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The horizontal magnetic primitive equations approximation of the anisotropic MHD equations in a thin 3D domain 薄三维域中各向异性 MHD 方程的水平磁性基元方程近似值
IF 1.7 2区 数学 Q2 MATHEMATICS, APPLIED Pub Date : 2024-06-11 DOI: 10.1088/1361-6544/ad5131
Jie Zhang and Wenjun Liu
In this paper, we give a rigorous justification for the derivation of the primitive equations with only horizontal viscosity and magnetic diffusivity (PEHM) as the small aspect ratio limit of the incompressible three-dimensional scaled horizontal viscous magnetohydrodynamics (SHMHD) equations. Choosing an aspect ratio parameter , we consider the case that if the orders of the horizontal and vertical viscous coefficients µ and ν are and , and the orders of magnetic diffusion coefficients κ and σ are and , with α > 2, then the limiting system is the PEHM as ɛ goes to zero. For -initial data, we prove that the global weak solutions of the SHMHD equations converge strongly to the local-in-time strong solutions of the PEHM, as ɛ tends to zero. For -initial data with additional regularity , we slightly improve the well-posed result in Cao et al (2017 J. Funct. Anal.272 4606–41) to extend the local-in-time strong convergences to the global-in-time one. For -initial data, we show that the global-in-time strong solutions of the SHMHD equations converge strongly to the global-in-time strong solutions of the PEHM, as ɛ goes to zero. Moreover, the rate of convergence is of the order , where with . It should be noted that in contrast to the case α > 2, the case α = 2 has been investigated by Du and Li in (2022 arXiv:2208.01985), in which they consider the primitive equations with magnetic field (PEM) and the rate of global-in-time convergences is of the order .
在本文中,我们给出了推导仅有水平粘性和磁扩散性的原始方程(PEHM)作为不可压缩三维缩放水平粘性磁流体动力学(SHMHD)方程的小纵横比极限的严格理由。选择纵横比参数为 ,我们考虑的情况是,如果水平和垂直粘性系数 µ 和 ν 的阶数分别为 和 ,磁扩散系数 κ 和 σ 的阶数分别为 和 ,且 α > 2,那么当 ɛ 变为零时,极限系统就是 PEHM。对于初始数据,我们证明当 ɛ 趋于零时,SHMHD方程的全局弱解强烈收敛于PEHM的局部实时强解。对于具有额外正则性的-初始数据,我们略微改进了 Cao 等(2017 J. Funct. Anal.272,4606-41)中的好求结果,将局部时间内强收敛扩展为全局时间内强收敛。对于-初始数据,我们证明当ɛ变为零时,SHMHD方程的全局-时间强解强烈收敛于PEHM的全局-时间强解。值得注意的是,与 α > 2 的情况相反,杜和李在(2022 arXiv:2208.01985)中研究了 α = 2 的情况,其中他们考虑了带磁场的基元方程(PEM),全局时间内收敛率为 。
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引用次数: 0
Existence and multiplicity of peaked bound states for nonlinear Schrödinger equations on metric graphs 度量图上非线性薛定谔方程峰值边界态的存在性和多重性
IF 1.7 2区 数学 Q2 MATHEMATICS, APPLIED Pub Date : 2024-06-07 DOI: 10.1088/1361-6544/ad5133
Haixia Chen, Simone Dovetta, Angela Pistoia, Enrico Serra
We establish existence and multiplicity of one-peaked and multi-peaked positive bound states for nonlinear Schrödinger equations on general compact and noncompact metric graphs. Precisely, we construct solutions concentrating at every vertex of odd degree greater than or equal to 3. We show that these solutions are not minimizers of the associated action and energy functionals. To the best of our knowledge, this is the first work exhibiting solutions concentrating at vertices with degree different than 1. The proof is based on a suitable Ljapunov–Schmidt reduction.
