Journal of Differential Equations最新文献

英文中文

Stability and convergence of in time approximations of hyperbolic elastodynamics via stepwise minimization 通过逐步最小化实现双曲弹性力学及时近似的稳定性和收敛性

IF 2.4 2区数学 Q1 MATHEMATICS

Journal of Differential Equations

Pub Date : 2024-09-24 DOI: 10.1016/j.jde.2024.09.034

We study step-wise time approximations of non-linear hyperbolic initial value problems. The technique used here is a generalization of the minimizing movements method, using two time-scales: one for velocity, the other (potentially much larger) for acceleration. The main applications are from elastodynamics, namely so-called generalized solids, undergoing large deformations. The evolution follows an underlying variational structure exploited by step-wise minimization. We show for a large family of (elastic) energies that the introduced scheme is stable; allowing for non-linearities of highest order. If the highest order can be assumed to be linear, we show that the limit solutions are regular and that the minimizing movements scheme converges with optimal linear rate. Thus this work extends numerical time-step minimization methods to the realm of hyperbolic problems.

我们研究非线性双曲初值问题的分步时间逼近。这里使用的技术是最小化运动法的一般化，使用两个时间尺度：一个用于速度，另一个（可能更大）用于加速度。主要应用于弹性动力学，即发生大变形的所谓广义固体。其演化过程遵循一种潜在的变分结构，并通过逐步最小化的方式加以利用。我们针对大量（弹性）能量证明，引入的方案是稳定的；允许最高阶的非线性。如果可以假定最高阶为线性，我们将证明极限解是有规律的，最小化运动方案将以最佳线性速率收敛。因此，这项工作将数值时间步最小化方法扩展到了双曲问题领域。

引用次数: 0

Study on the behaviors of rupture solutions for a class of elliptic MEMS equations in R2 R2 中一类椭圆 MEMS 方程的破裂解行为研究

IF 2.4 2区数学 Q1 MATHEMATICS

Journal of Differential Equations

Pub Date : 2024-09-23 DOI: 10.1016/j.jde.2024.09.035

This study examines nonnegative solutions to the problem

{\begin{matrix} Δ u = \frac{λ | x |^{α}}{u^{p}} & in R^{2} ∖ {0}, \\ u (0) = 0 and u > 0 & in R^{2} ∖ {0}, \end{matrix}

where

λ > 0

α > - 2

, and

p > 0

are constants. The possible asymptotic behaviors of

u (x)

| x | = 0

and

| x | = \infty

are classified according to

(α, p)

. In particular, the results show that for some

(α, p)

u (x)

exhibits only “isotropic” behavior at

| x | = 0

and

| x | = \infty

. However, in other cases,

u (x)

may exhibit the “anisotropic” behavior at

| x | = 0

| x | = \infty

. Furthermore, the relation between the limit at

| x | = 0

and the limit at

| x | = \infty

for a global solution is investigated.

本研究探讨了问题{Δu=λ|x|αup inR2∖{0},u(0)=0andu>0 inR2∖{0}的非负解，其中λ>0,α>-2,p>0为常数。根据 (α,p) 对 u(x) 在 |x|=0 和 |x|=∞ 时的可能渐近行为进行了分类。结果特别表明，对于某些 (α,p) 情况，u(x) 只在|x|=0 和 |x|=∞时表现出 "各向同性 "行为。然而，在其他情况下，u(x) 可能会在|x|=0 或 |x|=∞处表现出 "各向异性 "行为。此外，还研究了全局解在 |x|=0 时的极限与 |x|=∞ 时的极限之间的关系。

引用次数: 0

Global dynamics for a two-species chemotaxis-competition system with loop and nonlocal kinetics 具有循环和非局部动力学的双物种趋化竞争系统的全局动力学

