Journal of Differential Equations最新文献

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Regularity results for mixed local and nonlocal double phase functionals 混合局部和非局部双相函数的正则性结果

IF 2.4 2区数学 Q1 MATHEMATICS

Journal of Differential Equations

Pub Date : 2024-11-04 DOI: 10.1016/j.jde.2024.10.028

Sun-Sig Byun , Ho-Sik Lee , Kyeong Song

We investigate the De Giorgi-Nash-Moser theory for minimizers of mixed local and nonlocal functionals modeled after

v \mapsto \int_{R^{n}} \int_{R^{n}} \frac{| v (x) - v (y) |^{p}}{| x - y |^{n + s p}} d x d y + \int_{Ω} a (x) | D v |^{q} d x,

where

0 < s < 1 < p \leq q

and

a (\cdot) \geq 0

. In particular, we prove Hölder regularity and Harnack inequality under possibly sharp assumptions on

s, p, q

and

a (\cdot)

我们研究了建模在v↦∫Rn∫Rn|v(x)-v(y)|p|x-y|n+spdxdy+∫Ωa(x)|Dv|qdx之后的局部和非局部混合函数最小化的德乔治-纳什-莫泽理论，其中0<s<1<p≤q和a(⋅)≥0。我们特别证明了在对 s、p、q 和 a(⋅) 可能有尖锐假设的情况下的荷尔德正则性和哈纳克不等式。

引用次数: 0

A unified theory for inertial manifolds, saddle point property and exponential dichotomy 惯性流形、鞍点特性和指数二分法的统一理论

IF 2.4 2区数学 Q1 MATHEMATICS

Journal of Differential Equations

Pub Date : 2024-10-31 DOI: 10.1016/j.jde.2024.10.029

Alexandre N. Carvalho , Phillipo Lappicy , Estefani M. Moreira , Alexandre N. Oliveira-Sousa

Inertial manifold theory, saddle point property and exponential dichotomy have been treated as different topics in the literature with different proofs. As a common feature, they all have the purpose of ‘splitting’ the space to understand the dynamics. We present a unified proof for the inertial manifold theorem, which as a local consequence yields the saddle-point property with a fine structure of invariant manifolds and the roughness of exponential dichotomy. In particular, we use these tools in order to establish the hyperbolicity of certain global solutions for non-autonomous parabolic partial differential equations.

惯性流形理论、鞍点性质和指数二分法在文献中被视为不同的主题，并有不同的证明。它们的共同特点是都以 "分割 "空间来理解动力学为目的。我们提出了惯性流形定理的统一证明，作为局部结果，它产生了具有不变流形精细结构的鞍点性质和指数二分法的粗糙度。特别是，我们利用这些工具建立了非自治抛物线偏微分方程某些全局解的双曲性。

引用次数: 0

The Leray-Lions existence theorem under general growth conditions 一般增长条件下的勒雷-狮子存在定理

IF 2.4 2区数学 Q1 MATHEMATICS

Journal of Differential Equations

Pub Date : 2024-10-31 DOI: 10.1016/j.jde.2024.10.025

Giovanni Cupini , Paolo Marcellini , Elvira Mascolo

We prove an existence (and regularity) result of weak solutions

u \in W_{0}^{1, p} (Ω) \cap W_{loc}^{1, q} (Ω)

, to a Dirichlet problem for a second order elliptic equation in divergence form, under general and

p, q -

growth conditions of the differential operator. This is a first attempt to extend to general growth the well known Leray-Lions existence theorem, which holds under the so-called natural growth conditions with

q = p

. We found a way to treat the general context with explicit dependence on

(x, u)

, other than on the gradient variable

ξ = D u

; these aspects require particular attention due to the

p, q

-context, with some differences and new difficulties compared to the standard case

p = q

我们证明了在微分算子的一般和 p,q 增长条件下，发散形式二阶椭圆方程的 Dirichlet 问题的弱解 u∈W01,p(Ω)∩Wloc1,q(Ω) 的存在性（和正则性）结果。这是首次尝试将众所周知的勒雷-狮子存在定理扩展到一般增长，该定理在 q=p 的所谓自然增长条件下成立。除了梯度变量ξ=Du之外，我们还找到了一种明确依赖 (x,u) 的一般情况下的处理方法；由于 p,q 条件，这些方面需要特别注意，与标准情况 p=q 相比，存在一些差异和新的困难。

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引用次数: 0

Blowup dynamics for the mass critical half-wave equation in 2D 二维质量临界半波方程的爆炸动力学

IF 2.4 2区数学 Q1 MATHEMATICS

Journal of Differential Equations

Pub Date : 2024-10-31 DOI: 10.1016/j.jde.2024.10.031

Vladimir Georgiev , Yuan Li

We consider the two-dimensional half-wave equation

i u_{t} = D u - | u | u

. For the initial data

u_{0} (x) \in H^{s} (R^{2})

s \in (\frac{3}{4}, 1)

, we obtain the existence of non-radial ground state mass blow-up solutions with the blow-up rate

{‖ D^{\frac{1}{2}} u (t) ‖}_{L^{2}} \sim \frac{1}{| t |}

t \to 0^{-}

. This work extends the recent study by Georgiev and Li (2022) [9], which focused on constructing radial ground state mass blow-up solutions.

