L1 amplification near a self-similar vortex collapse in 2D Navier-Stokes
Abstract
We establish quantitative L1 instability for the two-dimensional Navier--Stokes equations near a self-similar three-vortex collapse as viscosity tends to zero. For every 0<γ<1/2 and t0∈(0,1), we construct solutions that are O(ν1−γ)-close in L1(ℝ2) at t0 but exhibit order-one L1 separation in a shrinking neighborhood of the collapse point at times approaching collapse as ν→0. By time translation, we also obtain pairs of solutions with initial L1 distance O(ν1−γ) that develop order-one separation at a fixed positive time. In both constructions, the solutions have the same viscosity and total circulation and are uniformly bounded in L1(ℝ2). The proof exploits the amplification of small time delays, the fast speed of the vortex centers near collapse, together with sharp inviscid approximation estimates for the Navier--Stokes equations established in~\cite{Ga11,NguyenCollapse}.
Related
- The inviscid limit of 2D Navier-Stokes near a self-similar vortex collapse
- Self-similar stationary Navier--Stokes flows in a half-space with arbitrarily large swirl forcing
- Self-similar solutions to the steady Navier-Stokes equations in arbitrarily high dimensions
- Finite-Time Blowup for Navier-Stokes with Smooth Forcing, Part I: Construction of Self-Similar Solutions with Admissible Stress and Flat Remainder
- Homothetic Self-Similar Solutions to the Incompressible Navier-Stokes Equations
BibTeX
@misc{nguyen2026amplification,
title = {{$L^1$ amplification near a self-similar vortex collapse in 2D Navier-Stokes}},
author = {Trinh T. Nguyen},
howpublished = {arXiv},
year = {2026},
month = nov,
doi = {10.48550/arxiv.2611.01433},
eprint = {2611.01433},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2611.01433},
}
RIS
TY - UNPB
TI - L^1 amplification near a self-similar vortex collapse in 2D Navier-Stokes
AU - Trinh T. Nguyen
PY - 2026
DA - 2026/11/01
PB - arXiv
DO - 10.48550/arxiv.2611.01433
UR - https://arxiv.org/abs/2611.01433
AB - We establish quantitative L^1 instability for the two-dimensional Navier--Stokes equations near a self-similar three-vortex collapse as viscosity tends to zero. For every 0<γ<1/2 and t_0∈(0,1), we construct solutions that are O(ν^1−γ)-close in L^1(ℝ^2) at t_0 but exhibit order-one L^1 separation in a shrinking neighborhood of the collapse point at times approaching collapse as ν→0. By time translation, we also obtain pairs of solutions with initial L^1 distance O(ν^1−γ) that develop order-one separation at a fixed positive time. In both constructions, the solutions have the same viscosity and total circulation and are uniformly bounded in L^1(ℝ^2). The proof exploits the amplification of small time delays, the fast speed of the vortex centers near collapse, together with sharp inviscid approximation estimates for the Navier--Stokes equations established in~\cite{Ga11,NguyenCollapse}.
ER -