arXiv 1502.045942015-02-16math.AP

Codimension one threshold manifold for the critical gKdV equation

Yvan Martel, Frank Merle, Kenji Nakanishi, Pierre Raphael

DOI：: 10.1007/s00220-015-2509-3

英文摘要：

We construct the 'threshold manifold' near the soliton for the mass critical gKdV equation, completing results obtained in arXiv:1204.4625 and arXiv:1204.4624. In a neighborhood of the soliton, this C1 manifold of codimension one separates solutions blowing up in finite time and solutions in the 'exit regime'. On the manifold, solutions are global in time and converge locally to a soliton. In particular, the soliton behavior is strongly unstable by blowup.

机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。

阅读原文

阅读 PDF ↗

arXiv 页面 ↗HTML 原文 ↗

本文内容

英文摘要相关论文

机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。

继续探索

相关论文

修正型和三次广义Korteweg-de Vries（gKdV）方程的粗糙数据的瞬时解析光滑化

Instantaneous analytic smoothing of rough data for the modified and cubic gKdV equations

Instituto de Matemática, Estatística e Computação Científica (IMECC), Universidade Estadual de Campinas (UNICAMP)（坎皮纳斯州立大学数学、统计和科学计算研究所）；Universidade Federal de Ouro Preto (UFOP)（欧罗普雷托联邦大学）

该研究针对修正型和三次gKdV方程，证明满足Nelson型条件的低正则奇异初始数据对应的解，在t≠0时瞬时实解析，扩展了KdV的光滑效应，采用Lorentz空间细化避免伪微分演算。

2026-07-30math.AP

One phase problem for two positive harmonic function: below the codimension $1$ threshold

What can be said about the domain $\Om$ in $\bR^n$ for which its Green's function $G(z)$ satisfies $G(z)\asymp \dist (z, \pd\Om)^δ$? What can we say about $\Om$ if the Boundary Harnack Principle holds in the form $u/v=\text{real analytic}$ on the part $E$ of its boundary? Here $u, v$ are positive harmonic functions on $\Om$ vanishing on $E$. Is this part of the boundary also nice? We discuss these questions below and give answers in very special cases.

2022-05-20math.APmath.CAmath.CV

Threshold for blowup for equivariant wave maps in higher dimensions

We consider equivariant wave maps from $\mathbb{R}^{d+1}$ to $\mathbb{S}^d$ in supercritical dimensions $3\leq d\leq 6$. Using mixed numerical and analytic methods, we show that the threshold of blowup is given by the codimension-one stable manifold of a self-similar solution with one instability.

2016-08-27math.AP

Dynamics at the threshold for blowup for supercritical wave equations outside a ball

We consider spherically symmetric supercritical focusing wave equations outside a ball. Using mixed analytical and numerical methods, we show that the threshold for blowup is given by a codimension-one stable manifold of the unique static solution with exactly one unstable direction. We analyze in detail the convergence to this critical solution for initial data fine-tuned to the threshold.

2019-09-04math.AP

一维聚焦三次 Klein–Gordon 方程孤子的余维一稳定性

On codimension one stability of the soliton for the 1D focusing cubic Klein-Gordon equation

本文针对一维聚焦三次 Klein–Gordon 方程孤子的偶扰动，克服阈值共振导致的小除数困难，结合超对称方法与适配泛函框架证明余维一扰动在指数时间尺度内的衰减估计。

2024-03-01math.AP

调和函数梯度在余维二附近的小性传播

Propagation of smallness near codimension two for gradients of harmonic functions

本文研究调和函数梯度的小性传播，证明在具有正 $(n-2+δ)$ 维 Hausdorff content 的集合上梯度 $ε$-小时，可推出球内梯度上界，并达到维数阈值。

2025-08-28math.AP

更多相关论文收起更多论文

二维三次Zakharov-Kuznetsov方程的有限或无限时间爆破

Blow-up in finite or infinite time of the 2D cubic Zakharov-Kuznetsov equation

本文证明了2D三次聚焦Zakharov-Kuznetsov方程的近阈值负能量解会在有限或无限时间内爆破，通过非线性Liouville定理和局部-virial时空估计排除非平凡线性解，并引入新方法处理二维情形。

