Navier-Stokes方程在R^n中的前向自相似解

中文摘要 AI 辅助

我们研究了具有齐次初值 $u_0(x)=\sigma(x/|x|)/|x|$, $x\in\mathbb R^n$ 的不可压缩Navier--Stokes方程的前向自相似解，重点关注临界四维情形。对于维数 $3\leq n\leq5$ 的存在性理论，我们为Leray剖面问题开发了一个Galerkin格式，该格式包含自相似数据的分解以及跨空间尺度的非局部热核效应。利用数据的结构性质和Leray算子的强制性，我们导出了均匀的\textit{先验}估计，取极限，并获得了弱前向自相似解。在4D情形中，我们证明了线性Leray系统的新Stokes型估计。逆Leray算子允许作为奇异积分算子的积分表示；其$L^p$有界性由Calderón--Zygmund理论得出。结合对流项的4D紧致性估计，这闭合了正则性迭代。对于空间衰减，我们在加权空间中重新表述问题。低频核的不可积性导致在$\sigma\in W^{1,\infty}(\mathbb S^3)$下的对数损失。当$\sigma\in C^{1,\gamma}(\mathbb S^3)$, $0<\gamma<1$时，频率局部化估计消除了这一损失并产生了尖锐衰减。

英文摘要

We investigate forward self-similar solutions to the incompressible Navier--Stokes equations with homogeneous initial data $u_0(x)=σ(x/|x|)/|x|$, $x\in\mathbb R^n$, focusing on the critical four-dimensional case. For the existence theory in dimensions $3\leq n\leq5$, we develop a Galerkin scheme for the Leray profile problem, featuring a decomposition of the self-similar data and nonlocal heat kernel effects across spatial scales. Exploiting the structural properties of the data and the coercivity of the Leray operator, we derive uniform \textit{a priori} estimates, pass to the limit, and obtain weak forward self-similar solutions. In the 4D case, we prove new Stokes-type estimates for the linear Leray system. The inverse Leray operator admits an integral representation as a singular integral operator; its $L^p$-boundedness follows from Calderón--Zygmund theory. Combined with a 4D compactness estimate for the convective term, this closes the regularity iteration. For spatial decay, we reformulate the problem in weighted spaces. The non-integrability of the low-frequency kernel causes a logarithmic loss under $σ\in W^{1,\infty}(\mathbb S^3)$. With $σ\in C^{1,γ}(\mathbb S^3)$, $0<γ<1$, frequency-localized estimates remove this loss and yield sharp decay.

发表机构

• Hunan Normal University（湖南师范大学）
• Institute of Applied Physics and Computational Mathematics（应用物理与计算数学研究所）
• Beihang University（北京航空航天大学）
• Key Laboratory of Mathematics, Informatics and Behavioral Semantics, Ministry of Education（教育部数学、信息学与行为语义重点实验室）

机构由 AI 辅助整理，请以论文原文为准。

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56 pages,0 figure