General validity of the missing transverse regularity

Determine whether every weak solution of the two-dimensional anisotropic incompressible Navier–Stokes equations satisfies the additional regularity $\partial_{x_2}u_1\in L^2(0,T;L^2(\mathbb{R}^2))$.

Background

The anisotropic system controls only the derivative ∂x1u\partial_{x_1}u at the natural energy level. Incompressibility yields control of ∂x2u2\partial_{x_2}u_2, but does not directly provide control of the transverse derivative ∂x2u1\partial_{x_2}u_1.

The authors explicitly state that their proof of energy rigidity does not establish this missing regularity and suggest that it may fail in general. Instead, they obtain energy equality by proving square-integrability of the pressure and a renormalization property for the horizontal component u1u_1. Thus, whether the transverse derivative ∂x2u1\partial_{x_2}u_1 belongs to the natural space for every weak solution remains unresolved.

References

The general idea is not to show the missing regularity $\partial_{x_2}u_1\in L2(0,T;L2(2))$, it likely does not even hold in general, instead we work out a way such that it is not required.

— Energy rigidity and weak-strong uniqueness for the 2D anisotropic Navier-Stokes equations  (2608.19931 - Demmel et al., 20 Aug 2026) in Introduction, subsection “Proof strategy”