Boundary Vorticity Estimates for NavierStokes and Application to the Inviscid Limit
Abstract
Consider the steady solution to the incompressible Euler equation $\bar u=Ae_1$ in the periodic tunnel $\Omega=\mathbb T^{d1}\times(0,1)$ in dimension $d=2,3$. Consider now the family of solutions $u^\nu$ to the associated NavierStokes equation with the noslip 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 LerayHopf solutions of the NavierStokes 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 nonuniqueness of the limit predicted by the convex integration theory. The result relies on a new boundary vorticity estimate for the NavierStokes equation. This new estimate, inspired by previous work on higher regularity estimates for NavierStokes, provides a nonlinear control scalable through the inviscid limit.
 Publication:

arXiv eprints
 Pub Date:
 October 2021
 arXiv:
 arXiv:2110.02426
 Bibcode:
 2021arXiv211002426V
 Keywords:

 Mathematics  Analysis of PDEs;
 Physics  Fluid Dynamics;
 76D05;
 35B65
 EPrint:
 29 pages, 5 figures. We improved our conclusion and estimated the layer separation of any weak inviscid limit