An Anisotropic Onsager Criterion for the Two-Dimensional Navier--Stokes Equations with Horizontal Viscosity

We consider distributional weak solutions $u\in L^\infty_tL^2_x$ of the two-dimensional Navier--Stokes equations with horizontal viscosity on $\mathbb T^2$ or $\mathbb R^2$, without assuming $\partial_1u\in L^2_tL^2_x$. We show that the Lions integrability condition $u\in L^4_tL^4_x$ together with O