Boundary Vorticity Estimates for Navier-Stokes and Application to the Inviscid Limit

Abstract

Consider the steady solution to the incompressible Euler equation $\bar u=Ae_1$ in the periodic tunnel $\Omega=\mathbb T^{d-1}\times(0,1)$ in dimension $d=2,3$. 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 $\nu=A/\mathsf{Re}$, and initial values in $L^2$. We are interested in the weak inviscid limits up to subsequences $u^\nu\rightharpoonup u^\infty$ when both the viscosity $\nu$ converges to 0, and the initial value $u^\nu_0$ converges to $Ae_1$ in $L^2$. Under a conditional assumption on the energy dissipation close to the boundary, Kato showed in 1984 that $u^\nu$ converges to $Ae_1$ strongly in $L^2$ uniformly in time under this double limit. It is still unknown whether this inviscid limit is unconditionally true. The convex integration method produces solutions $u _E$ to the Euler equation with the same initial values $Ae_1$ which verify at time $0<T<T_0$: $\|u_E(T)-Ae_1\|_{L^2(\Omega)}^2\approx A^3T$. This predicts the possibility of a layer separation with an energy of order $A^3 T$. We show in this paper that the energy of layer separation associated with any asymptotic $u^\infty$ obtained via double limits cannot be more than $\|u^\infty(T)-Ae_1\|_{L^2 (\Omega)}^2\lesssim A^3T$. This result holds unconditionally for any weak limit of Leray-Hopf solutions of the Navier-Stokes equation. 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 nonlinear control scalable through the inviscid limit.