我们为一般紧凑和非紧凑度量图上的非线性薛定谔方程建立了单峰和多峰正约束状态的存在性和多重性。确切地说,我们构建了集中于每个奇数度大于或等于 3 的顶点的解。我们证明了这些解不是相关作用和能量函数的最小值。据我们所知,这是第一部展示集中在阶数大于 1 的顶点上的解的著作。证明基于合适的 Ljapunov-Schmidt 还原。
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引用次数: 0
Stochastic electromechanical bidomain model * 随机机电双域模型 *
IF 1.7 2区 数学 Q2 MATHEMATICS, APPLIED Pub Date : 2024-06-07 DOI: 10.1088/1361-6544/ad5132
M Bendahmane, K H Karlsen, F Mroué
We analyse a system of nonlinear stochastic partial differential equations (SPDEs) of mixed elliptic-parabolic type that models the propagation of electric signals and their effect on the deformation of cardiac tissue. The system governs the dynamics of ionic quantities, intra and extra-cellular potentials, and linearised elasticity equations. We introduce a framework called the active strain decomposition, which factors the material gradient of deformation into an active (electrophysiology-dependent) part and an elastic (passive) part, to capture the coupling between muscle contraction, biochemical reactions, and electric activity. Under the assumption of linearised elastic behaviour and a truncation of the nonlinear diffusivities, we propose a stochastic electromechanical bidomain model, and establish the existence of weak solutions for this model. To prove existence through the convergence of approximate solutions, we employ a stochastic compactness method in tandem with an auxiliary non-degenerate system and the Faedo–Galerkin method. We utilise a stochastic adaptation of de Rham’s theorem to deduce the weak convergence of the pressure approximations.
我们分析了一个椭圆-解析混合型非线性随机偏微分方程(SPDE)系统,该系统模拟电信号的传播及其对心脏组织变形的影响。该系统控制离子量、细胞内外电位和线性化弹性方程的动态。我们引入了一个称为主动应变分解的框架,它将变形的材料梯度分为主动(依赖电生理学)部分和弹性(被动)部分,以捕捉肌肉收缩、生化反应和电活动之间的耦合。在线性化弹性行为和非线性扩散截断的假设下,我们提出了随机机电双域模型,并建立了该模型的弱解存在性。为了通过近似解的收敛性来证明存在性,我们采用了随机紧凑性方法、辅助非退化系统和 Faedo-Galerkin 方法。我们利用德拉姆定理的随机调整来推导压力近似的弱收敛性。
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引用次数: 0
Consistent spectral approximation of Koopman operators using resolvent compactification 利用解析压缩实现库普曼算子的一致谱近似
IF 1.7 2区 数学 Q2 MATHEMATICS, APPLIED Pub Date : 2024-06-03 DOI: 10.1088/1361-6544/ad4ade
Dimitrios Giannakis and Claire Valva
Koopman operators and transfer operators represent dynamical systems through their induced linear action on vector spaces of observables, enabling the use of operator-theoretic techniques to analyze nonlinear dynamics in state space. The extraction of approximate Koopman or transfer operator eigenfunctions (and the associated eigenvalues) from an unknown system is nontrivial, particularly if the system has mixed or continuous spectrum. In this paper, we describe a spectrally accurate approach to approximate the Koopman operator on L2 for measure-preserving, continuous-time systems via a ‘compactification’ of the resolvent of the generator. This approach employs kernel integral operators to approximate the skew-adjoint Koopman generator by a family of skew-adjoint operators with compact resolvent, whose spectral measures converge in a suitable asymptotic limit, and whose eigenfunctions are approximately periodic. Moreover, we develop a data-driven formulation of our approach, utilizing data sampled on dynamical trajectories and associated dictionaries of kernel eigenfunctions for operator approximation. The data-driven scheme is shown to converge in the limit of large training data under natural assumptions on the dynamical system and observation modality. We explore applications of this technique to dynamical systems on tori with pure point spectra and the Lorenz 63 system as an example with mixing dynamics.
