the well-posedness of the formulation and the asymptotic behaviour of the solutions, by proving the existence of a finite-dimensional global attractor for a conservative variant of the two-temperature phase field system with homogeneous Neumann boundary conditions. The inclusion of two temperatures in the definition of the enthalpy of the system is a necessity in the case of a non-simple material. In a simple material, once the phase-change temperature has been reached, the temperature of the system remains constant until the material has completely changed state. This is not true in the case of a non-simple material, where an increase in the temperature of the system is observed even after the phase-change temperature has been reached. To conclude the work, we present a method for numerically approximating the solution and carry out some numerical tests.

相场系统是近年来备受数学家关注的问题类型之一。其应用领域包括材料科学,在材料科学中,合金中的相分离、晶体形成和热焊接等现象层出不穷。在这些相变系统中,保守系统家族深受工业界欢迎。事实上,最大限度地减少生产系统中的损失是其盈利能力的一个主要问题。在本文中,我们通过证明具有同质 Neumann 边界条件的双温相场系统的保守变体存在有限维全局吸引子,研究了公式的拟合性和解的渐近行为。在非简单材料的情况下,必须在系统焓的定义中包含两个温度。在简单材料中,一旦达到相变温度,系统的温度就会保持不变,直到材料完全改变状态。而非简单材料则不然,即使达到了相变温度,系统的温度也会升高。最后,我们提出了一种数值近似解法,并进行了一些数值测试。
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引用次数: 0
Probabilistic Degenerate Fubini Polynomials Associated with Random Variables 与随机变量相关的概率退化富比尼多项式
IF 0.7 4区 物理与天体物理 Q2 MATHEMATICS, APPLIED Pub Date : 2024-07-10 DOI: 10.1007/s44198-024-00210-3
Rongrong Xu, Taekyun Kim, Dae San Kim, Yuankui Ma

Let Y be a random variable such that the moment generating function of Y exists in a neighborhood of the origin. The aim of this paper is to study probabilistic versions of the degenerate Fubini polynomials and the degenerate Fubini polynomials of order r, namely the probabilisitc degenerate Fubini polynomials associated with Y and the probabilistic degenerate Fubini polynomials of order r associated with Y. We derive some properties, explicit expressions, certain identities and recurrence relations for those polynomials. As special cases of Y, we treat the gamma random variable with parameters (alpha ,beta > 0), the Poisson random variable with parameter (alpha > 0), and the Bernoulli random variable with probability of success p.

设 Y 是随机变量,且 Y 的矩生成函数存在于原点附近。本文旨在研究退化富比尼多项式和 r 阶退化富比尼多项式的概率版本,即与 Y 相关的概率退化富比尼多项式和与 Y 相关的概率退化富比尼多项式。作为 Y 的特例,我们处理了参数为 (alpha ,beta > 0) 的伽马随机变量、参数为 (alpha > 0) 的泊松随机变量和成功概率为 p 的伯努利随机变量。
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引用次数: 0
A General Coupled Derivative Nonlinear Schrödinger System: Darboux Transformation and Soliton Solutions 一般耦合衍生非线性薛定谔系统:达尔布变换和孤子解
IF 0.7 4区 物理与天体物理 Q2 MATHEMATICS, APPLIED Pub Date : 2024-07-09 DOI: 10.1007/s44198-024-00212-1
Yonghui Kuang

In this work we present a general coupled derivative nonlinear Schrödinger system. We construct the corresponding N-fold Darboux transform and generalized Darboux transform. Under this construction, we give different soliton solutions and plot their figures describing the soliton characteristics and dynamical behaviors, including higher-order soliton and rouge wave solution etc.

在这项工作中,我们提出了一个一般耦合导数非线性薛定谔系统。我们构建了相应的 N 折达布克斯变换和广义达布克斯变换。在这一构造下,我们给出了不同的孤子解,并绘制了描述孤子特性和动力学行为的图形,包括高阶孤子和胭脂波解等。
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引用次数: 0
On Common Borel Direction of Entire Function f and Its q-Difference Operator 论全函数 f 及其 q-差分算子的共 Borel 方向
IF 0.7 4区 物理与天体物理 Q2 MATHEMATICS, APPLIED Pub Date : 2024-07-02 DOI: 10.1007/s44198-024-00205-0
Zhigao Qin, Jianren Long, Ling Wang

The common Borel direction of entire function f(z) and its q-difference operator is studied by using Nevanlinna theory. Some conditions of the existence of common Borel direction for entire function f(z) and its q-difference operator are obtained in this paper.

本文利用 Nevanlinna 理论研究了全函数 f(z) 及其 q-差分算子的共 Borel 方向。本文获得了全函数 f(z) 及其 q 差算子的共 Borel 方向存在的一些条件。
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引用次数: 0
On the Temporal Tweezing of Cavity Solitons 关于腔隙孤子的时空镊合
IF 0.7 4区 物理与天体物理 Q2 MATHEMATICS, APPLIED Pub Date : 2024-07-02 DOI: 10.1007/s44198-024-00193-1
Julia Rossi, Sathyanarayanan Chandramouli, Ricardo Carretero-González, Panayotis G. Kevrekidis

