Journal of Functional Analysis最新文献

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Absence of anomalous dissipation for vortex sheets 涡旋片不存在异常耗散

IF 1.6 2区数学 Q1 MATHEMATICS

Journal of Functional Analysis

Pub Date : 2025-12-08 DOI: 10.1016/j.jfa.2025.111304

Tarek M. Elgindi , Milton C. Lopes Filho , Helena J. Nussenzveig Lopes

A family of solutions of the incompressible Navier-Stokes equations is said to present anomalous dissipation if energy dissipation due to viscosity does not vanish in the limit of small viscosity. In this article we present a proof of absence of anomalous dissipation for 2D flows on the torus, with an arbitrary non-negative measure plus an integrable function as initial vorticity and square-integrable initial velocity. Our result applies to flows with forcing and provides an explicit estimate for the dissipation at small viscosity. The proof relies on a new refinement of a classical inequality due to J. Nash.

当不可压缩的Navier-Stokes方程的能量耗散在小黏度的极限下不消失时，其解就会出现反常耗散。本文用一个任意的非负测度加上一个可积函数作为初始涡量和初始速度的平方可积，给出了二维流在环面上不存在异常耗散的证明。我们的结果适用于有强迫的流动，并提供了小粘度下耗散的显式估计。这个证明依赖于纳什对一个经典不等式的新改进。

引用次数: 0

Clustering type solutions for critical elliptic system in dimension two 二维临界椭圆系统的聚类解

IF 1.6 2区数学 Q1 MATHEMATICS

Journal of Functional Analysis

Pub Date : 2025-12-08 DOI: 10.1016/j.jfa.2025.111299

Xu Zhang , Ying Zhang , Rui Zhu

<div><div>We are concerned with clustering-peak solutions to the following stationary Hamiltonian elliptic system<math><mrow><mo>{</mo><mtable><mtr><mtd></mtd><mtd><mo>−</mo><msup><mrow><mi>ε</mi></mrow><mrow><mn>2</mn></mrow></msup><mi>Δ</mi><mi>u</mi><mo>+</mo><mi>V</mi><mo>(</mo><mi>x</mi><mo>)</mo><mi>u</mi><mo>=</mo><mi>g</mi><mo>(</mo><mi>v</mi><mo>)</mo><mspace></mspace><mspace></mspace><mrow><mtext>in</mtext><mspace></mspace></mrow><mspace></mspace><msup><mrow><mi>R</mi></mrow><mrow><mn>2</mn></mrow></msup><mo>,</mo></mtd></mtr><mtr><mtd></mtd><mtd><mo>−</mo><msup><mrow><mi>ε</mi></mrow><mrow><mn>2</mn></mrow></msup><mi>Δ</mi><mi>v</mi><mo>+</mo><mi>V</mi><mo>(</mo><mi>x</mi><mo>)</mo><mi>v</mi><mo>=</mo><mi>f</mi><mo>(</mo><mi>u</mi><mo>)</mo><mspace></mspace><mspace></mspace><mrow><mtext>in</mtext><mspace></mspace></mrow><mspace></mspace><msup><mrow><mi>R</mi></mrow><mrow><mn>2</mn></mrow></msup><mo>,</mo></mtd></mtr><mtr><mtd></mtd><mtd><mspace></mspace><mi>u</mi><mo>(</mo><mi>x</mi><mo>)</mo><mo>→</mo><mn>0</mn><mo>,</mo><mspace></mspace><mspace></mspace><mi>v</mi><mo>(</mo><mi>x</mi><mo>)</mo><mo>→</mo><mn>0</mn><mspace></mspace><mrow><mi>as</mi></mrow><mspace></mspace><mo>|</mo><mi>x</mi><mo>|</mo><mo>→</mo><mo>∞</mo><mo>.</mo></mtd></mtr></mtable></mrow></math> Here <math><mi>ε</mi><mo>></mo><mn>0</mn></math> is a small parameter, V has a local maximum point, and <math><mi>f</mi><mo>,</mo><mi>g</mi></math> are assumed to be of critical growth in the sense of the Trudinger–Moser inequality. Differently from most results that consider solutions for the critical equation near the ground state level <math><msub><mrow><mi>c</mi></mrow><mrow><mn>0</mn></mrow></msub></math>, we attempt to construct high-energy solutions at levels close to <math><mi>k</mi><msub><mrow><mi>c</mi></mrow><mrow><mn>0</mn></mrow></msub></math> for any integer k. The solutions possess k peaks that cluster around a local maximum of V as <math><mi>ε</mi><mo>→</mo><mn>0</mn></math>. Due to the critical growth of the nonlinear terms, in order to handle the compactness problem, we make uniform estimates of the exponential integral of functions in a suitable neighborhood. Since there is no non-degeneracy property for the ground state solutions of the limit system, the variational method is applied here, and a sensitive lower gradient estimate should be made when the local centroids of the functions are away from the local maximum of V. We introduce a new method to obtain this estimate, which differs from the ideas of Del Pino and Felmer (2002) [27], and Byeon and Tanaka (2013, 2014) [9], [10]. In addition, the fact that the energy functional corresponding to Hamiltonian elliptic systems is strongly indefinite poses additional difficulties for

