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## Prandtl’s layer expansions for steady Navier-Stokes

In 1904, Prandtl conjectured that slightly viscous flows can be decomposed into the inviscid flows away from the boundary and a so-called Prandtl’s layer near the boundary. While various instabilities indicate the failure of the conjecture for unsteady flows (for instance, see Grenier 2000), recently with Y. Guo, we are able to prove that the conjecture holds for certain steady Navier-Stokes flows; see our paper which is to appear on Annals of PDEs.

Precisely, consider

\displaystyle \begin{aligned} u^\epsilon \cdot \nabla u^\epsilon + \nabla p^\epsilon = \epsilon \Delta u^\epsilon , \qquad \nabla \cdot u^\epsilon = 0, \end{aligned}

with ${u^\epsilon \in \mathbb{R}^2}$, ${(x,y) \in [0,L]\times \mathbb{R}_+}$, imposing the no-slip boundary condition on the moving plate: ${u^\epsilon_{\vert_{y=0}} = [u_b,0]}$, with the moving speed ${u_b>0}$. Here, ${\epsilon}$ denotes the inverse of the high Reynolds number.

The interest is to analyze the asymptotic limit as ${\epsilon \rightarrow 0}$. We are in particular interested in the behavior near the boundary ${y=0}$, about which Prandtl’s ansatz reads

$\displaystyle u^\epsilon \approx u^\mathrm{Euler}(x,y) + u^\mathrm{Prandtl}(x,y/\sqrt \epsilon)$

in which ${u^\mathrm{Euler}}$ solves the Euler equation (that is, Navier-Stokes with ${\epsilon=0}$), and ${u^\mathrm{Prandtl}(x,z)}$ solves the so-called Prandtl equation , which simply reads

$\displaystyle ( u_1 \partial_x + u_2 \partial_z) u_1 - \partial_z^2 u_1 = -p^\mathrm{Euler}(x), \qquad u_2 = -\int_0^y \partial_x u_1 \; dy$

for the Prandtl’s layers ${[u_1, \sqrt \epsilon u_2]}$, plus appropriate boundary conditions to correct the mismatch of Euler and Navier-Stokes flows on the boundary.

Remarkably, the Prandtl’s equation has self-similar solutions (for instance, Blasius solutions), and the equation can be solved, either by Crocco’s transformation or von Mises’ transformation; see   Its much simplification to the real flows allows leads to tremendous applications and advances in science and engineering. Justification of the validity of Prandtl’s layers is needed.

In my aforementioned paper, we are able to prove that indeed there holds the Prandtl’s asymptotic expansion near parallel Euler flows, at least for a short plate. Precisely,

$\displaystyle u^\epsilon \approx [u^\mathrm{E}(y),0] + u^\mathrm{Pr}(x,y/\sqrt \epsilon) + \sqrt\epsilon u^\mathrm{E} (x,y) + \sqrt \epsilon u^{\mathrm{Pr},1}(x,y/\sqrt \epsilon) + \mathcal{O}_{L^\infty}(\epsilon^{\gamma+\frac12})$

for ${x\in [0,L]}$, with small ${L}$.

In particular, the stability of Prandtl’s expansion yields the validity of the inviscid limit for such a flow. It’s worth noting that it appears not possible, in general, to extend for a longer length of the plate, due to the boundary layer separation phenomenon. However, when the mismatch between Euler and Navier-Stokes flows is sufficiently small, most recently S. Iyer is able to extend our work for the expansion to be valid for all ${L\ge 0}$. In the above expansion, except the first Euler flow which is given, the Prandtl’s layers and next Euler flows solve a parabolic or elliptic equations, respectively, which are introduced to correct the error created from the previous step.

The delicacy in dealing with this asymptotic problem is not only at constructing the ansatz solutions, but also at deriving a stability estimate to control the remainder (that is, to confirm that the error indeed remains small in the inviscid limit). Let me briefly discuss the latter. Indeed, as in the unsteady case, the convection is extremely large: ${\nabla u^\mathrm{app} \sim \frac{1}{\sqrt\epsilon}}$, which prevents us to solve the following linear equation for ${u}$:

$\displaystyle u^\mathrm{app}\cdot \nabla u + u \cdot \nabla u^\mathrm{app} - \epsilon \Delta u = R^\mathrm{app}$

The usual energy or elliptic estimate fails, due to the large convection. However, the energy estimate roughly yields

$\displaystyle \| \nabla u_1 \|_{L^2} \lesssim \iint_{[0,L]\times \mathbb{R}_+} |\nabla u^\mathrm{app}| |u_1 u_2| \lesssim L \| \nabla u_2\|_{L^2}$

noting that the ${L^1}$ norm of ${\nabla u^\mathrm{app} }$ is in fact bounded.

To bound ${\nabla u_2}$ in term of ${\nabla u_1}$, we still need to be able to inverse the linear operator! We then turn to the vorticity equation, which within the boundary layer reads

$\displaystyle u_1^\mathrm{app} \Delta \phi - \phi \Delta u_1^\mathrm{app} - \epsilon \Delta^2 \phi \approx \mathrm{curl}~R^\mathrm{app}$

in which vorticity ${\omega = \Delta \phi}$.

Our crucial observation is that the following operator is indeed positive

$\displaystyle -\Delta +\frac{\Delta u^\mathrm{app}}{u^\mathrm{app}} \ge 0$

as long as ${u^\mathrm{app}}$ remains positive. Thus, multiplying the vorticity equation with ${u_2}$ and noting that ${\int u_2 \omega =0}$, we are able to derive the gradient estimate for ${u_2}$ in term of ${u_1}$, and thus close the stability estimate, when ${L\ll1}$.

Our next difficulty to overcome is the apparent loss of derivatives in the asymptotic expansions: precisely, there is a loss of three derivatives in ${x}$ due to the fact that ${\epsilon \partial_x^2 u_2}$ is treated as a remainder in the Navier-Stokes equations, but not in the Prandtl’s, upon recalling ${u_2 = -\int_0^y \partial_x u_1\; dy}$. This three-derivative loss is serious, due to the possible lack of regularity of solutions to the elliptic problem in domains with corners. We treat this loss by adding yet another boundary layer, this time in ${x}$ near the corners, leaving some small and controllable errors in the momentum equations. The nonlinear iteration can be closed, combining the stability estimate, ${L^\infty}$ estimates, and a higher-order elliptic estimates.