Journal de Mathematiques Pures et Appliquees最新文献

英文中文

From KP-I lump solution to travelling waves of Gross-Pitaevskii equation 从Gross-Pitaevskii方程的KP-I块解到行波

IF 2.3 1区数学 Q1 MATHEMATICS

Journal de Mathematiques Pures et Appliquees

Pub Date : 2025-10-03 DOI: 10.1016/j.matpur.2025.103801

Yong Liu , Zhengping Wang , Juncheng Wei , Wen Yang

Let q be a nondegenerate lump type solution to the KP-I (Kadomtsev-Petviashvili-I) equation

\partial_{x}^{4} q - 2 \sqrt{2} \partial_{x}^{2} q - 3 \sqrt{2} \partial_{x} ({(\partial_{x} q)}^{2}) - 2 \partial_{y}^{2} q = 0 .

We show the existence of travelling wave solutions with the form

u_{ε} (x - c t, y)

, for the GP (Gross-Pitaevskii) equation

i \partial_{t} Ψ + Δ Ψ + (1 - | Ψ |^{2}) Ψ = 0 in R^{2},

with travelling speed

c = \sqrt{2} - ε^{2}

, and

u_{ε} = 1 + i ε q + O (ε^{2})

. This proves the existence of finite energy solutions in the so-called Jones-Roberts program within the transonic regime

c \in (\sqrt{2} - ε^{2}, \sqrt{2})

. The main ingredients in our proof are detailed point-wise estimates for the Green functions associated to a family of fourth order hypoelliptic operators. In view of the classification of lump type solutions of the KP-I equation, our proof also indicates that for fixed small ε, there should exist a sequence of travelling wave solutions to GP equation, with energy tends to infinity.

设q是KP-I （Kadomtsev-Petviashvili-I）方程∂x4q−22∂x2q−32∂x((∂xq)2)−2∂y2q=0的非简并块型解。我们证明了GP （Gross-Pitaevskii）方程∂tΨ+ΔΨ+（1−|Ψ|2）Ψ=0inR2的行波解的存在性，其行波解的形式为uε(x−ct,y)，行进速度c=2−ε2，且uε=1+iεq+O（ε2）。这证明了所谓的Jones-Roberts规划在跨声速区间c∈(2−ε2,2)内有限能量解的存在性。我们证明的主要成分是与一组四阶半椭圆算子相关的Green函数的详细点估计。鉴于KP-I方程块状解的分类，我们的证明还表明，对于固定的小ε， GP方程的行波解应该存在一个序列，且能量趋于无穷。

引用次数: 0

Global axisymmetric solution to the 3D incompressible anisotropic Navier–Stokes equations 三维不可压缩各向异性Navier-Stokes方程的全局轴对称解

IF 2.3 1区数学 Q1 MATHEMATICS

Journal de Mathematiques Pures et Appliquees

Pub Date : 2025-10-01 DOI: 10.1016/j.matpur.2025.103807

Hui Chen , Zijin Li , Ping Zhang

In this paper, we prove the global existence and uniqueness of axisymmetric solution to the 3D incompressible anisotropic Navier–Stokes equations in a cylindrical domain with Navier boundary condition provided that the swirl component of the initial velocity is sufficiently small. The main idea of the proof is to perform energy estimates for the pair

(J, Ω^{c})

, where

and

is a corrector of

. In order to close the energy estimates, we introduced the derivative-reduction technique and new elliptic estimates of the pressure function, which are established to overcome difficulties arising from the lower-order terms in the Navier boundary condition. We also consider the global regularity of the axisymmetric solution to the Navier–Stokes equations with full viscosity subject to the total-slip Navier boundary condition. Several new inequalities are established to address the challenges posed by the weak horizontal diffusion of the swirl component.

本文在初始速度的旋流分量足够小的条件下，证明了三维不可压缩各向异性Navier - stokes方程在圆柱域上轴对称解的整体存在唯一性。证明的主要思想是对（J，Ωc）进行能量估计，其中和是的校正器。为了关闭能量估计，我们引入了导数约简技术和新的压力函数椭圆估计，这是为了克服Navier边界条件中低阶项所带来的困难而建立的。同时考虑了全滑移Navier边界条件下具有全黏度的Navier - stokes方程轴对称解的全局正则性。建立了几个新的不等式来解决涡流分量的弱水平扩散所带来的挑战。

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引用次数: 0

Fabes-Stroock approach to higher integrability of Green's functions and ABP estimates with Ld drift Green函数高可积性的Fabes-Stroock方法及Ld漂移下的ABP估计