我们考虑二维半波方程 iut=Du-|u|u。对于初始数据 u0(x)∈Hs(R2), s∈(34,1)，我们得到了非径向基态质量炸毁解的存在，其炸毁率‖D12u(t)‖L2∼1|t|为 t→0-。这项工作扩展了 Georgiev 和 Li（2022 年）[9]的最新研究，后者的重点是构建径向基态质量炸毁解。

引用次数: 0

Boundedness in a chemotaxis system with weak singular sensitivity and logistic kinetics in any dimensional setting 具有弱奇异敏感性和逻辑动力学的趋化系统在任意维度环境中的有界性

IF 2.4 2区数学 Q1 MATHEMATICS

Journal of Differential Equations

Pub Date : 2024-10-31 DOI: 10.1016/j.jde.2024.10.027

Halil Ibrahim Kurt

<div><div>This paper deals with the following parabolic-elliptic chemotaxis competition system with weak singular sensitivity and logistic source<span><span><span>(0.1)</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><mi>χ</mi><mi>∇</mi><mo>⋅</mo><mo>(</mo><mfrac><mrow><mi>u</mi></mrow><mrow><msup><mrow><mi>v</mi></mrow><mrow><mi>λ</mi></mrow></msup></mrow></mfrac><mi>∇</mi><mi>v</mi><mo>)</mo><mo>+</mo><mi>r</mi><mi>u</mi><mo>−</mo><mi>μ</mi><msup><mrow><mi>u</mi></mrow><mrow><mn>2</mn></mrow></msup><mo>,</mo><mspace></mspace></mtd><mtd><mi>x</mi><mo>∈</mo><mi>Ω</mi><mo>,</mo></mtd></mtr><mtr><mtd><mn>0</mn><mo>=</mo><mi>Δ</mi><mi>v</mi><mo>−</mo><mi>α</mi><mi>v</mi><mo>+</mo><mi>β</mi><mi>u</mi><mo>,</mo><mspace></mspace></mtd><mtd><mi>x</mi><mo>∈</mo><mi>Ω</mi><mo>,</mo></mtd></mtr><mtr><mtd><mfrac><mrow><mo>∂</mo><mi>u</mi></mrow><mrow><mo>∂</mo><mi>ν</mi></mrow></mfrac><mo>=</mo><mfrac><mrow><mo>∂</mo><mi>v</mi></mrow><mrow><mo>∂</mo><mi>ν</mi></mrow></mfrac><mo>=</mo><mn>0</mn><mo>,</mo><mspace></mspace></mtd><mtd><mi>x</mi><mo>∈</mo><mo>∂</mo><mi>Ω</mi><mo>,</mo></mtd></mtr></mtable></mrow></math></span></span></span> where <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> is a smooth bounded domain, the parameters <span><math><mi>χ</mi><mo>,</mo><mspace></mspace><mi>r</mi><mo>,</mo><mspace></mspace><mi>μ</mi><mo>,</mo><mspace></mspace><mi>α</mi><mo>,</mo><mspace></mspace><mi>β</mi></math></span> are positive constants and <span><math><mi>λ</mi><mo>∈</mo><mo>(</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo>)</mo></math></span>.</div><div>It is well known that for parabolic-elliptic chemotaxis systems including singularity, a uniform-in-time positive pointwise lower bound for <em>v</em> is vitally important for establishing the global boundedness of classical solutions since the cross-diffusive term becomes unbounded near <span><math><mi>v</mi><mo>=</mo><mn>0</mn></math></span>. To this end, a key step in the literature is to establish a proper positive lower bound for the mass functional <span><math><msub><mrow><mo>∫</mo></mrow><mrow><mi>Ω</mi></mrow></msub><mi>u</mi></math></span>, which, due to the presence of logistic kinetics, is not preserved and hence it turns in for <em>v</em>. In contrast to this approach, in this article, the boundedness of classical solutions of (0.1) is obtained without using the uniformly positive lower bound of <em>v</em>.</div><div>Among others, it has been proven that without establishing a uniform-in-time positive pointwise lower bound for <em>v</em>, if <span><math><mi>λ</mi><mo>∈</mo><mo>(</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo>)</mo></math></span>, then there exists <span><math><mi>μ</mi><mo>></mo><msup><mrow><mi>μ</mi></mrow><mrow><mo>⁎</mo></mrow></msup></math></span> such that for all suitably smooth in

