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

We consider distributional weak solutions $u\in L_t^{\mathrm{\infty }}{L}_x^2$ of the two-dimensional Navier–Stokes equations with horizontal viscosity on ${\mathbb{T}}^2$ or ${\mathbb{R}}^2$, without assuming ${\mathrm{\partial }}_{1}u\in L_t^2{L}_x^2$. We show that the Lions integrability condition $u\in L_t^4{L}_x^4$ together with Onsager-critical regularity in the nondissipative direction, $u\in L_t^3{B}_{3,\mathrm{\infty }}^{1\mathrm{/}3,v}$ with the Besov regularity imposed only in the vertical direction, implies ${\mathrm{\partial }}_{1}u\in L_t^2{L}_x^2$, with a quantitative bound on the horizontal dissipation. Thus the dissipative regularity is a consequence of the equation and need not be part of the definition of the solution. If moreover $u\in L_t^3{B}_{3,c_0}^{1\mathrm{/}3,v}$, then $u$ belongs to the isotropic critical Onsager space $L_t^3{B}_{3,c_0}^{1\mathrm{/}3}$, satisfies the energy equality, and is continuous in $L^2$. Here $c_0$ indicates that the corresponding dyadic Besov sequence tends to zero at high frequencies. The proof combines absorption of the horizontal energy flux into the dissipation with a one-dimensional commutator estimate for the vertical energy flux.