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Particle paths in equatorial flows 赤道气流中的粒子路径
Pub Date : 2022-06-23 DOI: 10.3934/cpaa.2022041
Tony Lyons
We investigate particle trajectories in equatorial flows with geophysical corrections caused by the earth's rotation. Particle trajectories in the flows are constructed using pairs of analytic functions defined over the labelling space used in the Lagrangian formalism. Several classes of flow are investigated, and the physical regime in which each is valid is determined using the pressure distribution function of the flow, while the vorticity distribution of each flow is also calculated and found to be effected by earth's rotation.
我们用地球自转引起的地球物理修正来研究赤道流中的粒子轨迹。流动中的粒子轨迹是用拉格朗日形式主义中在标记空间上定义的解析函数对来构造的。本文研究了几类流动,并利用流动的压力分布函数确定了每一类流动有效的物理状态,同时计算了每一类流动的涡度分布,发现它们受地球自转的影响。
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引用次数: 0
Gagliardo-Nirenberg-Sobolev inequalities on planar graphs 平面图上的Gagliardo-Nirenberg-Sobolev不等式
Pub Date : 2021-10-14 DOI: 10.3934/cpaa.2022051
M. Esteban
In this paper we study a family of the interpolation Gagliardo-Nirenberg-Sobolev inequalities on planar graphs. We are interested in knowing when the best constants in the inequalities are achieved. The inequalities being equivalent to some minimization problems, we also analyse the set of solutions of the Euler-Lagrange equations satisfied by extremal functions, or equivalently, by minimizers.
本文研究了平面图上的一类插值Gagliardo-Nirenberg-Sobolev不等式。我们感兴趣的是知道不等式中的最佳常数是什么时候得到的。由于不等式等价于一些最小化问题,我们还分析了由极值函数或等效的最小化器满足的欧拉-拉格朗日方程的解集。
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引用次数: 1
Multiplicity results for nonhomogeneous elliptic equations with singular nonlinearities 奇异非线性非齐次椭圆方程的多重性结果
Pub Date : 2021-09-07 DOI: 10.3934/cpaa.2022056
R. Arora

This paper is concerned with the study of multiple positive solutions to the following elliptic problem involving a nonhomogeneous operator with nonstandard growth of begin{document}$ p $end{document}-begin{document}$ q $end{document} type and singular nonlinearities

where begin{document}$ Omega $end{document} is a bounded domain in begin{document}$ mathbb{R}^N $end{document} with begin{document}$ C^2 $end{document} boundary, begin{document}$ N geq 1 $end{document}, begin{document}$ lambda >0 $end{document} is a real parameter,

begin{document}$ 1, begin{document}$ gamma in (0,1) $end{document}, and begin{document}$ f $end{document} is a continuous nondecreasing map satisfying suitable conditions. By constructing two distinctive pairs of strict sub and super solution, and using fixed point theorems by Amann [1], we prove existence of three positive solutions in the positive cone of begin{document}$ C_delta(overline{Omega}) $end{document} and in a certain range of begin{document}$ lambda $end{document}.

本文涉及的研究多个正解如下椭圆问题涉及非齐次与非标准增长运营商开始{文档}$ p $ {文档}-{文档}开始结束问美元}{文档类型和奇异非线性开始{文档}$ 左{开始{alignedat} {2} {} - mathcal {L} _ {p, q} u &{} = λ压裂{f (u)}{你^ 伽马}, u > 0 & & 四 mbox的{},ω u &{} = 0 & & 四 mbox{在}部分ω, {alignedat} 正确的结束。$end{document}其中$ begin{document}$ Omega $end{document}是begin{document}$ mathbb{R}^N $end{document}与begin{document}$ C^2 $end{document}边界中的有界域,begin{document}$ N geq 1 $end{document}, begin{document}$ lambda >0 $end{document}是实参数,begin{document}$ mathcal{L}_{p,q} u:= {rm{div}}(|nabla u|^{p-2} nabla u + |nabla u|^{q-2} nabla u), $end{document} begin{document}$ 1, begin{document}$ gamma in (0,1) $end{document},和begin{document}$ f $end{document}是满足适当条件的连续非递减映射。利用Amann[1]的不动点定理,构造了严格下解和上解的两个不同对,证明了在begin{document}$ C_delta(overline{Omega}) $end{document}的正锥和begin{document}$ lambda $end{document}的一定范围内存在三个正解。
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引用次数: 4
Curvature-driven front propagation through planar lattices in oblique directions 曲率驱动锋面斜向通过平面晶格传播
Pub Date : 2021-07-09 DOI: 10.3934/cpaa.2022055
Mia Juki'c, H. Hupkes