我们关注以下平稳哈密顿椭圆系统{−ε2Δu+V(x)u=g(V)inR2,−ε2Δv+V(x) V =f(u)inR2,u(x)→0，V(x)→0as|x|→∞的聚类峰解。这里ε>；0是一个小参数，V有一个局部极大点，f，g在Trudinger-Moser不等式意义上被假定为临界增长。与大多数考虑临界方程在基态能级c0附近的解的结果不同，我们试图在接近kc0的能级上构造任意整数k的高能解。解具有k个峰，这些峰聚集在V的局部最大值ε→0附近。由于非线性项的临界增长，为了处理紧性问题，我们在合适的邻域内对函数的指数积分作了一致估计。由于极限系统的基态解不具有非简并性，本文采用变分方法，当函数的局部质心远离v的局部最大值时，需要进行敏感的低梯度估计。本文引入了一种不同于Del Pino and Felmer(2002)[27]和Byeon and Tanaka(2013, 2014)[9]，[10]的新方法来获得这种估计。此外，哈密顿椭圆系统对应的能量泛函是强不定的，这给我们的证明带来了额外的困难。通过考虑外部区域上的辅助极大极小问题和对初始路径能量的精确估计，得到了该泛函在合适邻域内的连接结构。结合前面提到的梯度估计和应用局部变形的方法，我们得到了系统期望解的存在性。

引用次数: 0

On the existence of extensions for manifold-valued Sobolev maps on perforated domains 穿孔域上流形值Sobolev映射扩展的存在性

IF 1.6 2区数学 Q1 MATHEMATICS

Journal of Functional Analysis

Pub Date : 2025-12-01 Epub Date: 2025-07-23 DOI: 10.1016/j.jfa.2025.111142

Chiara Gavioli , Leon Happ , Valerio Pagliari

Motivated by manifold-constrained homogenization problems, we construct suitable extensions for Sobolev functions defined on a perforated domain and taking values in a compact, connected

C^{2}

-manifold without boundary. The proof combines a by now classical extension result for the unconstrained case with a retraction argument that heavily relies on the topological properties of the manifold. With the ultimate goal of providing necessary conditions for the existence of extensions for Sobolev maps between manifolds, we additionally investigate the relationship between this problem and the surjectivity of the trace operator for such functions.