本文讨论了以下抛物线-椭圆趋化竞争系统，该系统具有弱奇异敏感性和逻辑源（0.1){ut=Δu-χ∇-u(λv∇v)+ru-μu2,x∈Ω,0=Δv-αv+βu,x∈Ω,∂u∂ν=∂v∂ν=0∈∂Ω，其中Ω⊂RN(N≥1)为光滑有界域，参数χ,r,μ,αβ为正常数，λ∈(0,1)。众所周知，对于包含奇异性的抛物线-椭圆趋化系统，由于交叉扩散项在 v=0 附近变得无界，因此 v 的时间均匀正向点式下界对于确定经典解的全局有界性至关重要。为此，文献中的一个关键步骤是为质量函数∫ωu 建立适当的正下界，由于逻辑动力学的存在，质量函数∫ωu 是不保留的，因此它在 v 时会变为有界。与这种方法不同，本文在不使用 v 的均匀正下界的情况下得到了 (0.1) 经典解的有界性。其中，本文证明了在不建立 v 的时间均匀正点下限的情况下，若 λ∈(0,1)，则存在 μ>μ⁎，从而对于所有适当光滑的初始数据，任何全局定义正解的 Lp-norm（对于任意 p≥2）都是有界的；此外，问题（0.1）具有唯一的全局定义经典解。此外，在附加假设λ<12+1NwithN≥2 条件下，解也证明是均匀有界的。

引用次数: 0

On the propagation of flatness for second order hypoelliptic operators 论二阶次椭圆算子的平坦性传播

IF 2.4 2区数学 Q1 MATHEMATICS

Journal of Differential Equations

Pub Date : 2024-10-30 DOI: 10.1016/j.jde.2024.10.034

Paolo Albano

For a class of hypoelliptic operators with real-analytic coefficients, we provide a criterion ensuring a partial analyticity result. As a consequence, even when the “elliptic” strong unique continuation (i.e. a solution of the homogeneous equation which vanishes of infinite order at a point is zero near such a point) fails, a weaker form of “propagation” of zeroes still holds.

对于一类具有实解析系数的次椭圆算子，我们提供了一个确保部分解析性结果的准则。因此，即使 "椭圆 "强唯一延续（即在某一点上无穷阶消失的同次方程解在该点附近为零）失效，零点 "传播 "的较弱形式仍然成立。

引用次数: 0

Minimal P-cyclic periodic brake orbits in semi-positive Hamiltonian system 半正向哈密顿系统中的最小 P 循环周期性制动轨道

IF 2.4 2区数学 Q1 MATHEMATICS

Journal of Differential Equations

Pub Date : 2024-10-30 DOI: 10.1016/j.jde.2024.10.021

Ruiliang Gao , Xiaorui Li , Duanzhi Zhang

In this paper, the authors study the minimal period problems for brake orbits in two types of first order Hamiltonian systems in

R^{2 n}

, namely the superquadratic type and asymptotically linear type. In both cases the Hamiltonian systems are assumed to be reversible, semi-positive, and symmetric with respect to certain orthogonal symplectic linear transformation P generating a p-order cyclic subgroup acting freely on

R^{2 n} ∖ {0}

. The authors prove that if

P = R (θ_{1}) ⋄ \dots ⋄ R (θ_{n})

and

θ_{i} \in [0, π]

for each

1 \leq i \leq n

, then for each

T > 0

there exists a pT-periodic P-cyclic brake orbit with minimal period belonging to an finite set with the form

{\frac{p T}{l p + q} : l, q \in Z, 0 \leq l \leq n + 1, \gcd (q, p) = 1, 1 \leq q \leq p - 1}

for both cases, which is an generalization of the results in [10]. The main tools involved are the iteration inequalities for Maslov-type indices, the saddle point reduction method and the Galerkin approximation method under the corresponding Lagrangian boundary condition.

