Alexis Vasseur, University of Texas Austin

Abstract: Consider the steady solution to the incompressible Euler equation $Ae_1$ in the periodic tunnel $\Omega=[0,1]\times \mathbb T^2$. Consider now the family of solutions $U_\nu$ to the associated Navier-Stokes equation with 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 converge to $A e_1$. 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 $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 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.

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