Forward self-similar solutions of the Navier-Stokes equations in R^n
Abstract
We investigate forward self-similar solutions to the incompressible Navier--Stokes equations with homogeneous initial data u0(x)=σ(x/|x|)/|x|, x∈ℝn, focusing on the critical four-dimensional case. For the existence theory in dimensions 3≤ n≤5, 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 Lp-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∈ W1,∞(𝕊3). With σ∈ C1,γ(𝕊3), 0<γ<1, frequency-localized estimates remove this loss and yield sharp decay.
Related
- Large forward discretely self-similar solutions to the two-dimensional Navier-Stokes equations with rough data
- Self-similar solutions to the steady Navier-Stokes equations in arbitrarily high dimensions
- Homothetic Self-Similar Solutions to the Incompressible Navier-Stokes Equations
- Finite-Time Blowup for Navier-Stokes with Smooth Forcing, Part I: Construction of Self-Similar Solutions with Admissible Stress and Flat Remainder
- Wasserstein regularity of vorticity solutions to the 2D Navier-Stokes equations
BibTeX
@misc{lai2026forward,
title = {{Forward self-similar solutions of the Navier-Stokes equations in R^n}},
author = {Baishun Lai and Changxing Miao and Xiaoxin Zheng},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.33316},
eprint = {2609.33316},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.33316},
}
RIS
TY - UNPB
TI - Forward self-similar solutions of the Navier-Stokes equations in R^n
AU - Baishun Lai
AU - Changxing Miao
AU - Xiaoxin Zheng
PY - 2026
DA - 2026/09/27
PB - arXiv
DO - 10.48550/arxiv.2609.33316
UR - https://arxiv.org/abs/2609.33316
AB - We investigate forward self-similar solutions to the incompressible Navier--Stokes equations with homogeneous initial data u_0(x)=σ(x/|x|)/|x|, x∈ℝ^n, focusing on the critical four-dimensional case. For the existence theory in dimensions 3≤ n≤5, 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∈ W^1,∞(𝕊^3). With σ∈ C^1,γ(𝕊^3), 0<γ<1, frequency-localized estimates remove this loss and yield sharp decay.
ER -