在本文中，作者研究了 R2n 中两类一阶哈密顿系统制动轨道的最小周期问题，即超二次型和渐近线性型。在这两种情况下，哈密顿系统都被假定为可逆的、半正的和对称的，相对于某些正交交映线性变换 P，产生一个自由作用于 R2n∖{0} 的 p 阶循环子群。作者证明，如果 P=R(θ1)⋄...⋄R(θn)，且θi∈[0,π]为每个 1≤i≤n，那么对于每个 T>0，存在一个具有最小周期的 pT 周期 P 循环制动轨道，属于一个有限集，其形式为{pTlp+q：l,q∈Z,0≤l≤n+1,gcd(q,p)=1,1≤q≤p-1}，这是对 [10] 中结果的概括。所涉及的主要工具是马斯洛夫指数迭代不等式、鞍点还原法和相应拉格朗日边界条件下的伽勒金近似法。

引用次数: 0

Orbital stability of smooth solitons in H1 ∩ W1,4 for the modified Camassa-Holm equation 修正卡马萨-霍尔姆方程 H1 ∩ W1,4 中光滑孤子的轨道稳定性

IF 2.4 2区数学 Q1 MATHEMATICS

Journal of Differential Equations

Pub Date : 2024-10-30 DOI: 10.1016/j.jde.2024.10.032

Qian Zhang, Guangming Zhu

We analyze the stability of smooth solitary waves in the modified Camassa-Holm equation, a quasilinear, integrable model for shallow water wave propagation. Through phase portrait analysis, we identify a unique smooth solitary wave within a certain range of the dispersive parameter. Using variational methods, we prove the orbital stability of this wave under small disturbances, solving a minimization problem with constraints. We strengthen the

H^{1} \cap W^{1, 4}

stability result in Li and Liu (2021) [8].

我们分析了修正的卡马萨-霍尔姆方程中光滑孤波的稳定性，该方程是一种准线性、可积分的浅水波传播模型。通过相位肖像分析，我们确定了在一定分散参数范围内的唯一平滑孤波。利用变分法，我们证明了这种波在小扰动下的轨道稳定性，求解了一个带约束条件的最小化问题。我们加强了 Li 和 Liu (2021) [8] 中的 H1∩W1,4 稳定性结果。

引用次数: 0

The optimal time-decay estimates for 2-D inhomogeneous Navier-Stokes equations 二维非均质纳维-斯托克斯方程的最佳时间衰减估计值

IF 2.4 2区数学 Q1 MATHEMATICS

Journal of Differential Equations

Pub Date : 2024-10-24 DOI: 10.1016/j.jde.2024.10.026

Yanlin Liu

In this paper, we derive the optimal time-decay estimates for 2-D inhomogeneous Navier-Stokes equations. In particular, we prove that

{‖ u (t) ‖}_{{\dot{B}}_{p, 1}^{θ} (R^{2})} = O (t^{\frac{1}{p} - \frac{3}{2} - \frac{θ}{2}})

t \to \infty

for any

p \in [2, \infty [, θ \in [0, 2]

if initially

ρ_{0} u_{0} \in {\dot{B}}_{2, \infty}^{- 2} (R^{2})

. This is optimal even for the classical homogeneous Navier-Stokes equations. Different with Schonbek and Wiegner's Fourier splitting device, our method here seems more direct, and can adapt to many other equations as well. Moreover, our method allows us to work in the

L^{p}

-based spaces.

本文推导了二维非均质纳维-斯托克斯方程的最优时间衰减估计。特别是，我们证明了 "u(t) "B˙p,1θ(R2)=O(t1p-32-θ2) as t→∞ for any p∈[2,∞[,θ∈[0,2] if initially ρ0u0∈B˙2,∞-2(R2) if initially ρ0u0∈B˙2, ∞-2(R2)。即使对于经典的均质纳维-斯托克斯方程来说，这也是最优的。与 Schonbek 和 Wiegner 的傅立叶分裂装置不同，我们的方法似乎更直接，也能适用于许多其他方程。此外，我们的方法允许我们在基于 Lp 的空间中工作。

引用次数: 0

Sequential stability of weak martingale solutions to stochastic compressible Navier-Stokes equations with viscosity vanishing on vacuum 真空上粘度消失的随机可压缩纳维-斯托克斯方程弱鞅解的连续稳定性

IF 2.4 2区数学 Q1 MATHEMATICS

Journal of Differential Equations

Pub Date : 2024-10-24 DOI: 10.1016/j.jde.2024.10.016

Zdzisław Brzeźniak , Gaurav Dhariwal , Ewelina Zatorska

In this paper, we investigate the compressible Navier-Stokes equations with degenerate, density-dependent, viscosity coefficient driven by multiplicative stochastic noise. We consider three-dimensional periodic domain and prove that the family of weak martingale solutions is sequentially compact.

在本文中，我们研究了由乘法随机噪声驱动的可压缩纳维-斯托克斯方程，该方程具有退化的、与密度相关的粘性系数。我们考虑了三维周期域，并证明弱鞅解的族是连续紧凑的。

引用次数: 0

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