Global Stability and Decay for the 2D Anisotropic Navier-Stokes Equations with Fractional Horizontal Dissipation
Abstract
Although the classical Yudovich theory guarantees global existence and uniqueness of two-dimensional Euler solutions with bounded vorticity, their higher-order norms may still grow in time. This raises a fundamental question: how much dissipation we need to ensure global stability and decay? While the full Laplacian dissipation is sufficient, the stabilizing effect of weaker, anisotropic dissipation is considerably more subtle. In this paper, we investigate the global stability and long-time decay of the two-dimensional incompressible Navier-Stokes equations with fractional horizontal dissipation Λ12su, where s<1. A central challenge is to recover sufficient decay of the second velocity component, whose weak dissipation obstruct the closure of conventional anisotropic energy estimates. Previous work by Wang-Wu-Zhu [arXiv:2601.16110] resolved this difficulty whereas the spatial weighted energy functionals must be used when s approaches $1$. The main purpose of our manuscript is to remove this restriction. We introduce a hierarchy of carefully designed time-weighted anisotropic energy functionals which captures the interplay between horizontal fractional dissipation and the decay of velocity components. Under such initial assumptions, the characteristics of anisotropic dissipation is presented more straightforwardly, which also reveals the more essential intrinsic structure of fluid stability.
Related
- Sharp linear stability and the absence of enhanced dissipation for Kolmogorov flow in the 2D Navier--Stokes equations with horizontal dissipation
- Global well-posedness for the 2D primitive equations with subcritical horizontal dissipation
- Stability of 2D MHD with Only Horizontal Dissipation near a Background Magnetic Field on ℝ2
- An Anisotropic Onsager Criterion for the Two-Dimensional Navier--Stokes Equations with Horizontal Viscosity
- The local Cauchy problem for hyperbolic systems of equations with fractional dissipation
BibTeX
@misc{shen2026global,
title = {{Global Stability and Decay for the 2D Anisotropic Navier-Stokes Equations with Fractional Horizontal Dissipation}},
author = {Chenyan Shen and Dan Wu and Yi Zhu},
howpublished = {arXiv},
year = {2026},
month = nov,
doi = {10.48550/arxiv.2611.03224},
eprint = {2611.03224},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2611.03224},
}
RIS
TY - UNPB
TI - Global Stability and Decay for the 2D Anisotropic Navier-Stokes Equations with Fractional Horizontal Dissipation
AU - Chenyan Shen
AU - Dan Wu
AU - Yi Zhu
PY - 2026
DA - 2026/11/02
PB - arXiv
DO - 10.48550/arxiv.2611.03224
UR - https://arxiv.org/abs/2611.03224
AB - Although the classical Yudovich theory guarantees global existence and uniqueness of two-dimensional Euler solutions with bounded vorticity, their higher-order norms may still grow in time. This raises a fundamental question: how much dissipation we need to ensure global stability and decay? While the full Laplacian dissipation is sufficient, the stabilizing effect of weaker, anisotropic dissipation is considerably more subtle. In this paper, we investigate the global stability and long-time decay of the two-dimensional incompressible Navier-Stokes equations with fractional horizontal dissipation Λ_1^2su, where s<1. A central challenge is to recover sufficient decay of the second velocity component, whose weak dissipation obstruct the closure of conventional anisotropic energy estimates. Previous work by Wang-Wu-Zhu [arXiv:2601.16110] resolved this difficulty whereas the spatial weighted energy functionals must be used when s approaches $1$. The main purpose of our manuscript is to remove this restriction. We introduce a hierarchy of carefully designed time-weighted anisotropic energy functionals which captures the interplay between horizontal fractional dissipation and the decay of velocity components. Under such initial assumptions, the characteristics of anisotropic dissipation is presented more straightforwardly, which also reveals the more essential intrinsic structure of fluid stability.
ER -