In this paper we investigate the long-term behaviour of solutions to the discrete Allen-Cahn equation posed on a two-dimensional lattice. We show that front-like initial conditions evolve towards a planar travelling wave modulated by a phaseshift begin{document}$ gamma_l(t) $end{document} that depends on the coordinate begin{document}$ l $end{document} transverse to the primary direction of propagation. This direction is allowed to be general, but rational, generalizing earlier known results for the horizontal direction. We show that the behaviour of begin{document}$ gamma $end{document} can be asymptotically linked to the behaviour of a suitably discretized mean curvature flow. This allows us to show that travelling waves propagating in rational directions are nonlinearly stable with respect to perturbations that are asymptotically periodic in the transverse direction.

In this paper we investigate the long-term behaviour of solutions to the discrete Allen-Cahn equation posed on a two-dimensional lattice. We show that front-like initial conditions evolve towards a planar travelling wave modulated by a phaseshift begin{document}$ gamma_l(t) $end{document} that depends on the coordinate begin{document}$ l $end{document} transverse to the primary direction of propagation. This direction is allowed to be general, but rational, generalizing earlier known results for the horizontal direction. We show that the behaviour of begin{document}$ gamma $end{document} can be asymptotically linked to the behaviour of a suitably discretized mean curvature flow. This allows us to show that travelling waves propagating in rational directions are nonlinearly stable with respect to perturbations that are asymptotically periodic in the transverse direction.
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引用次数: 1
Two-sided estimates of total bandwidth for Schrödinger operators on periodic graphs 周期图上Schrödinger算子总带宽的双边估计
Pub Date : 2021-06-16 DOI: 10.3934/cpaa.2022042
E. Korotyaev, N. Saburova
We consider Schrödinger operators with periodic potentials on periodic discrete graphs. Their spectrum consists of a finite number of bands. We obtain two-sided estimates of the total bandwidth for the Schrödinger operators in terms of geometric parameters of the graph and the potentials. In particular, we show that these estimates are sharp. It means that these estimates become identities for specific graphs and potentials. The proof is based on the Floquet theory and trace formulas for fiber operators. The traces are expressed as finite Fourier series of the quasimomentum with coefficients depending on the potentials and cycles of the quotient graph from some specific cycle sets. In order to obtain our results we estimate these Fourier coefficients in terms of geometric parameters of the graph and the potentials.
考虑周期离散图上具有周期势的Schrödinger算子。它们的光谱由有限的波段组成。我们根据图和势的几何参数获得了Schrödinger算子总带宽的双边估计。特别是,我们表明这些估计是尖锐的。这意味着这些估计成为特定图形和势的恒等式。该证明基于Floquet理论和光纤算子的迹公式。这些迹被表示为准动量的有限傅立叶级数,其系数取决于来自特定循环集的商图的势和循环。为了得到我们的结果,我们用图形和势的几何参数来估计这些傅立叶系数。
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引用次数: 3
A potential approach for planning mean-field games in one dimension 一种在一维空间中规划平均场博弈的潜在方法
Pub Date : 2021-04-25 DOI: 10.3934/cpaa.2022054
T. Bakaryan, Rita Ferreira, D. Gomes