IF 2.3 1区数学 Q1 MATHEMATICS

Journal de Mathematiques Pures et Appliquees

Pub Date : 2025-10-01 DOI: 10.1016/j.matpur.2025.103805

Pilgyu Jung , Kwan Woo

We explore the higher integrability of Green's functions associated with the second-order elliptic equation

a^{i j} D_{i j} u + b^{i} D_{i} u = f

in a bounded domain

Ω \subset R^{d}

, and establish an enhanced version of Aleksandrov's maximum principle. In particular, we consider the drift term

b = (b^{1}, \dots, b^{d})

L_{d}

and the source term

f \in L_{p}

for some

p < d

. This provides an alternative and analytic proof of a result by N.V. Krylov (Ann. Probab., 2021) concerning

L_{d}

drifts. The key step involves deriving a Gehring-type inequality for Green's functions by using the Fabes-Stroock approach (Duke Math. J., 1984).

我们探索了二阶椭圆方程aijDiju+biDiu=f在有界域Ω∧Rd上的格林函数的高可积性，并建立了Aleksandrov极大原理的增强版本。特别地，我们考虑Ld中的漂移项b=（b1，…，bd）和某些p<；d的源项f∈Lp。这为N.V. Krylov (Ann。Probab。（2021）关于Ld漂移。关键的一步是通过使用Fabes-Stroock方法（杜克数学）推导格林函数的格林型不等式。J。,1984)。

引用次数: 0

Derivation of Hartree theory for two-dimensional attractive Bose gases in almost Gross–Pitaevskii regime 几乎Gross-Pitaevskii状态下二维吸引玻色气体Hartree理论的推导

IF 2.3 1区数学 Q1 MATHEMATICS

Journal de Mathematiques Pures et Appliquees

Pub Date : 2025-10-01 DOI: 10.1016/j.matpur.2025.103800

Lukas Junge , François L.A. Visconti

We study the ground state energy of trapped two-dimensional Bose gases with mean-field type interactions that can be attractive. We prove the stability of second kind of the many-body system and the convergence of the ground state energy per particle to that of a non-linear Schrödinger (NLS) energy functional. Notably, we can take any polynomial scaling of the interaction, and even exponential scalings arbitrarily close to the Gross–Pitaevskii regime, which is a drastic improvement on the best-known result for systems with attractive interactions. As a consequence of the stability of second kind we also obtain Bose–Einstein condensation for the many-body ground states for a much improved range of the diluteness parameter.

我们研究了具有平均场相互作用的二维玻色气体的基态能量。我们证明了第二类多体系统的稳定性和每粒子的基态能量收敛于非线性Schrödinger （NLS）能量泛函。值得注意的是，我们可以对相互作用进行任何多项式缩放，甚至可以任意接近Gross-Pitaevskii状态的指数缩放，这是对具有吸引相互作用的系统的最著名结果的巨大改进。由于第二类的稳定性，我们还获得了多体基态的玻色-爱因斯坦凝聚，其稀释度参数的范围大大提高。

引用次数: 0

Qualitative properties of the spreading speed of a population structured in space and in phenotype 一个种群在空间和表型结构上传播速度的定性性质

IF 2.3 1区数学 Q1 MATHEMATICS

Journal de Mathematiques Pures et Appliquees

Pub Date : 2025-09-30 DOI: 10.1016/j.matpur.2025.103804

Nathanaël Boutillon

We consider a nonlocal Fisher-KPP equation that models a population structured in space and in phenotype. The population lives in a heterogeneous periodic environment: the diffusion coefficient, the mutation coefficient and the fitness of an individual may depend on its spatial position and on its phenotype.

We first prove a Freidlin-Gärtner formula for the spreading speed of the population. We then study the behaviour of the spreading speed in different scaling limits (small and large period, small and large mutation coefficient). Finally, we exhibit new phenomena arising thanks to the phenotypic dimension.

Our results are also valid when the phenotype is seen as another spatial variable along which the population does not spread.

