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## Green function for linearized Navier-Stokes around a boundary layer profile: near critical layers

Emmanuel Grenier and I have just submitted this 84-page! long paper, also posted on arxiv (arXiv:1705.05323). This work is a continuation and completion of the program (initiated in Grenier-Toan1 and Grenier-Toan2) to derive pointwise estimates on the Green function and sharp bounds on the semigroup of linearized Navier-Stokes around a stationary boundary layer profile.

## Sublayer of Prandtl boundary layers

The aim of this paper (arXiv:1705.04672), with E. Grenier, is to investigate the stability of Prandtl boundary layers in the vanishing viscosity limit: ${\nu \rightarrow 0}$.  In his CPAM2000 paper, Grenier proved that there exists no Prandtl’s asymptotic expansion involving one Prandtl’s boundary layer with thickness of order ${\sqrt\nu}$, which describes the inviscid limit of Navier-Stokes equations. The instability gives rise to a viscous boundary sublayer whose thickness is of order ${\nu^{3/4}}$. In this paper, we point out how the stability of the classical Prandtl’s layer is linked to the stability of this sublayer. In particular, we prove that the two layers cannot both be nonlinearly stable in ${L^\infty}$.  That is, either the Prandtl’s layer or the boundary sublayer is nonlinearly unstable in the sup norm.