This manuscript discusses planning problems for first- and second-order one-dimensional mean-field games (MFGs). These games are comprised of a Hamilton–Jacobi equation coupled with a Fokker–Planck equation. Applying Poincaré's Lemma to the Fokker–Planck equation, we deduce the existence of a potential. Rewriting the Hamilton–Jacobi equation in terms of the potential, we obtain a system of Euler–Lagrange equations for certain variational problems. Instead of the mean-field planning problem (MFP), we study this variational problem. By the direct method in the calculus of variations, we prove the existence and uniqueness of solutions to the variational problem. The variational approach has the advantage of eliminating the continuity equation.We also consider a first-order MFP with congestion. We prove that the congestion problem has a weak solution by introducing a potential and relying on the theory of variational inequalities. We end the paper by presenting an application to the one-dimensional Hughes' model.
本文讨论一阶和二阶一维平均场对策(mfg)的规划问题。这些博弈由Hamilton-Jacobi方程和Fokker-Planck方程组成。将庞加莱引理应用于福克-普朗克方程,推导出势的存在性。将Hamilton-Jacobi方程改写成势的形式,得到了某变分问题的欧拉-拉格朗日方程组。本文用变分问题来代替平均场规划问题(MFP)。利用变分学中的直接方法,证明了变分问题解的存在唯一性。变分方法的优点是消除了连续性方程。我们还考虑了一阶MFP的拥塞问题。利用变分不等式理论,引入一个势,证明了拥塞问题有一个弱解。最后,我们给出了一维休斯模型的一个应用。
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引用次数: 4
Exact travelling solution for a reaction-diffusion system with a piecewise constant production supported by a codimension-1 subspace 具有余维1子空间支持的分段常数生产的反应扩散系统的精确行进解
Pub Date : 2019-06-02 DOI: 10.3934/cpaa.2022030
Anton S. Zadorin
A generalisation of reaction diffusion systems and their travelling solutions to cases when the productive part of the reaction happens only on a surface in space or on a line on plane but the degradation and the diffusion happen in bulk are important for modelling various biological processes. These include problems of invasive species propagation along boundaries of ecozones, problems of gene spread in such situations, morphogenesis in cavities, intracellular reaction etc. Piecewise linear approximations of reaction terms in reaction-diffusion systems often result in exact solutions of propagation front problems. This article presents an exact travelling solution for a reaction-diffusion system with a piecewise constant production restricted to a codimension-1 subset. The solution is monotone, propagates with the unique constant velocity, and connects the trivial solution to a nontrivial nonhomogeneous stationary solution of the problem. The properties of the solution closely parallel the properties of monotone travelling solutions in classical bistable reaction-diffusion systems.
当反应的生产部分只发生在空间中的一个表面或平面上的一条直线上,而降解和扩散发生在整体上时,反应扩散系统的推广及其旅行解对于模拟各种生物过程是重要的。这些问题包括入侵物种沿生态带边界繁殖的问题、在这种情况下基因传播的问题、腔内形态发生的问题、细胞内反应等。对反应扩散系统中的反应项进行分段线性近似,往往能得到传播前问题的精确解。本文给出了一类反应扩散系统的精确旅行解,该系统具有限制于余维1子集的分段常数生产。解是单调的,以唯一的恒定速度传播,并将问题的平凡解与问题的非平凡非齐次平稳解联系起来。该解的性质与经典双稳反应扩散体系中单调行解的性质非常相似。
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引用次数: 0
Parabolic Systems with measurable coefficients in weighted Sobolev spaces 加权Sobolev空间中系数可测的抛物系统
Pub Date : 2018-09-05 DOI: 10.3934/cpaa.2022062
Doyoon Kim, Kyeong-Hun Kim, Kijung Lee

We present a weighted begin{document}$ L_p $end{document}-theory of parabolic systems on a half space begin{document}$ {mathbb{R}}^d_+ $end{document}. The leading coefficients are assumed to be only measurable in time begin{document}$ t $end{document} and have small bounded mean oscillations (BMO) with respect to the spatial variables begin{document}$ x $end{document}, and the lower order coefficients are allowed to blow up near the boundary.

