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## Sharp bounds on linear semigroup of Navier Stokes with boundary layer norms

I’ve just uploaded this paper, with E. Grenier, on the arXiv (arXiv:1703.00881), entitled Sharp bounds on linear semigroup of Navier Stokes with boundary layer norms, aiming a better understanding of the classical Prandtl’s boundary layers. Indeed, one of the key difficulties in dealing with boundary layers is the creation of (unbounded) vorticity in the inviscid limit.

More precisely, for incompressible Navier-Stokes flows with the no-slip boundary condition, the vorticity near the boundary is expected to behave as follows:

$\displaystyle \omega \approx \omega^\mathrm{Euler} + \nu^{-1/2} \omega^\mathrm{Prandtl},$

for small viscosity ${\nu\ll1}$. That is, one expects the vorticity to become unbounded, of order ${\nu^{-1/2}}$, in the inviscid limit (for instance, this is indeed the case for data with analytic regularity).

The novelty of this paper is to introduce boundary layer norms that capture the precise boundary layer behavior of the linearized vorticity and to derive sharp semigroup bounds with respect to the boundary layer norms for the linearized Navier-Stokes around an unstable boundary layer. Such a result is possible, thanks to the precise estimates on the Green function for the classical Orr-Sommerfeld problem (see arXiv:1702.07924).

As an immediate application, we construct approximate solutions that exhibit an ${L^\infty}$ instability of the classical Prandtl’s layers.