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

Abstract

Consider the steady solution to the incompressible Euler equation $\bar{u}=A{e}_1$ 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 $u_0^{\nu }$ converges to $A{e}_1$ in $L^2$. Under a conditional assumption on the energy dissipation close to the boundary, Kato showed in 1984 that $u^{\nu }$ converges to $A{e}_1$ 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 $A{e}_1$ which verify at time $0<T<T_0$: $\Vert u_E(T)-A{e}_1{\Vert }_{L^2(\Omega )}^2\approx A^{3}T$. 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 $\Vert u^{\infty }(T)-A{e}_1{\Vert }_{L^2(\Omega )}^2\lesssim A^{3}T$. 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.

29 pages, 5 figures. We improved our conclusion and estimated the layer separation of any weak inviscid limit

References in corpus (2)

• Vanishing Viscosity Limits and Boundary Layers for Circularly Symmetric 2D Flows
• Remarks on high Reynolds numbers hydrodynamics and the inviscid limit

Cited by in corpus (3)

• Layer separation of the 3D incompressible Navier-Stokes equation in a bounded domain
• Hölder regularity of the pressure for weak solutions of the 3D Euler equations in bounded domains
• Vorticity interior trace estimates and higher derivative estimates via blow-up method