We present a weighted begin{document}$ L_p $end{document}-theory of parabolic systems on a half space begin{document}$ {mathbb{R}}^d_+ $end{document}. The leading coefficients are assumed to be only measurable in time begin{document}$ t $end{document} and have small bounded mean oscillations (BMO) with respect to the spatial variables begin{document}$ x $end{document}, and the lower order coefficients are allowed to blow up near the boundary.
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引用次数: 2
Multiple localized nodal solutions of high topological type for Kirchhoff-type equation with double potentials 双势kirchhoff型方程的高拓扑型多局部节点解
Pub Date : 1900-01-01 DOI: 10.3934/cpaa.2022058
Zhi-Guo Wu, Wen Guan, Da-Bin Wang

We are concerned with sign-changing solutions and their concentration behaviors of singularly perturbed Kirchhoff problem

where begin{document}$ varepsilon $end{document} is a small positive parameter, begin{document}$ a, b>0 $end{document} and begin{document}$ V, Pin C^{1}(mathbb{R}^{3}, mathbb{R}) $end{document}. Without using any non-degeneracy conditions, we obtain multiple localized sign-changing solutions of higher topological type for this problem. Furthermore, we also determine a concrete set as the concentration position of these sign-changing solutions. The main methods we use are penalization techniques and the method of invariant sets of descending flow. It is notice that, when nonlinear potential begin{document}$ P $end{document} is a positive constant, our result generalizes the result obtained in [5] to Kirchhoff problem.

We are concerned with sign-changing solutions and their concentration behaviors of singularly perturbed Kirchhoff problem begin{document}$ begin{equation*} -(varepsilon^{2}a+ varepsilon bint _{mathbb{R}^{3}}|nabla v|^{2}dx)Delta v+V(x)v = P(x)f(v); ; {rm{in}}; mathbb{R}^{3}, end{equation*} $end{document} where begin{document}$ varepsilon $end{document} is a small positive parameter, begin{document}$ a, b>0 $end{document} and begin{document}$ V, Pin C^{1}(mathbb{R}^{3}, mathbb{R}) $end{document}. Without using any non-degeneracy conditions, we obtain multiple localized sign-changing solutions of higher topological type for this problem. Furthermore, we also determine a concrete set as the concentration position of these sign-changing solutions. The main methods we use are penalization techniques and the method of invariant sets of descending flow. It is notice that, when nonlinear potential begin{document}$ P $end{document} is a positive constant, our result generalizes the result obtained in [5] to Kirchhoff problem.
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引用次数: 2
The number of limit cycles from the perturbation of a quadratic isochronous system with two switching lines 具有两条切换线的二次等时系统摄动的极限环数
Pub Date : 1900-01-01 DOI: 10.3934/cpaa.2022047
Ai Ke, Maoan Han, Wei-Jian Geng

In this paper, we give an upper bound (for begin{document}$ ngeq3 $end{document}) and the least upper bound (for begin{document}$ n = 1,2 $end{document}) of the number of limit cycles bifurcated from period annuli of a quadratic isochronous system under the piecewise polynomial perturbations of degree begin{document}$ n $end{document}, respectively. The results improve the conclusions in [19].

In this paper, we give an upper bound (for begin{document}$ ngeq3 $end{document}) and the least upper bound (for begin{document}$ n = 1,2 $end{document}) of the number of limit cycles bifurcated from period annuli of a quadratic isochronous system under the piecewise polynomial perturbations of degree begin{document}$ n $end{document}, respectively. The results improve the conclusions in [19].
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引用次数: 3
期刊
Communications on Pure &amp; Applied Analysis
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