IF 2.4 2区数学 Q1 MATHEMATICS

Journal of Differential Equations

Pub Date : 2024-09-23 DOI: 10.1016/j.jde.2024.09.027

<div><div>In this paper, we consider the two-species chemotaxis-competition system with loop and nonlocal kinetics<span><span><span><math><mrow><mo>{</mo><mtable><mtr><mtd><msub><mrow><mi>u</mi></mrow><mrow><mi>t</mi></mrow></msub><mo>=</mo><mi>Δ</mi><mi>u</mi><mo>−</mo><msub><mrow><mi>χ</mi></mrow><mrow><mn>11</mn></mrow></msub><mi>∇</mi><mo>⋅</mo><mo>(</mo><mi>u</mi><mi>∇</mi><mi>v</mi><mo>)</mo><mo>−</mo><msub><mrow><mi>χ</mi></mrow><mrow><mn>12</mn></mrow></msub><mi>∇</mi><mo>⋅</mo><mo>(</mo><mi>u</mi><mi>∇</mi><mi>z</mi><mo>)</mo><mo>+</mo><msub><mrow><mi>f</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>(</mo><mi>u</mi><mo>,</mo><mi>w</mi><mo>)</mo><mo>,</mo><mspace></mspace></mtd><mtd><mi>x</mi><mo>∈</mo><mi>Ω</mi><mo>,</mo><mspace></mspace><mi>t</mi><mo>></mo><mn>0</mn><mo>,</mo></mtd></mtr><mtr><mtd><mn>0</mn><mo>=</mo><mi>Δ</mi><mi>v</mi><mo>−</mo><mi>v</mi><mo>+</mo><mi>u</mi><mo>+</mo><mi>w</mi><mo>,</mo><mspace></mspace></mtd><mtd><mi>x</mi><mo>∈</mo><mi>Ω</mi><mo>,</mo><mspace></mspace><mi>t</mi><mo>></mo><mn>0</mn><mo>,</mo></mtd></mtr><mtr><mtd><msub><mrow><mi>w</mi></mrow><mrow><mi>t</mi></mrow></msub><mo>=</mo><mi>Δ</mi><mi>w</mi><mo>−</mo><msub><mrow><mi>χ</mi></mrow><mrow><mn>21</mn></mrow></msub><mi>∇</mi><mo>⋅</mo><mo>(</mo><mi>w</mi><mi>∇</mi><mi>v</mi><mo>)</mo><mo>−</mo><msub><mrow><mi>χ</mi></mrow><mrow><mn>22</mn></mrow></msub><mi>∇</mi><mo>⋅</mo><mo>(</mo><mi>w</mi><mi>∇</mi><mi>z</mi><mo>)</mo><mo>+</mo><msub><mrow><mi>f</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>(</mo><mi>u</mi><mo>,</mo><mi>w</mi><mo>)</mo><mo>,</mo><mspace></mspace></mtd><mtd><mi>x</mi><mo>∈</mo><mi>Ω</mi><mo>,</mo><mspace></mspace><mi>t</mi><mo>></mo><mn>0</mn><mo>,</mo></mtd></mtr><mtr><mtd><mn>0</mn><mo>=</mo><mi>Δ</mi><mi>z</mi><mo>−</mo><mi>z</mi><mo>+</mo><mi>u</mi><mo>+</mo><mi>w</mi><mo>,</mo><mspace></mspace></mtd><mtd><mi>x</mi><mo>∈</mo><mi>Ω</mi><mo>,</mo><mspace></mspace><mi>t</mi><mo>></mo><mn>0</mn><mo>,</mo></mtd></mtr></mtable></mrow></math></span></span></span> subject to homogeneous Neumann boundary conditions in a smooth bounded domain <span><math><mi>Ω</mi><mo>⊂</mo><msup><mrow><mi>R</mi></mrow><mrow><mi>n</mi></mrow></msup><mo>(</mo><mi>n</mi><mo>≥</mo><mn>1</mn><mo>)</mo></math></span>, where <span><math><msub><mrow><mi>χ</mi></mrow><mrow><mi>i</mi><mi>j</mi></mrow></msub><mo>></mo><mn>0</mn><mo>(</mo><mi>i</mi><mo>,</mo><mi>j</mi><mo>=</mo><mn>1</mn><mo>,</mo><mn>2</mn><mo>)</mo></math></span>, <span><math><msub><mrow><mi>f</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>(</mo><mi>u</mi><mo>,</mo><mi>w</mi><mo>)</mo><mo>=</mo><mi>u</mi><mrow><mo>(</mo><msub><mrow><mi>a</mi></mrow><mrow><mn>0</mn></mrow></msub><mo>−</mo><msub><mrow><mi>a</mi></mrow><mrow><mn>1</mn></mrow></msub><mi>u</mi><mo>−</mo><msub><mrow><mi>a</mi></mrow><mrow><mn>2</mn></mrow></msub><mi>w</mi><mo>−</mo><msub><mrow><mi>a</mi></mrow><mrow><mn>3</mn></mrow></msub><msub><mrow><mo>∫</mo></mrow><mrow><mi>Ω</mi></mrow></msub><mi>u</mi><mi>d</mi><mi>

