consider now the family of solutions $u_\nu$ to the associated navier-stokes equation with the no-slip condition on the flat boundaries, for small viscosities $\nu=1/ re$, and initial values close in $l^2$ to $ae_1$. under a conditional assumption on the energy dissipation close to the boundary, kato showed in 1984 that $u_\nu$ converges to $ae_1$ when the viscosity converges to 0 and the initial value converges to $a e_1$. it is still unknown whether this inviscid is unconditionally valid. actually, the convex integration method predicts the possibility of layer separation. it produces solutions to the euler equation with initial values $ae_1 $, but with layer separation energy at time t up to:

 $$\|u(t)-ae_1\|^2_{l^2}\equiv a^3t.$$

in this work, we prove that at the double limit for the inviscid asymptotic $\bar{u}$, where both the reynolds number $re$ converges to infinity and the initial value $u_{\nu}$ converges to $ae_1$ in $l^2$, the energy of layer separation cannot be more than:

$$\| \bar{u}(t)-ae_1\|^2_{l^2}\lesssim a^3t.$$

especially, it shows that, even if the limit is not unique, the shear flow pattern is observable up to time $1/a$. this provides a notion of stability despite the possible non-uniqueness of the limit predicted by the convex integration theory.

the result relies on a new boundary vorticity estimate for the navier-stokes equation. this new estimate, inspired by previous work on higher regularity estimates for navier-stokes, provides a non-linear control scalable through the inviscid limit.