Motivated by the work of Jang et al., Nat Commun 6:7370 (2015), where the authors experimentally tweeze cavity solitons in a passive loop of optical fiber, we study the amenability to tweezing of cavity solitons as the properties of a localized tweezer are varied. The system is modeled by the Lugiato-Lefever equation, a variant of the complex Ginzburg-Landau equation. We produce an effective, localized, trapping tweezer potential by assuming a Gaussian phase-modulation of the holding beam. The potential for tweezing is then assessed as the total (temporal) displacement and speed of the tweezer are varied, and corresponding phase diagrams are presented. As the relative speed of the tweezer is increased we find two possible dynamical scenarios: successful tweezing and release of the cavity soliton. We also deploy a non-conservative variational approximation (NCVA) based on a Lagrangian description which reduces the original dissipative partial differential equation to a set of coupled ordinary differential equations for the cavity soliton parameters. We illustrate the ability of the NCVA to accurately predict the separatrix between successful and failed tweezing. This showcases the versatility of the NCVA to provide a low-dimensional description of the experimental realization of the temporal tweezing.

在 Jang 等人的研究成果(Nat Commun 6:7370 (2015))的激励下,我们研究了当局部镊子的特性发生变化时,空穴孤子是否可以被镊子镊断。该系统由 Lugiato-Lefever 方程建模,它是复杂金兹堡-兰道方程的一个变体。我们通过假设保持光束的高斯相位调制,产生了一个有效的局部诱捕镊子电势。然后,随着镊子总(时间)位移和速度的变化,对镊子的电位进行评估,并给出相应的相图。随着镊子相对速度的增加,我们发现了两种可能的动态情况:成功镊取和释放空穴孤子。我们还采用了基于拉格朗日描述的非保守变分近似(NCVA)方法,将原来的耗散偏微分方程简化为空穴孤子参数的一组耦合常微分方程。我们展示了 NCVA 准确预测成功和失败镊合之间分离矩阵的能力。这展示了 NCVA 的多功能性,它提供了时空镊合实验实现的低维描述。
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引用次数: 0
Existence of Regular Solutions for a Class of Incompressible Non-Newtonian MHD Equations Coupled to the Heat Equation 一类与热方程耦合的不可压缩非牛顿多流体力学方程常规解的存在性
IF 0.7 4区 物理与天体物理 Q2 MATHEMATICS, APPLIED Pub Date : 2024-07-02 DOI: 10.1007/s44198-024-00211-2
Rina Su, Changjia Wang

We consider a system of PDE’s describing the steady flow of an electrically conducting fluid in the presence of a magnetic field. The system of governing equations composes of the stationary non-Newtonian incompressible MHD equations coupled to the heat equation wherein the influence of buoyancy is taken into account in the momentum equation and the Joule heating and viscous heating terms are included. We proved the existence of (C^{1,gamma }({bar{Omega }})times W^{2,r}(Omega )times W^{2,2}{(Omega )}) solutions of the systems for (1< p<2) corresponding to a small data and we show that this solution is unique in case (6/5< p < 2). Moreover, we also proved the higher regularity properties of this solution.

我们考虑了一个描述导电流体在磁场作用下稳定流动的 PDE 系统。支配方程系统由静态非牛顿不可压缩多流体力学方程和热方程组成,其中动量方程考虑了浮力的影响,焦耳加热和粘性加热项也包括在内。我们证明了在 (1< p<2) 对应于一个小数据的系统中存在 (C^{1,gamma }({bar{Omega }})times W^{2,r}(Omega )times W^{2,2}{(Omega )}) 解,并证明了在 (6/5< p< 2) 的情况下这个解是唯一的。此外,我们还证明了这个解的高正则性。
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引用次数: 0
Spectral Solutions of Specific Singular Differential Equations Using A Unified Spectral Galerkin-Collocation Algorithm 使用统一谱 Galerkin-Collocation 算法的特定奇异微分方程谱解法
IF 0.7 4区 物理与天体物理 Q2 MATHEMATICS, APPLIED Pub Date : 2024-06-27 DOI: 10.1007/s44198-024-00194-0
H. M. Ahmed, W. M. Abd-Elhameed

This paper presents a novel numerical approach to addressing three types of high-order singular boundary value problems. We introduce and consider three modified Chebyshev polynomials (CPs) of the third kind as proposed basis functions for these problems. We develop new derivative operational matrices for the three modified CPs of the third kind by deriving formulas for their first derivatives. Our approach follows a unified method for numerically handling singular differential equations (DEs). To transform these equations into algebraic systems suitable for numerical treatment, we employ the collocation method in combination with the introduced operational matrices of derivatives of the modified CPs of the third kind. We address the convergence examination for the three expansions in a unified manner. We present numerous numerical examples to demonstrate the accuracy and efficiency of our unified numerical approach.

本文提出了一种解决三类高阶奇异边界值问题的新颖数值方法。我们引入并考虑了三个修正的第三类切比雪夫多项式 (CP),作为这些问题的拟议基函数。我们通过推导这三个修正的第三类切比雪夫多项式的一阶导数公式,为它们建立了新的导数运算矩阵。我们的方法遵循数值处理奇异微分方程(DE)的统一方法。为了将这些方程转化为适合数值处理的代数系统,我们结合引入的修正第三类 CP 的导数运算矩阵,采用了搭配法。我们以统一的方式解决了三种展开的收敛性问题。我们列举了大量数值示例,以证明我们的统一数值方法的准确性和高效性。
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引用次数: 0
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Journal of Nonlinear Mathematical Physics
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