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

Abstract

Consider the steady solution to the incompressible Euler equation ū = Ae 1 in the periodic tunnel Ω = T d−1 × (0, 1) in dimension d = 2, 3. Consider now the family of solutions u ν to the associated Navier-Stokes equation with the no-slip condition on the flat boundaries, for small viscosities ν = A/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 ν converges to Ae 1 when the viscosity converges to 0 and the initial value converges to Ae 1 . It is still unknown whether this inviscid limit is unconditionally true. The convex integration method predicts the possibility of a layer separation with energy at time T up to u ν (T ) − Ae 1 2 L 2 (Ω) ≈ A 3 T. In this work, we prove that at the double limit for the inviscid asymptotic, where both the Reynolds number Re converges to infinity and the initial value u ν (0) converges to Ae 1 in L 2 , the energy of layer separation cannot be more than u ν (T ) − Ae 1 2A 3 T (log Re) N . This result holds unconditionally for any limit of Leray-Hopf solutions of Navier-Stokes equation. Especially, it shows that, even if the limit is not unique, the shear flow pattern is observable up to time A −1 (log Re) −N . 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 rescalable through the inviscid limit.

Layer separation of the 3D incompressible Navier-Stokes equation in a bounded domain

“…Indeed, if ū vanishes on the boundary, then A = 0 and LS(ū) = 0, which can also be verified by elementary computation. This result is a generalization of the previous work by the authors [VY21] which studied the setting when ū is a static shear flow in a finite channel without force.…”

“…Proof. The proof is the same as the one in [VY21], with only some mild modifications to resolve the curved boundary issue. Without loss of generality, assume by linearity that ∇u…”

“…Their construction is based on self-similar solutions [JS15] and Euler instability [Vis18a, Vis18b, ABC + 21]. Moreover, at the Euler level, even near a constant plug flow ū = Ae 1 , Széklyhidi [Szé11,VY21] constructed nonunique Euler solutions ũ using convex integrations with a layer separation of…”