首页/论文/2011.08133

arXiv 2011.081332020-11-16math.AP

On the commutativity of flows of rough vector fields

Maria Colombo, Riccardo Tione

英文摘要：

In the class of Sobolev vector fields in $\mathbb{R}^n$ of bounded divergence, for which the theory of DiPerna and Lions provides a well defined notion of flow, we characterize the vector fields whose flow commute in terms of the Lie bracket and of a regularity condition on the flows themselves. This extends a classical result of Frobenius in the smooth setting.

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

Smooth approximation is not a selection principle for the transport equation with rough vector field

In this paper we analyse the selection problem for weak solutions of the transport equation with rough vector field. We answer in the negative the question whether solutions of the equation with a regularized vector field converge to a unique limit, which would be the selected solution of the limit problem. To this aim, we give a new example of a vector field which admits infinitely many flows. Then we construct a smooth approximating sequence of the vector field for which the corresponding solutions have subsequences converging to different solutions of the limit equation.

2019-11-05math.AP

On smooth approximations of rough vector fields and the selection of flows

In this work we deal with the selection problem of flows of an irregular vector field. We first summarize an example from \cite{CCS} of a vector field $b$ and a smooth approximation $b_ε$ for which the sequence $X^ε$ of flows of $b_ε$ has subsequences converging to different flows of the limit vector field $b$. Furthermore, we give some heuristic ideas on the selection of a subclass of flows in our specific case.

2019-02-02math.AP

二维粗糙向量场消失李括号的几何解释

Geometric interpretation of the vanishing Lie Bracket for two-dimensional rough vector fields

本文证明了实平面上具有有界散度的连续 Sobolev 向量场在零李括号条件下其流可交换，从而去除了 Colombo-Tione（2021）所需的弱李可微性附加假设，并探讨了向 BV 情形的推广。

2025-03-11math.AP

高阶无界粗糙驱动项

Higher Order Unbounded Rough Drivers

University of Agder（阿格德尔大学）

该研究构建了高阶无界粗糙驱动项的框架，将其应用于含两类不同驱动项的粗糙输运方程，证明了该方程的适定性。

2026-09-07math.APmath.PR

在粗糙平面集合上消失的无散向量场的逼近

Approximation of divergence-free vector fields vanishing on rough planar sets

本文研究二维有界域中无散 Sobolev 向量场能否被紧支撑的无散光滑向量场逼近，针对不同 $p$ 值给出边界正则性条件，并证明 Hölder 空间中一般有界域均成立，结果与 Stokes 方程弱解唯一性相关。

2024-11-20math.AP

粗糙向量场积分曲线的随机选择

On the stochastic selection of integral curves of a rough vector field

本文证明一类有界无散粗糙向量场的积分曲线上存在唯一不可压缩测度，并借助 Depauw 向量场表明该唯一测度可具有随机性。

2024-07-04math.APmath.CAmath.PR

更多相关论文收起更多论文

Critical non Sobolev regularity for continuity equations with rough force fields

We present a divergence free vector field in the Sobolev space $H^1$ such that the flow associated to the field does not belong to any Sobolev space. The vector field is deterministic but constructed as the realization of a random field combining simple elements. This construction illustrates the optimality of recent quantitative regularity estimates as it gives a straightforward example of a well-posed flow which has nevertheless only very weak regularity.

2015-07-17math.APmath.CA

Effective wall-laws for the Stokes equations over curved rough boundaries

We derive effective wall-laws for Stokes systems with inhomogeneous boundary conditions in three dimensional bounded domains with curved rough boundaries. No-slip boundary condition is given on the locally periodic rough boundary parts with characteristic roughness size $ε$ and boundary data is assumed to be supported in the nonoscillatory smooth boundary. Based on the analysis of a boundary layer cell problem depending on geometry of the fictitious boundary and roughness shape, boundary layer approximations are constructed using orthogonal tangential vectors and normal vector on the fictitiou…

2013-11-05math-phmath.APmath.MP

具有粗糙速度场的对流-扩散方程的弱解与抛物解

Weak and parabolic solutions of advection-diffusion equations with rough velocity field

研究粗糙速度场下对流-扩散方程的弱解与抛物解，总结存在性、正则性与唯一性结果，提供原创证明并提出基于凸积分格式的开放问题。

2023-06-27math.AP

抛物型边值问题中粗糙边界条件的迹理论

Trace theory for parabolic boundary value problems with rough boundary conditions

Universität Konstanz（康斯坦茨大学）；Delft University of Technology（代尔夫特理工大学）

本文提出了一种适用于处理边界处导数发散的抛物型边值问题的迹理论，证明了热方程在粗糙边界数据下的适定性。

2025-12-17math.AP

沿 $\mathscr{C}^0$-向量场的测度传播与粗糙紧流形上的波可控性

Measure propagation along a $\mathscr{C}^0$-vector field and wave controllability on a rough compact manifold

本文在无边界紧流形的低正则性设定下，重新审视几何控制条件，通过研究连续系数 ODE 测度解的支集传播，导出了波动方程能观性与精确可控性的充分条件，并探讨了度量扰动下的稳定性与敏感性。

2024-09-24math.APmath.OC

带粗糙系数的对流扩散方程有限体积格式的误差估计

Error estimates for a finite volume scheme for advection-diffusion equations with rough coefficients

该研究针对低正则性DiPerna-Lions框架下带粗糙系数的对流扩散方程，分析隐式迎风有限体积格式的收敛性，证明非结构正则网格上收敛速率至少为一阶，并通过对数Kantorovich-Rubinstein距离给出弱收敛误差界。

2023-02-27cs.NAmath.APmath.NA

粗糙区域上散度测度张量场的高斯 - 格林公式

Gauss-Green formulas for divergence measure tensor fields on rough domains

Sapienza University of Rome（罗马第一大学）；University of Naples Federico II（那不勒斯费德里科二世大学）

研究粗糙区域上散度测度张量场的高斯 - 格林公式，通过引入新配对概念扩展经典理论，调整法向迹定义，建立低正则性集合上的公式，为不规则张量场和区域的分部积分提供统一强大框架。

2026-07-14math.APmath.FA

Noise prevents infinite stretching of the passive field in a stochastic vector advection equation

A linear stochastic vector advection equation is considered; the equation may model a passive magnetic field in a random fluid. When the driving velocity field is rough but deterministic, in particular just Hölder continuous and bounded, one can construct examples of infinite stretching of the passive field, arising from smooth initial conditions. The purpose of the paper is to prove that infinite stretching is prevented if the driving velocity field contains in addition a white noise component.

2014-02-28math-phmath.APmath.MP