Large forward discretely self-similar solutions to the two-dimensional Navier-Stokes equations with rough data
Abstract
We construct forward discretely self-similar (DSS) solutions of two-dimensional Navier-Stokes equations with arbitrarily large data in L2loc(ℝ2∖{0}). For such rough initial data, we first establish a sharp linear estimate under the discrete scaling, which identifies the L2-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 Ḣ1-bound for the remainder, where the natural L2loc-regularity of the initial data seems to be sharp. Since the two-dimensional Leray operator has no L2-coercivity, we recover uniform energy control for the remainder by exploiting Lp-estimates (1<p<2), together with a key bootstrap argument and periodicity. We then first construct smooth DSS solutions using further weighted estimates and a fixed-point argument as long as the initial data belong to C2loc(ℝ2∖{0}). These stronger weighted bounds are not uniform under approximation of rough data. Instead, the passage to L2loc data relies only on the unweighted energy estimates. Finally, we use strong time continuity of the limiting periodic profile to recover the uniform spatial tightness needed to verify the initial data.
Related
- Forward self-similar solutions of the Navier-Stokes equations in R^n
- Global forward self-similar solutions of the spatially inhomogeneous Boltzmann equation for cutoff Maxwell molecules
- Self-similar Imploding Solutions of the Relativistic Euler Equations
- Self-similar solutions for the gSQG patch equations
- Self-similar imploding solutions of the 1D compressible Euler equations with a far field cutoff
BibTeX
@misc{gui2026large,
title = {{Large forward discretely self-similar solutions to the two-dimensional Navier-Stokes equations with rough data}},
author = {Changfeng Gui and Hao Liu and Chunjing Xie},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.28157},
eprint = {2609.28157},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.28157},
}
RIS
TY - UNPB
TI - Large forward discretely self-similar solutions to the two-dimensional Navier-Stokes equations with rough data
AU - Changfeng Gui
AU - Hao Liu
AU - Chunjing Xie
PY - 2026
DA - 2026/09/23
PB - arXiv
DO - 10.48550/arxiv.2609.28157
UR - https://arxiv.org/abs/2609.28157
AB - We construct forward discretely self-similar (DSS) solutions of two-dimensional Navier-Stokes equations with arbitrarily large data in L^2_loc(ℝ^2∖{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 Ḣ^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<p<2), together with a key bootstrap argument and periodicity. We then first construct smooth DSS solutions using further weighted estimates and a fixed-point argument as long as the initial data belong to C^2_loc(ℝ^2∖{0}). These stronger weighted bounds are not uniform under approximation of rough data. Instead, the passage to L^2_loc data relies only on the unweighted energy estimates. Finally, we use strong time continuity of the limiting periodic profile to recover the uniform spatial tightness needed to verify the initial data.
ER -