在流形约束齐次化问题的激励下，我们构造了定义在穿孔区域上的Sobolev函数的适当扩展，并在无边界的紧致连通c2流形上取值。该证明结合了无约束情况下的经典扩展结果和严重依赖于流形拓扑性质的缩回论证。为了提供流形间Sobolev映射扩展存在的必要条件，我们进一步研究了该问题与此类函数的迹算子的满射性之间的关系。

引用次数: 0

A refined Lusin type theorem for gradients 梯度的一个改进的Lusin型定理

IF 1.6 2区数学 Q1 MATHEMATICS

Journal of Functional Analysis

Pub Date : 2025-12-01 Epub Date: 2025-07-23 DOI: 10.1016/j.jfa.2025.111152

Luigi De Masi, Andrea Marchese

We prove a refined version of the celebrated Lusin type theorem for gradients by Alberti, stating that any Borel vector field f coincides with the gradient of a

C^{1}

function g, outside a set E of arbitrarily small Lebesgue measure. We replace the Lebesgue measure with any Radon measure μ, and we obtain that the estimate on the

L^{p}

norm of Dg does not depend on

μ (E)

, if the value of f is μ-a.e. orthogonal to the decomposability bundle of μ. We observe that our result implies the 1-dimensional version of the flat chain conjecture by Ambrosio and Kirchheim on the equivalence between metric currents and flat chains with finite mass in

R^{n}

and we state a suitable generalization for k-forms, which would imply the validity of the conjecture in full generality.

我们证明了Alberti关于梯度的著名的Lusin类型定理的一个改进版本，说明任何Borel向量场f与任意小Lebesgue测度的集合E外的C1函数g的梯度重合。我们用任意Radon测度μ代替Lebesgue测度，得到Dg在Lp范数上的估计不依赖于μ(E)，如果f的值为μ-a.e。正交于μ的可分解束。我们观察到，我们的结果暗示了Ambrosio和Kirchheim关于度量流与有限质量的平面链在Rn中的等价性的平链猜想的一维版本，并且我们陈述了k形式的适当推广，这将暗示猜想在完全一般情况下的有效性。

引用次数: 0

Hölder regularity of solutions of the steady Boltzmann equation with soft potentials Hölder具有软势的稳定玻尔兹曼方程解的规律性

IF 1.6 2区数学 Q1 MATHEMATICS

Journal of Functional Analysis

Pub Date : 2025-12-01 Epub Date: 2025-07-23 DOI: 10.1016/j.jfa.2025.111146

Kung-Chien Wu , Kuan-Hsiang Wang

We consider the Hölder regularity of solutions to the steady Boltzmann equation with in-flow boundary condition in bounded and strictly convex domains

Ω \subset R^{3}

for gases with cutoff soft potential

(- 3 < γ < 0)

. We prove that there is a unique solution with a bounded

L^{\infty}

norm in space and velocity. This solution is Hölder continuous, and its order depends not only on the regularity of the incoming boundary data, but also on the potential power γ. The result for modulated soft potential case

- 2 < γ < 0

is similar to hard potential case

(0 \leq γ < 1)

since we have

C^{1}

velocity regularity from collision part. However, we observe that for very soft potential case

(- 3 < γ \leq - 2)

, the regularity in velocity obtained by the collision part is lower (Hölder only), but the boundary regularity still can transfer to solution (in both space and velocity) by transport and collision part under the restriction of γ.

我们考虑有界和严格凸域内具有流动边界条件的稳定玻尔兹曼方程解的Hölder正则性对于具有截止软势（−3<γ<0）的气体Ω∧R3。证明了在空间和速度上存在一个具有有界L∞范数的唯一解。该解是Hölder连续的，其阶数不仅取决于输入边界数据的规律性，还取决于势幂γ。调制软势情形−2<；γ<；0的结果与硬势情形（0≤γ<1）相似，因为我们从碰撞部分得到了C1速度规律。然而，我们观察到，对于极软势情况（−3<γ≤−2），碰撞部分得到的速度规律性较低（仅Hölder），但在γ的限制下，边界规律性仍然可以通过输运和碰撞部分传递到解（在空间和速度上）。

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引用次数: 0

Critical prescribed Q-curvature flow on closed even-dimensional manifolds with sign-changing functions 具有变符号函数的闭偶维流形上的临界规定q曲率流

