Global well-posedness for 2D Navier–Stokes equations with horizontal dissipation in only one component

Abstract

This paper concerns a special anisotropic Navier–Stokes equations on periodic boxes. The system only has horizontal dissipation in the horizontal component equation. Based on special properties of the periodic domain and decomposition techniques, we prove the global well-posedness and stability of the symmetric solution in $H^2$. Furthermore, exponential decay rates are obtained for $u_2$ and the oscillation ${\tilde{u}}_1^{\left(1\right)}$ in horizontal periodic. Our work extends the stability result in Dong et al. (2021) to weaker dissipation.