库普曼(Koopman)算子和转移算子通过其对观测变量向量空间的诱导线性作用来表示动态系统,从而使算子理论技术得以用于分析状态空间中的非线性动力学。从未知系统中提取近似库普曼或转移算子特征函数(及相关特征值)并非易事,尤其是在系统具有混合谱或连续谱的情况下。在本文中,我们介绍了一种频谱精确的方法,通过对生成器的解析量进行 "压缩",来近似 L2 上的保度量连续时间系统的库普曼算子。这种方法利用核积分算子,通过具有紧凑解析力的倾斜-关节算子族来近似倾斜-关节库普曼生成器,这些算子族的谱度量收敛于合适的渐近极限,其特征函数近似为周期性的。此外,我们还开发了一种数据驱动的方法,利用动态轨迹上的数据采样和相关的核特征函数字典进行算子近似。研究表明,在动态系统和观测模式的自然假设下,数据驱动方案在大量训练数据的极限情况下会收敛。我们探讨了这一技术在具有纯点谱的环上动力学系统中的应用,并以具有混合动力学的洛伦兹 63 系统为例。
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引用次数: 0
Exact global control of small divisors in rational normal form * 有理正则表达式中小除数的精确全局控制 *
IF 1.7 2区 数学 Q2 MATHEMATICS, APPLIED Pub Date : 2024-05-30 DOI: 10.1088/1361-6544/ad4cd2
Jianjun Liu and Duohui Xiang
Rational normal form is a powerful tool to deal with Hamiltonian partial differential equations without external parameters. In this paper, we build rational normal form with exact global control of small divisors. As an application to nonlinear Schrödinger equations in Gevrey spaces, we prove sub-exponentially long time stability results for generic small initial data.
有理正则表达式是处理无外部参数哈密顿偏微分方程的有力工具。在本文中,我们建立了精确全局控制小除数的有理正则表达式。作为 Gevrey 空间中非线性薛定谔方程的应用,我们证明了一般小初始数据的亚指数长时间稳定性结果。
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引用次数: 0
Asymptotics of the sticky particles evolution 粘性粒子演化的渐进性
IF 1.7 2区 数学 Q2 MATHEMATICS, APPLIED Pub Date : 2024-05-28 DOI: 10.1088/1361-6544/ad4c4a
Ryan Hynd and Adrian Tudorascu
We study the long-time asymptotic behavior of the Sticky Particles dynamics on the real line. The time average of the Sticky Particles Lagrangian map has a limit which arises as a general property of projections onto closed convex cones in Hilbert spaces. More notably, we prove that the map itself has an asymptotic limit in the case where the sticky particles dynamics is confined to a compact set.
我们研究了实线上粘性粒子动力学的长期渐近行为。粘性粒子拉格朗日映射的时间平均值有一个极限,它是投影到希尔伯特空间闭合凸锥上的一般性质。更值得注意的是,我们证明了在粘性粒子动力学局限于紧凑集合的情况下,映射本身具有渐近极限。
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引用次数: 0
The evolution problem associated with the fractional first eigenvalue 与分数第一特征值相关的演化问题
IF 1.7 2区 数学 Q2 MATHEMATICS, APPLIED Pub Date : 2024-05-28 DOI: 10.1088/1361-6544/ad4cd0
Begoña Barrios, Leandro Del Pezzo, Alexander Quaas and Julio D Rossi
In this paper we study the evolution problem associated with the first fractional eigenvalue. We prove that the Dirichlet problem with homogeneous boundary condition is well posed for this operator in the framework of viscosity solutions (the problem has existence and uniqueness of a solution and a comparison principle holds). In addition, we show that solutions decay to zero exponentially fast as with a bound that is given by the first eigenvalue for this problem that we also study.