IF 1.7 2区数学 Q1 MATHEMATICS

Journal of Functional Analysis

Pub Date : 2025-12-01 Epub Date: 2025-07-16 DOI: 10.1016/j.jfa.2025.111133

Leilei Cui , Changfeng Gui , Haicheng Yan , Wen Yang

In this article, we consider the prescribed Q-curvature equation

P u = ρ (\frac{h e^{2 n u}}{\int_{M} h e^{2 n u} d μ} - \frac{1}{| M |_{g_{0}}}) in M,

where

(M, g_{0})

is a closed 2n-dimensional Riemannian manifold,

P

represents the GJMS operator, which is (weakly) positive with a kernel of constant functions. The function h is smooth and sign-changing, while ρ is a positive constant. In the critical case with

ρ = 4^{n} (n - 1)! π^{n}

, we employ a negative gradient-like flow method to establish the existence of solutions to this prescribed Q-curvature equation. Our approach extends the work of Li-Xu [46], which focused on dimension 2, to general even dimensions. This result can also be viewed as a counterpart to [8] in the case where h is a sign-changing function.

本文考虑规定的q曲率方程pu =ρ(he2nu∫mhe2nu μ - 1|M|g0)inM，其中（M,g0）是一个封闭的2n维黎曼流形，P表示GJMS算子，它是一个（弱）正的常数函数核。函数h是光滑的，可以改变符号，而ρ是一个正常数。在临界情况下ρ=4n(n−1)！πn时，我们采用负梯度流动法来证明这个规定的q曲率方程解的存在性。我们的方法将Li-Xu[46]的工作从关注维度2扩展到一般的偶数维度。在h是一个改变符号的函数的情况下，这个结果也可以看作[8]的对应。

引用次数: 0

Dimension free estimates for the vector-valued Hardy–Littlewood maximal function on the Heisenberg group Heisenberg群上向量值Hardy-Littlewood极大函数的无维估计

IF 1.6 2区数学 Q1 MATHEMATICS

Journal of Functional Analysis

Pub Date : 2025-11-19 DOI: 10.1016/j.jfa.2025.111285

Pritam Ganguly , Abhishek Ghosh

In this article, we establish dimension-free Fefferman-Stein inequalities for the Hardy-Littlewood maximal function associated with averages over Korányi balls in the Heisenberg group. We also generalize the result to more general UMD lattices. As a key stepping stone, we establish the

L^{p}

- boundedness of the vector-valued Nevo-Thangavelu spherical maximal function, which plays a crucial role in our proofs of the main theorems.

在本文中，我们建立了与海森堡群中Korányi球上的平均值相关的Hardy-Littlewood极大函数的无量纲Fefferman-Stein不等式。我们还将结果推广到更一般的UMD格。作为一个重要的步骤，我们建立了向量值的Nevo-Thangavelu球面极大函数的Lp有界性，它在我们主要定理的证明中起着至关重要的作用。

引用次数: 0

Asymptotic stability of non-self-similar rarefaction wave for two-dimensional viscous Burgers equation 二维粘性Burgers方程非自相似稀疏波的渐近稳定性

