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In this paper we study the nonlinear stability of a shear layer profile forNavier Stokes equations near a boundary. This question plays a major role inthe study of the inviscid limit of Navier Stokes equations in a bounded domainas the viscosity goes to $0$.  The stability of a shear layer for Navier Stokes equations depends on itsstability for Euler equations. If it is linearly unstable for Euler, then it isknown that it is also nonlinearly unstable for Navier Stokes equations providedthe viscosity is small enough: an initial perturbation grows until it reaches$O(1)$ in $L^\infty$ norm. If it is linearly stable for Euler, the situation ismore complex, since the viscous instability is much slower, with growth ratesof order $O(\nu^{-1/4})$ only (instead of $O(1)$ in the first case). It is notclear whether linear instabilities fully develop till they reach a magnitude oforder $O(1)$ or whether they are damped by the nonlinearity and saturate at amuch smaller magnitude, or order $O(\nu^{1/4})$ for instance.  In this paper we study the effect of cubic interactions on the growth of thelinear instability. In the case of the exponential profile and Blasius profilewe obtain that the nonlinearity tame the linear instability. We thus conjecturethat small perturbations grow until they reach a magnitude $O(\nu^{1/4})$ only,forming small rolls in the critical layer near the boundary. The mathematicalproof of this conjecture is open.

Onset of nonlinear instabilities in monotonic viscous boundary layers