2025-11-02math.AP

二维三次Zakharov--Kuznetsov方程最小质量爆破解的构造

Construction of blow-up solution with minimal mass for 2D cubic Zakharov--Kuznetsov equation

针对二维三次Zakharov--Kuznetsov方程，构造其最小质量爆破解，研究低正则性解在质量阈值处的动力学并证明有限时间爆破可能发生，补充了能量空间中最小质量爆破元不存在的结果。

2025-08-23math.AP

关于周期非线性薛定谔和广义科赫尔-德维尔斯方程Sobolev范数增长的探讨

On growth of Sobolev norms for periodic nonlinear Schrödinger and generalised Korteweg-de Vries equations under critical Gibbs dynamics

研究在环状上聚焦质量临界NLS和gKdV方程的Sobolev范数对数增长界，利用Orlicz空间中的不变测度论证方法。

2025-12-22math.APmath.PR

临界广义KdV方程的慢行进无穷点爆破

Slowly travelling infinite point blow-up for the critical generalized KdV equation

Laboratoire de mathématiques de Versailles（凡尔赛数学实验室）；Université de Versailles St-Quentin en Y.（圣康坦凡尔赛大学）；CNRS（法国国家科学研究中心）；Université Paris-Saclay（巴黎萨克雷大学）

本文研究临界广义KdV方程的有限时间爆破，证明存在慢行进无穷点爆破解，速率阈值ν=1/2，采用指数尾项并调整能量-位力泛函，扩展了相关爆破解的构造结果。

2026-08-10math.AP

Higher dimensional generalization of the Benjamin-Ono equation: 2D case

We consider a higher-dimensional version of the Benjamin-Ono (HBO) equation in the 2D setting: $u_t- \mathcal{R}_1 Δu + \frac{1}{2}(u^2)_x=0, (x,y) \in \mathbb{R}^2$, which is $L^2$-critical, and investigate properties of solutions both analytically and numerically. For a generalized equation (fractional 2D gKdV) after deriving the Pohozaev identities, we obtain non-existence conditions for solitary wave solutions, then prove uniform bounds in the energy space or conditional global existence, and investigate the radiation region, a specific wedge in the negative $x$-direction. We then introduc…

2021-03-29cs.NAmath.APmath.NA

关于二维三次 Zakharov-Kuznetsov 方程的近孤子动力学

On the near soliton dynamics for the 2D cubic Zakharov-Kuznetsov equations

本文研究二维三次质量临界 Zakharov-Kuznetsov 方程在孤子附近解的渐近行为，证明解只有离开孤子管状邻域、有限时间爆破或全局存在并局部收敛到孤子三种可能，并借助调制估计和修正能量-virial 泛函给出证明。

2024-06-29math.AP

Bilinear control of a degenerate hyperbolic equation

We consider the linear degenerate wave equation, on the interval $(0, 1)$ $$ w_{tt} - (x^αw_x)_x = p(t) μ(x) w, $$ with bilinear control $p$ and Neumann boundary conditions. We study the controllability of this nonlinear control system, locally around a constant reference trajectory, the ground state. We prove that, generically with respect to $μ$, any target close to the ground state in the $H^3\times H^2$ topology (suitably adapted to the underlying degenerate operator) is reachable in time $T > \frac{4}{2-α}$, with controls in $L^2((0, T ),\mathbb R)$. Under some classical and generic assum…

2021-12-01math.APmath.OC

引力坍缩中的临界现象

Critical Phenomena in Gravitational Collapse

本文综述了广义相对论引力坍缩中的临界现象，涵盖其普适性与幂律缩放特征，并介绍了超越球对称的数值模拟及PDE奇点形成等最新数学进展。

2025-07-10gr-qcmath.AP

$\mathbb{R}^N$ 与紧致度量图乘积上的非线性薛定谔方程

The nonlinear Schrödinger equation on products of $\mathbb{R}^N$ and compact metric graphs

Università degli Studi di Torino（都灵大学）；Politecnico di Milano（米兰理工大学）

研究 $\mathbb{R}^N$ 与紧致度量图乘积上的质量约束聚焦非线性薛定谔方程，建立函数框架并证明基态存在性、临界阈值及维度交叉现象。

2026-09-21math-phmath.APmath.MP