我们考虑一个非局部Fisher-KPP方程，该方程模拟了在空间和表型上结构的种群。种群生活在异质周期性环境中：个体的扩散系数、突变系数和适合度可能取决于其空间位置和表型。我们首先证明了人口扩散速度的Freidlin-Gärtner公式。然后，我们研究了不同尺度极限（小周期和大周期，小突变系数和大突变系数）下的传播速度行为。最后，我们展示了由于表型维度而产生的新现象。当表型被视为另一个空间变量时，我们的结果也是有效的，种群不会沿着这个空间变量传播。

引用次数: 0

Long time classical solutions of quasilinear Klein-Gordon equations with small weakly decaying initial data 具有弱衰减小初始数据的拟线性Klein-Gordon方程的长时间经典解

IF 2.3 1区数学 Q1 MATHEMATICS

Journal de Mathematiques Pures et Appliquees

Pub Date : 2025-09-30 DOI: 10.1016/j.matpur.2025.103803

Fei Hou , Huicheng Yin

It is well known that for the quasilinear Klein-Gordon equation with quadratic nonlinearity and sufficiently decaying small initial data, there exists a global smooth solution if the space dimensions

d \geq 2

. When the initial data are of size

ε > 0

in the Sobolev space, for the semilinear Klein-Gordon equation satisfying the null condition, the authors in the article (Delort and Fang, 2000 [11]) prove that the solution exists in time

[0, T_{ε})

with

T_{ε} \geq C e^{C ε^{- μ}}

(

μ = 1

d \geq 3

μ = 2 / 3

d = 2

). In the present paper, we will focus on the general quasilinear Klein-Gordon equation without the null condition and further show that the existence time of the solution can be improved to

T_{ε} = + \infty

d \geq 3

and

T_{ε} \geq e^{C ε^{- 2}}

d = 2

. In addition, for

d = 2

and any fixed number

α > 0

, if the weighted

L^{2}

norm of the initial data with the weight

{(1 + | x |)}^{α}

is small, then the solution exists globally and scatters to a free solution. Our arguments are based on the introduction of a new good unknown, the Strichartz estimate, the weighted

L^{2}

-norm estimate and the resonance analysis.

众所周知，对于具有二次非线性和充分衰减的小初始数据的拟线性Klein-Gordon方程，当空间维数d≥2时，存在全局光滑解。在Sobolev空间中，当初始数据大小为ε>；0时，对于满足零条件的半线性Klein-Gordon方程，本文（Delort and Fang, 2000[11]）证明了解在时间[0，t)上存在，且t≥cee ε−μ（当d≥3时μ=1，当d=2时μ=2/3）。本文将重点讨论不带零条件的一般拟线性Klein-Gordon方程，并进一步证明当d≥3时，解的存在时间可提高到Tε=+∞，当d=2时，解的存在时间可提高到Tε≥eCε−2。另外，对于d=2和任意固定数α>；0，如果初始数据的权重为(1+|x|)α的加权L2范数较小，则该解全局存在并散射到一个自由解。我们的论点是基于引入一个新的好未知数、Strichartz估计、加权l2 -范数估计和共振分析。

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引用次数: 0

Waiting time solutions in gas dynamics 气体动力学中的等待时间解

IF 2.3 1区数学 Q1 MATHEMATICS

Journal de Mathematiques Pures et Appliquees

Pub Date : 2025-09-30 DOI: 10.1016/j.matpur.2025.103806

Juhi Jang , Jiaqi Liu , Nader Masmoudi

In this article, we construct a continuum family of self-similar waiting time solutions for the one-dimensional compressible Euler equations for the adiabatic exponent

γ \in (1, 3)

in the half-line with the vacuum boundary. The solutions are confined by a stationary vacuum interface for a finite time with at least

C^{1}

regularity of the velocity and the sound speed up to the boundary. Subsequently, the solutions undergo the change of the behavior, becoming only Hölder continuous near the singular point, and simultaneously transition to the solutions to the vacuum moving boundary Euler equations satisfying the physical vacuum condition. When the boundary starts moving, a weak discontinuity emanating from the singular point along the sonic curve emerges. The solutions are locally smooth in the interior region away from the vacuum boundary and the sonic curve.

本文构造了具有绝热指数γ∈(1,3)的一维可压缩欧拉方程在真空边界半直线上的自相似等待时间解的连续统族。解被一个固定的真空界面限制在有限时间内，速度和声速在边界处至少呈C1规律。随后，解发生行为变化，仅在奇点附近Hölder连续，同时过渡到满足物理真空条件的真空移动边界欧拉方程的解。当边界开始移动时，从奇异点沿声波曲线发出的弱不连续出现。解在远离真空边界和声波曲线的内部区域是局部光滑的。

引用次数: 0

Monotonicity formulas for capillary surfaces 毛细管表面的单调性公式

IF 2.3 1区数学 Q1 MATHEMATICS

Journal de Mathematiques Pures et Appliquees

Pub Date : 2025-09-30 DOI: 10.1016/j.matpur.2025.103802

Guofang Wang , Chao Xia , Xuwen Zhang

In this paper, we establish monotonicity formulas for capillary surfaces in the half-space

R_{+}^{3}

and in the unit ball

B^{3}

and extend the result of Volkmann (2016) [27] for surfaces with free boundary. As applications, we obtain Li-Yau-type inequalities for the Willmore energy of capillary surfaces, and extend Fraser-Schoen's optimal area estimate for minimal free boundary surfaces in

B^{3}

(2011) [10] to the capillary setting, which is different to another optimal area estimate proved by Brendle (2023) [5].