本文考虑了具有循环和非局部动力学的双物种趋化-竞争系统{ut=Δu-χ11∇⋅(u∇v)-χ12∇⋅(u∇z)+f1(u,w),x∈Ω,t>；0,0=Δv-v+u+w,x∈Ω,t>0,wt=Δw-χ21∇⋅(w∇v)-χ22∇⋅(w∇z)+f2(u,w),x∈Ω,t>；0,0=Δz-z+u+w,x∈Ω,t>0，在光滑有界域Ω⊂Rn(n≥1)中服从均质 Neumann 边界条件，其中 χij>；0（i,j=1,2），f1（u,w）=u（a0-a1u-a2w-a3∫Ωudx-a4∫Ωwdx），f2（u,w）=w（b0-b1u-b2w-b3∫Ωudx-b4∫Ωwdx），其中 ai,bi>0（i=0,1,2）,aj,bj∈R（j=3,4）。研究表明，如果参数满足某些条件，那么相应的初始边界值问题在任何空间维度上都有唯一的全局时间经典解，且该解均匀有界。此外，基于合适能量函数的构造，还考虑了共存和半共存稳态的全局渐近稳定问题。我们的结果概括并改进了之前文献中的一些结果。

引用次数: 0

On the locally self-similar blowup for the generalized SQG equation 论广义 SQG方程的局部自相似膨胀

IF 2.4 2区数学 Q1 MATHEMATICS

Journal of Differential Equations

Pub Date : 2024-09-23 DOI: 10.1016/j.jde.2024.09.025

We analyze finite-time blowup scenarios of locally self-similar type for the inviscid generalized surface quasi-geostrophic equation (gSQG) in

R^{2}

. Under an

L^{r}

growth assumption on the self-similar profile and its gradient, we identify appropriate ranges of the self-similar parameter where the profile is either identically zero, and hence blowup cannot occur, or its

L^{p}

asymptotic behavior can be characterized, for suitable

r, p

. Our results extend the work by Xue [38] regarding the SQG equation, and also partially recover the results proved by Cannone and Xue [3] concerning globally self-similar solutions of the gSQG equation.

我们分析了 R2 中不粘性广义表面准地转方程（gSQG）的局部自相似型有限时间炸裂情形。在自相似剖面及其梯度的 Lr 增长假设下，我们确定了自相似参数的适当范围，在这些范围内，对于合适的 r，p，剖面要么为完全相同的零，因此不会发生炸裂，要么可以描述其 Lp 渐近行为。我们的结果扩展了 Xue [38] 关于 SQG 方程的研究，也部分恢复了 Cannone 和 Xue [3] 关于 gSQG 方程全局自相似解的结果。

引用次数: 0

Normalized solutions for a nonlinear Dirac equation 非线性狄拉克方程的归一化解

IF 2.4 2区数学 Q1 MATHEMATICS

Journal of Differential Equations

Pub Date : 2024-09-23 DOI: 10.1016/j.jde.2024.09.029

We prove the existence of a normalized, stationary solution

ψ : R^{3} \to C^{4}

with frequency

ω > 0

of the nonlinear Dirac equation. The result covers the case in which the nonlinearity is the gradient of a function of the form

F (Ψ) = a | (Ψ, γ^{0} Ψ) |^{\frac{α}{2}} + b | (Ψ, γ^{1} γ^{2} γ^{3} Ψ) |^{\frac{α}{2}}

with

α \in (2, \frac{8}{3}]

b \geq 0

and

a > 0

sufficiently small. Here

γ^{i}

i = 0, \dots, 3

are the

4 \times 4

Dirac's matrices.

We find the solution as a critical point of a suitable functional restricted to the unit sphere in

L^{2}

, and ω turns out to be the corresponding Lagrange multiplier.

我们证明了非线性狄拉克方程存在一个频率为 ω>0 的归一化静止解 ψ:R3→C4。结果涵盖了这样一种情况：非线性是形式为F(Ψ)=a|(Ψ,γ0Ψ)|α2+b|(Ψ,γ1γ2γ3Ψ)|α2的函数的梯度，α∈(2,83]，b≥0且a>0足够小。这里 γi, i=0,..., 3 是 4×4 的狄拉克矩阵。我们发现解是限制在 L2 单位球内的合适函数的临界点，而 ω 就是相应的拉格朗日乘数。

引用次数: 0

The focusing complex mKdV equation with nonzero background: Large N-order asymptotics of multi-rational solitons and related Painlevé-III hierarchy 非零背景的聚焦复数 mKdV 方程：多有理孤子的大 N 阶渐近和相关的 Painlevé-III 层次结构