IF 1.6 2区数学 Q1 MATHEMATICS

Journal of Functional Analysis

Pub Date : 2025-11-19 DOI: 10.1016/j.jfa.2025.111286

Feimin Huang , Guiqin Qiu , Yi Wang , Xiaozhou Yang

<div><div>We investigate the large time behavior of solutions to the two-dimensional viscous Burgers equation <math><msub><mrow><mi>u</mi></mrow><mrow><mi>t</mi></mrow></msub><mo>+</mo><mi>u</mi><msub><mrow><mi>u</mi></mrow><mrow><mi>x</mi></mrow></msub><mo>+</mo><mi>u</mi><msub><mrow><mi>u</mi></mrow><mrow><mi>y</mi></mrow></msub><mo>=</mo><mi>Δ</mi><mi>u</mi></math>, toward a non-self-similar rarefaction wave of inviscid Burgers equation with two initial constant states, separated by a curve <math><mi>y</mi><mo>=</mo><mi>φ</mi><mo>(</mo><mi>x</mi><mo>)</mo></math>, and prove that the above 2D non-self-similar rarefaction wave is time-asymptotically stable. This is the first result on the nonlinear time-asymptotic stability of non-self-similar rarefaction waves. Furthermore, we can get the decay rate. Both the rarefaction wave strength and the initial perturbation can be large.</div><div>The main difficulty comes from the fact that the initial discontinuity of 2D non-self-similar rarefaction wave is a curve <math><mi>y</mi><mo>=</mo><mi>φ</mi><mo>(</mo><mi>x</mi><mo>)</mo></math>. Fortunately, we uncover a novel property that the non-self-similar inviscid rarefaction wave is also asymptotically stable with respect to the discontinuity curve <math><mi>y</mi><mo>=</mo><mi>φ</mi><mo>(</mo><mi>x</mi><mo>)</mo></math>. More precisely, let <math><msubsup><mrow><mi>u</mi></mrow><mrow><mi>i</mi></mrow><mrow><mi>R</mi></mrow></msubsup><mo>(</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo>,</mo><mi>t</mi><mo>)</mo><mo>,</mo><mi>i</mi><mo>=</mo><mn>1</mn><mo>,</mo><mn>2</mn></math> be the corresponding non-self-similar rarefaction wave with the initial discontinuity curve <math><mi>y</mi><mo>=</mo><msub><mrow><mi>φ</mi></mrow><mrow><mi>i</mi></mrow></msub><mo>(</mo><mi>x</mi><mo>)</mo></math>, then <math><msub><mrow><mo>‖</mo><msubsup><mrow><mi>u</mi></mrow><mrow><mn>1</mn></mrow><mrow><mi>R</mi></mrow></msubsup><mo>−</mo><msubsup><mrow><mi>u</mi></mrow><mrow><mn>2</mn></mrow><mrow><mi>R</mi></mrow></msubsup><mo>‖</mo></mrow><mrow><msup><mrow><mi>L</mi></mrow><mrow><mo>∞</mo></mrow></msup></mrow></msub><mo>≤</mo><mfrac><mrow><mi>C</mi></mrow><mrow><mi>t</mi></mrow></mfrac></math> if <math><msub><mrow><mo>‖</mo><msub><mrow><mi>φ</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>(</mo><mi>x</mi><mo>)</mo><mo>−</mo><msub><mrow><mi>φ</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>(</mo><mi>x</mi><mo>)</mo><mo>‖</mo></mrow><mrow><msup><mrow><mi>L</mi></mrow><mrow><mo>∞</mo></mrow></msup></mrow></msub></math> is bounded. Based on this property, we prove that the asymptotic stability of non-self-similar rarefaction wave corresponding to the general initial discontinuity <math><mi>y</mi><mo>=</mo><mi>φ</mi><mo>(</mo><mi>x</mi><mo>)</mo></math> is equivalent to that of the non-self-similar rarefaction wave with an initial discontinuity given by the modification curve

我们研究了二维粘性Burgers方程ut+uux+ uy=Δu的解在具有两个初始常数状态的无粘性Burgers方程的非自相似稀疏波上的大时间行为，并证明了该二维非自相似稀疏波是时间渐近稳定的。这是关于非自相似稀疏波的非线性时间渐近稳定性的第一个结果。进一步，我们可以得到衰减率。稀疏波强度和初始扰动都可能很大。主要困难在于二维非自相似稀疏波的初始不连续是一条y=φ(x)的曲线。幸运的是，我们发现了非自相似无粘稀薄波对于不连续曲线y=φ(x)也是渐近稳定的一个新性质。更精确地说，设uiR(x,y,t),i=1,2为初始不连续曲线y=φi(x)对应的非自相似稀疏波，若‖φ1(x)−φ2(x)‖L∞有界，则‖u1R−u2R‖L∞≤Ct。基于这一性质，证明了具有一般初始不连续y=φ(x)的非自相似稀疏波的渐近稳定性与具有初始不连续y=φ(x)的非自相似稀疏波的渐近稳定性等价于由折线的修正曲线给出的非自相似稀疏波的渐近稳定性，且折线的左右斜率为k±=limx→±∞(x)x。然后在上述修正曲线的基础上构造了粘性Burgers方程的近似光滑稀疏波，并通过新的非线性坐标变换将该波转化为自相似的平面稀疏波，同时将二维粘性Burgers方程转化为具有可变和混合导数粘度的抛物方程。另一个优点是新的抛物方程近似光滑稀疏产生的主要误差项在R2中是可积的。这些新方法使我们能够克服上述主要困难。最后，采用合适的时间加权lp -能量估计对粘性稀薄波和变换后的二维粘性Burgers方程进行了时间渐近稳定性分析。