本文建立了半空间R+3和单位球B3中毛细曲面的单调性公式，推广了Volkmann(2016)[27]关于自由边界曲面的结果。作为应用，我们得到了毛细表面Willmore能量的li - yau型不等式，并将B3(2011)[10]中最小自由边界表面的Fraser-Schoen最优面积估计推广到毛细环境，这与Brendle(2023)[5]证明的另一种最优面积估计不同。

引用次数: 0

On the magnetic Dirichlet to Neumann operator on the exterior of the disk – Diamagnetism, weak-magnetic field limit and flux effects 圆盘表面的狄利克雷-诺伊曼算子——抗磁性、弱磁场极限和磁通效应

IF 2.3 1区数学 Q1 MATHEMATICS

Journal de Mathematiques Pures et Appliquees

Pub Date : 2025-09-30 DOI: 10.1016/j.matpur.2025.103799

Bernard Helffer, François Nicoleau

In this paper, we analyze the magnetic Dirichlet-to-Neumann operator (D-to-N map)

\overset{ˇ}{Λ} (b, ν)

on the exterior of the disk with respect to a magnetic potential

A_{b, ν} = A^{b} + A_{ν}

where, for

b \in R

and

ν \in R

A^{b} (x, y) = b (- y, x)

and

A_{ν} (x, y)

is the Aharonov-Bohm potential centered at the origin of flux

2 π ν

. First, we show that the limit of

\overset{ˇ}{Λ} (b, ν)

b \to 0

is equal to the D-to-N map

\hat{Λ} (ν)

on the interior of the disk associated with the potential

A_{ν} (x, y)

. Secondly, we study the ground state energy of the D-to-N map

\overset{ˇ}{Λ} (b, ν)

and show that the strong diamagnetism property holds. Finally we slightly extend to the exterior case the asymptotic results as

b \to \infty

obtained in the interior case for general domains.

本文分析了圆盘外部磁势Ab，ν=Ab+ ν的磁Dirichlet-to-Neumann算子（D-to-N映射）Λ + (b,ν)，其中，对于b∈R和ν∈R， Ab(x,y)=b（- y,x）和ν(x,y)是以通量2πν为中心的Aharonov-Bohm势。首先，我们证明了Λ （b,ν）在b→0时的极限等于与势ν(x,y)相关联的磁盘内部的D-to-N映射Λ （ν）。其次，我们研究了D-to-N映射Λ （b,ν）的基态能量，证明了其强抗磁性。最后，我们将在一般区域的内情形下得到的b→∞渐近结果稍微推广到外情形。

引用次数: 0

On long time behavior of solutions of the Schrödinger-KdV system with and without resonant interactions 有和无共振相互作用时Schrödinger-KdV系统解的长时间行为

