Abstract. Consider the steady solution to the incompressible Euler equation Ae1 in the periodic tunnel Ω = [0, 1] × T2 . Consider now the family of solutions Uν to the associated Navier-Stokes equation with no-slip condition on the flat boundaries, for small viscosities ν = 1/Re, and initial values close in L2 to Ae1. Under a conditional assumption on the energy dissipation close to the boundary, Kato showed in 1984 that Uν converges to Ae1 when the viscosity converges to 0 and the initial value converge to Ae1. It is still unknown whether this inviscid is unconditionally true. Actually, the convex integration method predicts the possibility of a layer separation. It produces solutions to the Euler equation with initial values Ae1, but with layer separation energy at time T up to:
||U(T) − Ae1||2L2 ≡ A3T.
In this work we prove that at the double limit for the inviscid asymptotic U¯, where both the Reynolds number Re converges to infinity and the initial value Uν converges to Ae1 in L2 , the energy of layer separation cannot be more than:
||U¯(T) − Ae1||2L2 ≡ A3T.
Especially, it shows that, even if 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.
