Large forward discretely self-similar solutions to the two-dimensional Navier-Stokes equations with rough data
Year: 2026
Type: preprint
Abstract: We construct forward discretely self-similar (DSS) solutions of two-dimensional Navier-Stokes equations with arbitrarily large data in $L^2_{loc}(\mathbb{R}^2\setminus\{0\})$. For such rough initial data, we first establish a sharp linear estimate under the discrete scaling, which identifies the \(L^2\)-norm of the initial datum on a fundamental annulus with the Dirichlet energy of its caloric lift over one period in logarithmic time. This, together with a structural cancellation between the linear profile and the nonlinear remainder over one period in logarithmic time, yields a uniform \(\dot{H}^1\)-bound for the remainder, where the natural $L^2_{loc}$-regularity of the initial data seems to be sharp. Since the two-dimensional Leray operator has no \(L^2\)-coercivity, we recover uniform energy control for the remainder by exploiting \(L^p\)-estimates ($1
Source: arXiv (Cornell University)
Authors: Gui, Changfeng, Liu, Hao, Xie, Chunjing
Institution: University of Macau
Language: English
Subfield: Applied Mathematics
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7214220602