IF 2.3 1区数学 Q1 MATHEMATICS

Journal de Mathematiques Pures et Appliquees

Pub Date : 2025-09-12 DOI: 10.1016/j.matpur.2025.103792

Deqin Zhou , Felipe Linares

<div><div>We consider the long time behavior of the solutions of the coupled Schrödinger-KdV system<math><mrow><mo>{</mo><mtable><mtr><mtd><mi>i</mi><msub><mrow><mo>∂</mo></mrow><mrow><mi>t</mi></mrow></msub><mi>u</mi><mo>+</mo><msubsup><mrow><mo>∂</mo></mrow><mrow><mi>x</mi></mrow><mrow><mn>2</mn></mrow></msubsup><mi>u</mi><mo>=</mo><mi>α</mi><mi>u</mi><mi>v</mi><mo>+</mo><mi>β</mi><mi>u</mi><mo>|</mo><mi>u</mi><msup><mrow><mo>|</mo></mrow><mrow><mn>2</mn></mrow></msup><mo>,</mo><mspace></mspace><mo>(</mo><mi>x</mi><mo>,</mo><mi>t</mi><mo>)</mo><mo>∈</mo><mi>R</mi><mo>×</mo><msup><mrow><mi>R</mi></mrow><mrow><mo>+</mo></mrow></msup><mo>,</mo></mtd></mtr><mtr><mtd><msub><mrow><mo>∂</mo></mrow><mrow><mi>t</mi></mrow></msub><mi>v</mi><mo>+</mo><msubsup><mrow><mo>∂</mo></mrow><mrow><mi>x</mi></mrow><mrow><mn>3</mn></mrow></msubsup><mi>v</mi><mo>+</mo><mi>v</mi><msub><mrow><mo>∂</mo></mrow><mrow><mi>x</mi></mrow></msub><mi>v</mi><mo>=</mo><mi>γ</mi><msub><mrow><mo>∂</mo></mrow><mrow><mi>x</mi></mrow></msub><mo>(</mo><mo>|</mo><mi>u</mi><msup><mrow><mo>|</mo></mrow><mrow><mn>2</mn></mrow></msup><mo>)</mo><mo>,</mo><mspace></mspace><mo>(</mo><mi>x</mi><mo>,</mo><mi>t</mi><mo>)</mo><mo>∈</mo><mi>R</mi><mo>×</mo><msup><mrow><mi>R</mi></mrow><mrow><mo>+</mo></mrow></msup><mo>,</mo></mtd></mtr><mtr><mtd><mo>(</mo><mi>u</mi><mo>,</mo><mi>v</mi><mo>)</mo><msub><mrow><mo>|</mo></mrow><mrow><mi>t</mi><mo>=</mo><mn>0</mn></mrow></msub><mo>=</mo><mo>(</mo><msub><mrow><mi>u</mi></mrow><mrow><mn>0</mn></mrow></msub><mo>,</mo><msub><mrow><mi>v</mi></mrow><mrow><mn>0</mn></mrow></msub><mo>)</mo><mo>.</mo></mtd></mtr></mtable></mrow></math> We show that global solutions to this system satisfy locally energy decay in a suitable interval, growing unbounded in time, in two situations. In the first case, we regard the parameter vector <math><mo>(</mo><mi>α</mi><mo>,</mo><mi>β</mi><mo>,</mo><mi>γ</mi><mo>)</mo><mo>∈</mo><msup><mrow><mi>R</mi></mrow><mrow><mo>+</mo></mrow></msup><mo>×</mo><mover><mrow><msup><mrow><mi>R</mi></mrow><mrow><mo>+</mo></mrow></msup></mrow><mo>‾</mo></mover><mo>×</mo><msup><mrow><mi>R</mi></mrow><mrow><mo>+</mo></mrow></msup></math> without any size assumption on the initial data in <math><msup><mrow><mi>H</mi></mrow><mrow><mn>1</mn></mrow></msup><mo>(</mo><mi>R</mi><mo>)</mo><mo>×</mo><msup><mrow><mi>H</mi></mrow><mrow><mn>1</mn></mrow></msup><mo>(</mo><mi>R</mi><mo>)</mo></math>. In the second one, we consider the parameter vector <math><mo>(</mo><mi>α</mi><mo>,</mo><mi>β</mi><mo>,</mo><mi>γ</mi><mo>)</mo><mo>∈</mo><msup><mrow><mi>R</mi></mrow><mrow><mo>+</mo></mrow></msup><mo>×</mo><msup><mrow><mi>R</mi></mrow><mrow><mo>−</mo></mrow></msup><mo>×</mo><msup><mrow><mi>R</mi></mrow><mrow><mo>+</mo></mrow></msup></math>. In this case, we give a ‘‘smallness” criterion involving the product of the parameter −β and a constant depending on the initial data in <

我们考虑耦合Schrödinger-KdV系统{i∂tu+∂x2u=αuv+βu|u|2,(x,t)∈R×R+,∂tv+∂x3v+v∂xv=γ∂x(|u|2),(x,t)∈R×R+,(u,v)|t=0=(u0,v0)的长时间行为。在两种情况下，我们证明了该系统的全局解在一个适当的区间内满足局部能量衰减，并随时间无界增长。在第一种情况下，我们考虑参数向量(α,β,γ)∈R+×R+，对H1(R)×H1(R)中的初始数据没有任何大小假设。在第二个例子中，我们考虑参数向量(α,β,γ)∈R+×R−×R+。在这种情况下，我们给出了一个“小”准则，涉及参数- β和一个常数的乘积，这取决于H1(R)×H1(R)中的初始数据。我们的研究结果积极地回答了F. Linares， A. J. Mendez (2021) b[18]中提出的开放性问题。我们使用了与前一篇文章不同的新思路和技术。

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