L-infinity instability of Prandtl layers

In 1904, Prandtl introduced his famous boundary layer theory in order to describe the behavior of solutions of incompressible Navier Stokes equations near a boundary as the viscosity goes to $0$ . His Ansatz was that the solution of Navier Stokes equations can be described as a solution of Euler equations, plus a boundary layer corrector, plus a vanishing error term in $L^\infty$ in the inviscid limit.

In this post, I briefly announce my recent work with E. Grenier (ENS Lyon) on the Prandtl’s boundary layer theory, where we prove

the Prandtl’s Ansatz is false for shear profiles that are unstable to Rayleigh equations;
the Prandtl’s asymptotic expansion is invalid for shear profiles that are monotone and stable to Rayleigh equations.

Precisely, we consider incompressible Navier Stokes equations on the half plane $x \in \mathbb{R}$ and $y > 0$ , with a forcing $f^\nu$ :

$\displaystyle \partial_t u^\nu + (u^\nu \cdot \nabla) u^\nu + \nabla p^\nu = \nu \Delta u^\nu + f^\nu,$

$\displaystyle \nabla \cdot u^\nu = 0,$

$\displaystyle u^\nu = 0 \qquad \mbox{on} \qquad y = 0 .$

As the viscosity goes to $0$ , we expect $u^\nu$ to converge to a solution of Euler equations. The justification of this convergence is however very delicate, since the boundary conditions dramatically change. As a consequence, a boundary layer is expected near $y = 0$ in order to describe the transition between Navier Stokes boundary conditions and Euler boundary conditions. To take into account this transition, Prandtl back in 1904 introduced the following Ansatz

$u^\nu(t,x,y) = u^E(t,x,y) + u^P(t,x,y / \sqrt{\nu}) + o(1)_{L^\infty},$
where $u^P$ describes the behavior of $u^\nu$ in a boundary layer of size $\mathcal{O}(\sqrt \nu)$ , called the Prandtl’s boundary layer, and the remainder $o(1)_{L^\infty}$ tends to zero in the inviscid limit.

The existence of the Prandtl corrector $u^P$ has been proven for monotonic initial data by Oleinik in the 60s. For analytic initial data, the existence of the corrector together with the validity of the Prandtl’s Ansatz has been established by Caflisch and Sammartino in 1998. This latter result in particular proves that if a boundary layer Ansatz exists to describe the limiting behavior of $u^\nu$ , then it must be of the above Prandtl’s form. However, considering analytic initial data is too restrictive, since it precludes small but high frequencies perturbations, which are more physically relevant. Up to now, there were no result which proved, or disproved, the Prandtl’s Ansatz for Sobolev initial data, despite several efforts over the years, and our work is the first in this direction: namely, we prove that there exists particular regular initial data such that the Prandtl’s Ansatz is wrong.

Precisely, we consider a time-independent shear layer profile is a solution of the form

$\displaystyle U^\nu(t,x,y) = \begin{pmatrix}U(y/ \sqrt{\nu}) \\ 0 \end{pmatrix}$

where $U(z)$ is a sufficiently smooth function with $U(0) = 0$ and $U(z)$ converges when $z \to + \infty$ to a constant $U_\infty$ . To get a time independent shear layer profile, we add a stationary forcing term which compensates for the viscosity; precisely, we take

$\displaystyle F^\nu = \begin{pmatrix} - U''(y / \sqrt{\nu}) \\0\end{pmatrix} .$

These shear profiles are smooth solutions of Navier Stokes, Prandtl, and Heat equations, with the same forcing $F^\nu$ . As will be seen, our instability results occur at a vanishing time in $\nu \to 0$ , and thus some can also be extended to time-dependent shear profiles.

Our two main theorems read as follows:

Theorem 1 (Grenier-Toan 2018) Let $U(y/\sqrt\nu)$ be a time-independent, analytic, and Rayleigh unstable shear profile. Then, for any $N$ and $s$ arbitrarily large, there exists $\sigma_0 > 0$ , $C_0 > 0$ and a sequence of solutions $u^\nu$ of Navier Stokes equations with forcing $F^\nu + f^\nu$ , on some interval $[0,T^\nu]$ , such that

$\displaystyle \| u^\nu(0) - U^\nu(0) \|_{H^s} + \| f^\nu \|_{L^\infty([0,T^\nu],H^s)} \le \nu^N,$

but

$\displaystyle \| u^\nu(T^\nu) - U^\nu(T^\nu) \|_{L^\infty} \ge \sigma_0,$

and $latex
T^\nu = C_0 \sqrt{\nu} \log \nu^{-1} .
$

Theorem 2 (Grenier-Toan 2017) Let $U(t,y/\sqrt\nu)$ be a sufficiently smooth, time-dependent, monotone, and Rayleigh stable shear profile. Then, for any $N$ and $s$ arbitrarily large, there exists $\sigma_0 > 0$ , $C_0 > 0$ and a sequence of solutions $u^\nu$ of Navier Stokes equations with forcing $f^\nu$ , on some interval $[0,T^\nu]$ , such that

$\displaystyle \| u^\nu(0) - U^\nu(0) \|_{H^s} + \| f^\nu \|_{L^\infty([0,T^\nu],H^s)} \le \nu^N,$

but

$\displaystyle \| u^\nu(T^\nu) - U^\nu(T^\nu) \|_{L^\infty} \ge \sigma_0 \nu^{\frac14 +},$

and $latex
T^\nu = C_0 \nu^{\frac14} \log \nu^{-1} .
$

The Rayleigh stable or unstable shear profiles $U^\nu$ are those that are stable or unstable to the corresponding Rayleigh equation (or the linearized Euler equations). Theorem 1 disproves the Prandtl’s Ansatz. Physically, the Rayleigh unstable profile may correspond to a reverse flow and thus rules out the exponential profile $U_\infty (1 - e^{-y / C})$ . The Prandtl equation is well posed for $U^\nu$ and for neighboring analytic profiles. However we do not know whether the Prandtl equation is well posed for nearby profiles with only Sobolev regularity. When $U^\nu$ has no inflection point or is stable to the Rayleigh equation as in Theorem 2, we do not know how to prove the $L^\infty$ instability result of order one as in Theorem 1. Nevertheless, Theorem 2 proves that there exists no asymptotic expansion of Prandtl’s type. The question of the $L^\infty$ instability of monotonic shear profiles remains open.

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