引用次数: 0

On some bilinear Fourier multipliers with oscillating factors, I 在一些带有振荡因子的双线性傅立叶乘法器上，I

IF 1.6 2区数学 Q1 MATHEMATICS

Journal of Functional Analysis

Pub Date : 2025-11-19 DOI: 10.1016/j.jfa.2025.111289

Tomoya Kato , Akihiko Miyachi , Naoto Shida , Naohito Tomita

Bilinear Fourier multipliers of the form

e^{i (| ξ | + | η | + | ξ + η |)} σ (ξ, η)

are considered. It is proved that if

σ (ξ, η)

is in the Hörmander class

S_{1, 0}^{m} (R^{2 n})

with

m = - (n + 1) / 2

then the corresponding bilinear operator is bounded in

L^{\infty} \times L^{\infty} \to b m o

h^{1} \times L^{\infty} \to L^{1}

, and

L^{\infty} \times h^{1} \to L^{1}

. This improves a result given by Rodríguez-López, Rule and Staubach.

考虑形式为ei(|ξ|+|η|+|ξ+η|)σ(ξ,η)的双线性傅立叶乘子。证明了当σ(ξ,η)在Hörmander类S1,0m（R2n）中，且m= - (n+1)/2，则相应的双线性算子在L∞×L∞→bmo、h1×L∞→L1、L∞×h1→L1上有界。这改进了Rodríguez-López、Rule和Staubach给出的结果。

引用次数: 0

Special potentials for relativistic Laplacians I: Fractional Rollnik-class 相对论拉普拉斯算子I的特殊势：分数阶rollnik类

IF 1.6 2区数学 Q1 MATHEMATICS

Journal of Functional Analysis

Pub Date : 2025-11-19 DOI: 10.1016/j.jfa.2025.111282

Giacomo Ascione , Atsuhide Ishida , József Lőrinczi

We propose a counterpart of the classical Rollnik-class of potentials for fractional and massive relativistic Laplacians, and describe this space in terms of appropriate Riesz potentials. These definitions rely on precise resolvent estimates, which we present in detail. We obtain these classes for diverse ranges of fractional exponent in dimensions

1 \leq d \leq 3

, and for the physical operators with fractional exponent

α = 1

in dimensions one and two as limiting cases resulting under Γ-convergence. We show that Coulomb-type potentials are elements of fractional Rollnik-class up to but not including the critical singularity of the Hardy potential. In a second part of the paper we derive detailed results on the self-adjointness and spectral properties of relativistic Schrödinger operators obtained under perturbations by fractional Rollnik potentials. We also define an extended fractional Rollnik-class which is the maximal space for the Hilbert-Schmidt property of the related Birman-Schwinger operators.

我们提出了分数和质量相对论拉普拉斯算子的经典rollnik类势的对应，并用适当的Riesz势来描述这个空间。这些定义依赖于精确的解决方案估计，我们将详细介绍。我们得到了在1≤d≤3维中分数指数的不同范围，以及在Γ-convergence下得到的1维和2维中分数指数α=1的物理算子的极限情况。我们证明了库仑型势是分数阶rollnik类的元素，直到但不包括Hardy势的临界奇点。在论文的第二部分，我们得到了分数阶罗尼克势扰动下相对论Schrödinger算子的自伴随性和谱性质的详细结果。我们还定义了一个扩展分数rollnik类，它是相关Birman-Schwinger算子Hilbert-Schmidt性质的极大空间。

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