Nonlinear Analysis-Theory Methods & Applications最新文献

英文中文

Evolution of convex closed curves under the generalized gradient flow of anisoperimetric ratio 广义等径比梯度流下凸闭合曲线的演化

IF 1.3 2区数学 Q1 MATHEMATICS

Nonlinear Analysis-Theory Methods & Applications

Pub Date : 2025-10-21 DOI: 10.1016/j.na.2025.113980

Ze-Yu Ye, Xiao-Liu Wang

In this paper, we study a generalized gradient flow of anisoperimetric ratio, whose inner normal velocity contains a power of anisotropic curvature for convex closed curves. It is shown that for any embedded smooth closed convex initial curve, the flow exists globally and the curvature of evolving curves converges smoothly to the curvature of the boundary of the Wulff shape, which is determined by the given anisotropic function, as time goes to infinity.

本文研究了一类广义各向异性比梯度流，其内法向速度包含凸闭曲线各向异性曲率的幂次。结果表明，对于任意嵌入的光滑闭凸初始曲线，随着时间趋于无穷，流动是全局存在的，且演化曲线的曲率平滑地收敛于由给定各向异性函数决定的Wulff形状边界的曲率。

引用次数: 0

An inhomogeneous porous medium equation with non-integrable data: Asymptotics 具有不可积数据的非齐次多孔介质方程：渐近性

IF 1.3 2区数学 Q1 MATHEMATICS

Nonlinear Analysis-Theory Methods & Applications

Pub Date : 2025-10-15 DOI: 10.1016/j.na.2025.113979

Matteo Muratori , Troy Petitt , Fernando Quirós

We investigate the asymptotic behavior as

t \to + \infty

of solutions to a weighted porous medium equation in

R^{N}

, whose weight

ρ (x)

behaves at spatial infinity like

{| x |}^{- γ}

with subcritical power, namely

γ \in [0, 2)

. Inspired by some results (Alikakos and Rostamian, 1984; Kamin and Ughi, 1987) from the 1980s on the unweighted problem, we focus on solutions whose initial data

u_{0} (x)

are not globally integrable with respect to the weight and behave at infinity like

{| x |}^{- α}

, for

α \in (0, N - γ)

. In the special case

ρ (x) = {| x |}^{- γ}

and

u_{0} (x) = {| x |}^{- α}

we show that self-similar solutions of Barenblatt type, i.e. reminiscent of the usual source-type solutions, still exist, although they are no longer compactly supported. Moreover, they exhibit a transition phenomenon which is new even for the unweighted equation. We prove that such self-similar solutions are attractors for the original problem, and convergence takes place globally in suitable weighted

L^{p}

spaces for

p \in [1, \infty)

and even globally in

L^{\infty}

under some mild additional regularity assumptions on the weight. Among the fundamental tools that we exploit, it is worth mentioning a global smoothing effect for non-integrable data.

我们研究了RN中一个加权多孔介质方程解在t→+∞时的渐近行为，该方程的权值ρ(x)在空间无穷远处表现为具有次临界幂的|x|−γ，即γ∈[0,2]。受20世纪80年代关于无权问题的一些结果（Alikakos和Rostamian, 1984； Kamin和Ughi， 1987）的启发，我们重点研究了初始数据u0(x)对于权不是全局可积的解，并且对于α∈(0,N−γ)在无穷远处表现为|x|−α。在ρ(x)=|x|−γ和u0(x)=|x|−α的特殊情况下，我们证明了自相似的Barenblatt型解，即让人想起通常的源型解，仍然存在，尽管它们不再紧支持。此外，它们还表现出一种即使对未加权方程也是新的过渡现象。我们证明了这类自相似解是原问题的吸引子，并且在p∈[1，∞]的适当加权Lp空间中全局收敛，在一些对权的温和附加正则性假设下甚至在L∞上全局收敛。在我们利用的基本工具中，值得一提的是不可积数据的全局平滑效应。

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引用次数: 0

Travelling waves for Maxwell’s equations in nonlinear and symmetric media 非线性对称介质中麦克斯韦方程组的行波

IF 1.3 2区数学 Q1 MATHEMATICS

Nonlinear Analysis-Theory Methods & Applications

Pub Date : 2025-10-15 DOI: 10.1016/j.na.2025.113976

Jarosław Mederski , Jacopo Schino

We look for travelling wave fields

E (x, y, z, t) = U (x, y) cos (k z + ω t) + \tilde{U} (x, y) sin (k z + ω t), (x, y, z) \in R^{3}, t \in R,

satisfying Maxwell’s equations in a nonlinear and cylindrically symmetric medium. We obtain a sequence of solutions with diverging energy consisting of transverse magnetic field modes. In addition, we consider a general nonlinearity, controlled by an N-function.

