$L^1$ amplification near a self-similar vortex collapse in 2D Navier-Stokes

Abstract

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 $0<γ<\frac12$ and $t_0\in(0,1)$, we construct solutions that are $O(ν^{1-γ})$-close in $L^1(\mathbb{R}^2)$ at $t_0$ but exhibit order-one $L^1$ separation in a shrinking neighborhood of the collapse point at times approaching collapse as $ν\to0$. 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 $L^1(\mathbb{R}^2)$. 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}.