Feeds:
Posts

## Invalidity of Prandtl’s boundary layers

I’ve just submitted this paper with Grenier (ENS Lyon) which studies Prandtl’s boundary layer asymptotic expansions for incompressible fluids on the half-space in the inviscid limit. In 1904, Prandtl introduced his well known boundary layers in order to describe the transition from Navier-Stokes to Euler equations in the inviscid limit.

Formally, we expect the so-called Prandtl’s Ansatz:

$\displaystyle u^\nu(t,x,y) \approx u^\mathrm{Euler}(t,x,y) + u^\mathrm{Prandtl} \Big ( t,x, {y \over \sqrt{\nu}} \Big ) + \mathcal{O}(\sqrt{\nu})$

where ${u^\mathrm{Euler}}$ solves the Euler equations, and ${u^\mathrm{Prandtl}}$ is the Prandtl boundary layer corrector, denoting $\nu$ the small viscosity or the inverse of the large Reynolds number.

The Prandtl boundary layers have been intensively studied in the mathematical literature. Notably, solutions to the Prandtl equations have been constructed for monotonic data (Oleinik ’60s, or recently Masmoudi-Wong or Alexandre at al.), or data with Gevrey or analytic regularity (Sammartino-Caflisch ’98 or recently Gerard-Varet and Masmoudi). The validity of the Prandtl’s Ansatz has been established by Sammartino-Caflisch ’98 for initial data with analytic regularity. The Ansatz, with a specific boundary layer profile, has also been recently justified for data with Gevrey regularity by Gerard-Varet, Maekawa and Masmoudi.

When only data with Sobolev regularity are assumed, Grenier proved in his CPAM 2000 paper that such an asymptotic expansion is false, up to a remainder of order ${\nu^{1/4}}$ in ${L^\infty}$ norm. The invalidity of the expansion is proved near boundary layers with an inflection point or more precisely near those that are spectrally unstable for the Rayleigh equations.

In this paper, we prove the Prandtl’s Ansatz is also false for Rayleigh’s stable shear flows, giving a proof of the Conjecture stated in Section 4 of Grenier-Guo-Toan. Such shear flows are stable for Euler equations, but not for Navier-Stokes equations: adding a small viscosity destabilizes the flow.

Roughly speaking, given an arbitrary stable boundary layer, the two main results in this paper are

Theorem 1 (Grenier-Toan: a rough statement) In the case of time-dependent boundary layers, we construct Navier-Stokes solutions, with arbitrarily small forcing of order ${\mathcal{O}(\nu^\infty)}$, so that the Prandtl’s Ansatz is false near the boundary layer, up to a remainder of order ${\nu^{1/4+\epsilon}}$ in ${L^\infty}$ norm, ${\epsilon}$ being arbitrarily small.

Theorem 2 (Grenier-Toan: a rough statement) In the case of stationary boundary layers, we construct Navier-Stokes solutions, without forcing term, so that the Prandtl’s Ansatz is false, up to a remainder of order ${\nu^{5/8}}$ in ${L^\infty}$ norm.