我们寻找在非线性圆柱对称介质中满足麦克斯韦方程组的行波场E(x,y,z,t)=U(x,y)cos(kz+ωt)+U ~ (x,y)sin(kz+ωt),(x,y,z)∈R3,t∈R。我们得到了一系列由横向磁场模组成的发散能量的解。此外，我们考虑一个由n函数控制的一般非线性。

引用次数: 0

Traveling gravity-capillary waves with odd viscosity 行进重力-奇粘度毛细管波

IF 1.3 2区数学 Q1 MATHEMATICS

Nonlinear Analysis-Theory Methods & Applications

Pub Date : 2025-10-11 DOI: 10.1016/j.na.2025.113975

Diego Alonso-Orán , Claudia García , Rafael Granero-Belinchón

In this note, we study the existence of traveling waves of a surface model in a non-newtonian fluid with odd viscosity. The proof relies on nonlinear bifurcation techniques.

本文研究了奇黏度非牛顿流体中表面模型行波的存在性。该证明依赖于非线性分岔技术。

引用次数: 0

Schottky invariant diffusion on the transcendent p-adic upper half plane 超越p进上半平面上的Schottky不变扩散

IF 1.3 2区数学 Q1 MATHEMATICS

Nonlinear Analysis-Theory Methods & Applications

Pub Date : 2025-10-10 DOI: 10.1016/j.na.2025.113947

Patrick Erik Bradley

The transcendent part of the Drinfel’d

p

-adic upper half plane is shown to be a Polish space. Using Radon measures associated with regular differential 1-forms invariant under Schottky groups allows to construct self-adjoint diffusion operators as Laplacian integral operators with kernel functions determined by the

p

-adic absolute value on the complex

p

-adic numbers. Their spectra are explicitly calculated and the corresponding Cauchy problems for their associated heat equations are found to be uniquely solvable and to determine Markov processes having paths which are càdlàg. The heat kernels are shown to have explicitly given distribution functions, as well as boundary value problems associated with the heat equations under Dirichlet and von Neumann conditions are solved.

证明了德林菲尔p进上半平面的超越部分是一个波兰空间。利用与Schottky群下正则微分1-形式不变量相关的Radon测度，可以将自伴随扩散算子构造为拉普拉斯积分算子，其核函数由复p进数上的p进绝对值决定。明确地计算了它们的光谱，并发现其相关热方程的对应柯西问题是唯一可解的，并确定了路径为càdlàg的马尔可夫过程。结果表明，热核具有显式给定的分布函数，并解决了狄利克雷和冯·诺伊曼条件下与热方程相关的边值问题。

引用次数: 0

Resolvent estimates for the one-dimensional damped wave equation with unbounded damping 具有无界阻尼的一维阻尼波动方程的解析估计

IF 1.3 2区数学 Q1 MATHEMATICS

Nonlinear Analysis-Theory Methods & Applications

Pub Date : 2025-10-10 DOI: 10.1016/j.na.2025.113978

Antonio Arnal

We study the generator

G

of the one-dimensional damped wave equation with unbounded damping at infinity. We show that the norm of the corresponding resolvent operator,

‖ {(G - λ)}^{- 1} ‖

, is approximately constant as

| λ | \to + \infty

on vertical strips of bounded width contained in the closure of the left-hand side complex semi-plane,

{\bar{ℂ}}_{-} ≔ {λ \in ℂ : Re λ \leq 0}

. Our proof rests on a precise asymptotic analysis of the norm of the inverse of

T (λ)

, the quadratic operator associated with

G

研究了无穷远处具有无界阻尼的一维阻尼波动方程的产生器G。证明了对应的解析算子‖(G−λ)−1‖在左手边复半平面的闭包中包含的有界宽度的垂直线上，其范数近似为|λ|→+∞。我们的证明依赖于T（λ）的逆模的精确渐近分析，即与G相关的二次算子。

引用次数: 0

Optimal planar immersions of prescribed winding number and Arnold invariants 规定圈数和阿诺德不变量的最优平面浸入

IF 1.3 2区数学 Q1 MATHEMATICS

Nonlinear Analysis-Theory Methods & Applications

Pub Date : 2025-10-10 DOI: 10.1016/j.na.2025.113942

Anna Lagemann, Heiko von der Mosel

Vladimir Arnold defined three invariants for generic planar immersions, i.e. planar curves whose self-intersections are all transverse double points. We use a variational approach to study these invariants by investigating a suitably truncated knot energy, the tangent-point energy. We prove existence of energy minimizers for each truncation parameter

δ > 0

in a class of immersions with prescribed winding number and Arnold invariants, and establish Gamma convergence of the truncated tangent-point energies to a limiting renormalized tangent-point energy as

δ \to 0

. Moreover, we show that any sequence of minimizers subconverges in

C^{1}

, and the corresponding limit curve has the same topological invariants, self-intersects exclusively at right angles, and minimizes the renormalized tangent-point energy among all curves with right self-intersection angles. In addition, the limit curve is an almost-minimizer for all of the original truncated tangent-point energies as long as the truncation parameter

δ

is sufficiently small. Therefore, this limit curve serves as an “optimal” curve in the class of generic planar immersions with prescribed winding number and Arnold invariants.

