English (United Kingdom)

https://curated-unify.zendy.io/wp-json/zendy-region/v1/featured_content/oa?rat=en

https://curated-unify.zendy.io/wp-json/zendy-region/v1/highlighted_journal/

Global: Zendy Plus annual

Presents the access of premium content as premium feature

Premium Content

Presents the keyphrase highlighting as premium feature

Keyphrase Highlighting

Presents the summarisation as premium feature

Summarisation

Insights

Presents the pdf analysis as premium feature

PDF Analysis

Presents the zaia usage as premium feature

ZAIA

Global: Zendy Tools annual

Global: Zendy Free Plan

Global: Zendy Tools monthly

Global: Zendy Plus monthly

Consider the steady solution to the incompressible Euler equation $\baru=Ae_1$ 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 associatedNavier-Stokes equation with the no-slip condition on the flat boundaries, forsmall viscosities $\nu=A/\mathsf{Re}$, and initial values in $L^2$. We areinterested in the weak inviscid limits up to subsequences $u^\nu\rightharpoonupu^\infty$ 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 theenergy 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 doublelimit. It is still unknown whether this inviscid limit is unconditionally true.The convex integration method produces solutions $u _E$ to the Euler equationwith 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 ofa layer separation with an energy of order $A^3 T$. We show in this paper thatthe 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 limitof Leray-Hopf solutions of the Navier-Stokes equation. Especially, it showsthat, even if the limit is not unique, the shear flow pattern is observable upto time $1/A$. This provides a notion of stability despite the possiblenon-uniqueness of the limit predicted by the convex integration theory. Theresult relies on a new boundary vorticity estimate for the Navier-Stokesequation. This new estimate, inspired by previous work on higher regularityestimates for Navier-Stokes, provides a nonlinear control scalable through theinviscid limit.

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