在本文中,我们研究了与第一个分数特征值相关的演化问题。我们证明,在粘性解的框架内,具有同质边界条件的迪里夏特问题对这个算子来说是很好解决的(问题具有解的存在性和唯一性,并且比较原理成立)。此外,我们还证明了解以指数级的速度衰减为零,其约束条件由我们同时研究的该问题的第一个特征值给出。
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引用次数: 0
Time-reversibility and nonvanishing Lévy area 时间可逆性和不消失的莱维区
IF 1.7 2区 数学 Q2 MATHEMATICS, APPLIED Pub Date : 2024-05-28 DOI: 10.1088/1361-6544/ad4947
Georg A Gottwald and Ian Melbourne
We give a complete description and clarification of the structure of the Lévy area correction to Itô/Stratonovich stochastic integrals arising as limits of time-reversible deterministic dynamical systems. In particular, we show that time-reversibility forces the Lévy area to vanish only in very specific situations that are easily classified. In the absence of such obstructions, we prove that there are no further restrictions on the Lévy area and that it is typically nonvanishing and far from negligible.
我们对作为时间可逆确定性动力学系统极限而产生的伊托/斯特拉托诺维奇随机积分的勒维面积修正结构进行了完整的描述和澄清。我们特别指出,时间可逆性迫使莱维面积只在非常特殊的情况下消失,而这些情况很容易归类。如果不存在这种障碍,我们就能证明莱维面积没有进一步的限制,而且它通常是不消失的,远非可以忽略不计。
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引用次数: 0
Global existence of weak solutions and weak–strong uniqueness for nonisothermal Maxwell–Stefan systems * 非等温麦克斯韦-斯特凡系统弱解的全局存在性和弱-强唯一性 *
IF 1.7 2区 数学 Q2 MATHEMATICS, APPLIED Pub Date : 2024-05-27 DOI: 10.1088/1361-6544/ad4c49
Stefanos Georgiadis and Ansgar Jüngel
The dynamics of multicomponent gas mixtures with vanishing barycentric velocity is described by Maxwell–Stefan equations with mass diffusion and heat conduction. The equations consist of the mass and energy balances, coupled to an algebraic system that relates the partial velocities and driving forces. The global existence of weak solutions to this system in a bounded domain with no-flux boundary conditions is proved by using the boundedness-by-entropy method. A priori estimates are obtained from the entropy inequality which originates from the consistent thermodynamic modelling. Furthermore, a conditional weak–strong uniqueness property is shown by using the relative entropy method.
麦克斯韦-斯特凡方程用质量扩散和热传导描述了重心速度消失的多组分气体混合物的动力学。该方程由质量和能量平衡组成,并与一个代数系统相耦合,该代数系统将部分速度和驱动力联系起来。利用有界熵法证明了该系统在具有无流动边界条件的有界域中弱解的全局存在性。先验估计从熵不等式中获得,而熵不等式源于一致的热力学模型。此外,还利用相对熵方法证明了条件弱-强唯一性。
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引用次数: 0
Abelian covers of hyperbolic surfaces: equidistribution of spectra and infinite volume mixing asymptotics for horocycle flows 双曲面的阿贝尔盖:角环流的等分布谱和无限体积混合渐近线
IF 1.7 2区 数学 Q2 MATHEMATICS, APPLIED Pub Date : 2024-05-27 DOI: 10.1088/1361-6544/ad4aaf
Livio Flaminio and Davide Ravotti
We consider Abelian covers of compact hyperbolic surfaces. We establish an asymptotic expansion of the correlations for the horocycle flow on -covers, thus proving a strong form of Krickeberg mixing. We also prove that the spectral measures around 0 of the Casimir operators on any increasing sequence of finite Abelian covers converge weakly to an absolutely continuous measure.
我们考虑了紧凑双曲面的阿贝尔盖。我们建立了-盖上的角循环流的相关性渐近展开,从而证明了克里克伯格混合的强形式。我们还证明了在任何有限阿贝尔盖的递增序列上,卡西米尔算子在 0 附近的谱度量弱收敛于绝对连续度量。
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引用次数: 0
期刊
Nonlinearity
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