Vladimir Arnold定义了一般平面浸入式的三个不变量，即自交均为横向双点的平面曲线。我们使用变分的方法来研究这些不变量，通过研究一个适当截断的结能量，切点能量。在给定圈数和Arnold不变量的浸入式中，证明了每一个截断参数δ>；0的能量极小值的存在性，并建立了截断的切点能量的伽玛收敛到一个极限重归一化切点能量为δ→0。此外，我们还证明了任何最小值序列在C1中都是子收敛的，并且相应的极限曲线具有相同的拓扑不变量，在直角处完全自交，并且在所有自交角为直角的曲线中极小化了的切点能量。此外，只要截断参数δ足够小，对于所有原始截断的切点能量，极限曲线几乎是最小的。因此，该极限曲线可作为具有规定圈数和阿诺德不变量的一般平面浸没的一类“最优”曲线。

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引用次数: 0

Γ-convergence of higher-order phase transition models Γ-convergence的高阶相变模型

IF 1.3 2区数学 Q1 MATHEMATICS

Nonlinear Analysis-Theory Methods & Applications

Pub Date : 2025-10-09 DOI: 10.1016/j.na.2025.113971

Denis Brazke , Gianna Götzmann , Hans Knüpfer

We investigate the asymptotic behavior as

ɛ \to 0

of singularly perturbed phase transition models of order

n \geq 2

, given by

G_{ɛ}^{λ, n} [u] ≔ \int_{I} \frac{1}{ɛ} W (u) - λ ɛ^{2 n - 3} {(u^{(n - 1)})}^{2} + ɛ^{2 n - 1} {(u^{(n)})}^{2} d x, u \in W^{n, 2} (I),

where

λ > 0

is fixed,

I \subset R

is an open bounded interval, and

W \in C^{0} (R)

is a suitable double-well potential. We find that there exists a positive critical parameter depending on

W

and

n

, such that the

Γ

-limit of

G_{ɛ}^{λ, n}

with respect to the

L^{1}

-topology is given by a sharp interface functional in the subcritical regime. The cornerstone for the corresponding compactness property is a novel nonlinear interpolation inequality involving higher-order derivatives, which is based on Gagliardo–Nirenberg type inequalities.

研究了n≥2阶奇异摄动相变模型的渐近性，其中λ λ，n[u]是∫I1 ^ W(u)−λ ^ 2n−3(u(n−1))2+ ^ 2n−1(u(n))2dx,u∈Wn,2(I)，其中λ ^ gt；0是固定的，I∧R是一个开有界区间，W∈C0(R)是一个合适的双阱势。我们发现存在一个依赖于W和n的正临界参数，使得G λ，n关于l1拓扑的Γ-limit由亚临界区中的锐界面泛函给出。该紧性的基础是基于Gagliardo-Nirenberg型不等式的一种涉及高阶导数的非线性插值不等式。

引用次数: 0

Geometric analysis on weighted manifolds under lower 0-weighted Ricci curvature bounds 下0权Ricci曲率界下加权流形的几何分析

IF 1.3 2区数学 Q1 MATHEMATICS

Nonlinear Analysis-Theory Methods & Applications

Pub Date : 2025-10-09 DOI: 10.1016/j.na.2025.113965

Yasuaki Fujitani , Yohei Sakurai

We develop geometric analysis on weighted Riemannian manifolds under lower 0-weighted Ricci curvature bounds. Under such curvature bounds, we prove a first non-zero Steklov eigenvalue estimate of Wang–Xia type on compact weighted manifolds with boundary, and a first non-zero eigenvalue estimate of Choi–Wang type on closed weighted minimal hypersurfaces. We also produce an ABP estimate and a Sobolev inequality of Brendle type.

给出了下0权Ricci曲率界下加权黎曼流形的几何分析。在这样的曲率边界下，证明了紧加权流形上Wang-Xia型的第一个非零Steklov特征值估计，以及封闭加权极小超曲面上Choi-Wang型的第一个非零特征值估计。我们也得到了一个ABP估计和一个Brendle型的Sobolev不等式。

引用次数: 0

A priori estimates for anti-symmetric solutions to a fractional Laplacian equation in a bounded domain 有界域上分数阶拉普拉斯方程反对称解的先验估计

IF 1.3 2区数学 Q1 MATHEMATICS

Nonlinear Analysis-Theory Methods & Applications

Pub Date : 2025-10-08 DOI: 10.1016/j.na.2025.113970

Chenkai Liu , Shaodong Wang , Ran Zhuo

In this paper, we obtain a priori estimates for the set of anti-symmetric solutions to a fractional Laplacian equation in a bounded domain using a blowing-up and rescaling argument. In order to establish a contradiction to possible blow-ups, we apply a certain variation of the moving planes method in order to prove a monotonicity result for the limit equation after rescaling.

在有界区域上，利用放大和重标尺论证，给出了分数阶拉普拉斯方程的反对称解集的先验估计。为了证明极限方程在重新标度后的单调性，我们对运动平面法进行了一定的变换。

引用次数: 0

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