IF 2.4 2区数学 Q1 MATHEMATICS

Journal of Differential Equations

Pub Date : 2024-09-23 DOI: 10.1016/j.jde.2024.09.038

In this paper, we investigate the large-order asymptotics of multi-rational solitons of the focusing complex modified Korteweg-de Vries (c-mKdV) equation with nonzero background via the Riemann-Hilbert problems. First, based on the Lax pair, inverse scattering transform, and a series of deformations, we construct a multi-rational soliton of the c-mKdV equation via a solvable Riemann-Hilbert problem (RHP). Then, through a scale transformation, we construct a RHP corresponding to the limit function which is a new solution of the c-mKdV equation in the rescaled variables

X, T

, and prove the existence and uniqueness of the RHP's solution. Moreover, we also find that the limit function satisfies the ordinary differential equations (ODEs) with respect to space X and time T, respectively. The ODEs with respect to space X are identified with certain members of the Painlevé-III hierarchy. We study the large X and transitional asymptotic behaviors of near-field limit solutions, and we provide some part results for the case of large T. These results will be useful to understand and apply the large-order rational solitons in the nonlinear wave equations.

本文通过黎曼-希尔伯特（Riemann-Hilbert）问题研究了具有非零背景的聚焦复修正科特韦格-德弗里斯（c-mKdV）方程的多理性孤子的大阶渐近性。首先，基于拉克斯对、反散射变换和一系列变形，我们通过可解黎曼-希尔伯特问题（Riemann-Hilbert problem，RHP）构建了 c-mKdV 方程的多理性孤子。然后，通过尺度变换，我们构建了一个与极限函数相对应的 RHP，该极限函数是 c-mKdV 方程在重标度变量 X,T 中的新解，并证明了 RHP 解的存在性和唯一性。此外，我们还发现极限函数分别满足空间 X 和时间 T 的常微分方程。关于空间 X 的 ODE 与 Painlevé-III 层次结构的某些成员相一致。我们研究了近场极限解的大 X 和过渡渐近行为，并提供了大 T 情况下的部分结果。这些结果将有助于理解和应用非线性波方程中的大阶有理孤子。

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引用次数: 0

The isochronal phase of stochastic PDE and integral equations: Metastability and other properties 随机 PDE 和积分方程的等时相：可代谢性及其他特性

IF 2.4 2区数学 Q1 MATHEMATICS

Journal of Differential Equations

Pub Date : 2024-09-23 DOI: 10.1016/j.jde.2024.09.002

We study the dynamics of waves, oscillations, and other spatio-temporal patterns in stochastic evolution systems, including SPDE and stochastic integral equations. Representing a given pattern as a smooth, stable invariant manifold of the deterministic dynamics, we reduce the stochastic dynamics to a finite dimensional SDE on this manifold using the isochronal phase. The isochronal phase is defined by mapping a neighborhood of the manifold onto the manifold itself, analogous to the isochronal phase defined for finite-dimensional oscillators by A.T. Winfree and J. Guckenheimer. We then determine a probability measure that indicates the average position of the stochastic perturbation of the pattern/wave as it wanders over the manifold. It is proved that this probability measure is accurate on time-scales greater than

O (σ^{- 2})

, but less than

O (\exp (C σ^{- 2}))

, where

σ ≪ 1

is the amplitude of the stochastic perturbation. Moreover, using this measure, we determine the expected velocity of the difference between the deterministic and stochastic motion on the manifold.

我们研究随机演化系统（包括 SPDE 和随机积分方程）中波浪、振荡和其他时空模式的动力学。将给定模式表示为确定性动力学的平滑稳定不变流形，我们利用等时相将随机动力学简化为该流形上的有限维 SDE。等时相的定义是将流形的一个邻域映射到流形本身，类似于 A.T. Winfree 和 J. Guckenheimer 为有限维振荡器定义的等时相。然后，我们确定了一个概率度量，它表示图案/波在流形上游荡时随机扰动的平均位置。事实证明，在时间尺度大于 O(σ-2)，但小于 O(exp(Cσ-2))（其中 σ≪1 是随机扰动的振幅）的情况下，这种概率度量是准确的。此外，利用这一量度，我们还能确定流形上确定运动与随机运动之差的预期速度。

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引用次数: 0

Asymptotic behavior for the fast diffusion equation with absorption and singularity 具有吸收和奇异性的快速扩散方程的渐近行为

IF 2.4 2区数学 Q1 MATHEMATICS

Journal of Differential Equations

Pub Date : 2024-09-20 DOI: 10.1016/j.jde.2024.09.026

This paper is concerned with the weak solution for the fast diffusion equation with absorption and singularity in the form of $u_{t} = △ u^{m} - u^{p}$ . We first prove the existence and decay estimate of weak solution when the fast diffusion index satisfies $0 < m < 1$ and the absorption index is $p > 1$ . Then we show the asymptotic convergence of weak solution to the corresponding Barenblatt solution for $\frac{n - 1}{n} < m < 1$ and $p > m + \frac{2}{n}$ via the entropy dissipation method combining the generalized Shannon's inequality and Csiszár-Kullback inequality. The singularity of spatial diffusion causes us the technical challenges for the asymptotic behavior of weak solution.

本文关注有吸收和奇异性的快速扩散方程的弱解，其形式为 ut=△um-up。我们首先证明了当快速扩散指数满足 0<m<1 和吸收指数为 p>1 时弱解的存在性和衰减估计，然后通过熵耗散方法结合广义香农不等式和 Csiszár-Kullback 不等式证明了弱解在 n-1n<m<1 和 p>m+2n 时对相应的 Barenblatt 解的渐近收敛性。空间扩散的奇异性给弱解的渐近行为带来了技术挑战。

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引用次数: 0

Existence, uniqueness and interior regularity of viscosity solutions for a class of Monge-Ampère type equations 一类 Monge-Ampère 型方程的粘性解的存在性、唯一性和内部正则性

IF 2.4 2区数学 Q1 MATHEMATICS

Journal of Differential Equations

Pub Date : 2024-09-20 DOI: 10.1016/j.jde.2024.09.024

The Monge-Ampère type equations over bounded convex domains arise in a host of geometric applications. In this paper, we focus on the Dirichlet problem for a class of Monge-Ampère type equations, which can be degenerate or singular near the boundary of convex domains. Viscosity subsolutions and viscosity supersolutions to the problem can be constructed via comparison principle. Finally, we demonstrate the existence, uniqueness and a series of interior regularities (including $W^{2, p}$ with $p \in (1, + \infty)$ , $C^{1, μ}$ with $μ \in (0, 1)$ , and $C^{\infty}$ ) of the viscosity solution to the problem.

有界凸域上的 Monge-Ampère 型方程出现在大量几何应用中。在本文中，我们重点研究一类 Monge-Ampère 型方程的 Dirichlet 问题，这类方程在凸域边界附近可能退化或奇异。通过比较原理，我们可以构建该问题的粘性子解和粘性超解。最后，我们证明了问题的粘性解的存在性、唯一性和一系列内部正则性（包括 p∈(1,+∞)的 W2,p、μ∈(0,1)的 C1,μ 和 C∞）。

引用次数: 0

Uniform-in-time stability and continuous transition of the time-discrete infinite Kuramoto model 时间离散无限仓本模型的均匀时间稳定性和连续转换

IF 2.4 2区数学 Q1 MATHEMATICS

Journal of Differential Equations

Pub Date : 2024-09-19 DOI: 10.1016/j.jde.2024.09.021

We study a continuous transition from the discrete infinite Kuramoto model to the continuous counterpart in a whole time interval. The discrete infinite Kuramoto model corresponds to the discretization of the infinite Kuramoto model [18] via the first-order Euler discretization algorithm. For the proposed discrete infinite Kuramoto model, we study the emergent dynamics and uniform (-in-time) stability with respect to initial data under a suitable framework which is formulated in terms of system parameters and initial data. For a homogeneous ensemble with the same natural frequencies, we identify sufficient conditions for the existence of “quasi-stationary state” and complete synchronization. In contrast, for a heterogeneous ensemble, we also provide a weak emergent dynamics, namely “practical synchronization”. For the continuous transition in a zero time-step limit, we provide an improved truncation error estimate compared to the error estimate which can be obtained from the general theory for first-order discretized model using the uniform stability and emergent dynamics.

我们研究的是从离散无限仓本模型到连续对应模型在整个时间间隔内的连续转换。离散无限仓本模型对应于通过一阶欧拉离散算法对无限仓本模型的离散化[18]。对于所提出的离散无限库拉莫托模型，我们在一个合适的框架下（该框架由系统参数和初始数据构成）研究了与初始数据相关的突发动力学和均匀（-时间内）稳定性。对于具有相同固有频率的同质集合，我们确定了 "准稳态 "和完全同步存在的充分条件。相反，对于异质集合体，我们还提供了一种弱新兴动力学，即 "实用同步"。对于零时步极限的连续转换，我们提供了一种改进的截断误差估计，与利用均匀稳定性和突发动力学从一阶离散模型的一般理论中得到的误差估计相比，截断误差估计有所改进。

引用次数: 0

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