Inviscid limit for Navier-Stokes in a rough domain

ttn12 fluid dynamics, New papers

In this paper with Gérard-Varet, Lacave, and Rousset, we prove the inviscid limit of Navier-Stokes flows in domains with a rough or oscillating boundary. Precisely, we study the 2D incompressible Navier-Stokes flows with small viscosity {\nu}, posed on the following rough domain:

\displaystyle \Omega^\epsilon := \{ x = (x_1,x_2), \quad x_1 \in \mathbb{T}, \: x_2 > \epsilon^{1+\alpha} \eta(x_1/\epsilon) \}

whose boundary oscillates at wavelength {\epsilon}. Here, {\alpha>0}.

The inviscid limit problem of Navier-Stokes, with a boundary on which the classical no-slip condition is imposed, is essentially open due to various instabilities of Prandtl’s boundary layers (for instance, I discussed here). Near the boundary {x_2=0}, the Prandtl’s ansatz reads

\displaystyle u^\mathrm{app}(t,x) \approx u^\mathrm{Euler}(t,x) + u^\mathrm{Prandtl}(t,x_1,x_2/\sqrt{\nu})

which experiences a large gradient {\nabla u^\mathrm{app} \sim \frac{1}{\sqrt\nu}}. This prevents available analyses to obtain the inviscid limit: Navier-Stokes converges to Euler in certain norms, except the classical analyticity result by Sammartino-Caflisch ’98 (see also Maekawa ’14, where he obtains the inviscid limit, by allowing Sobolev regularity away from the boundary, but still analyticity needed near the boundary, or precisely Euler vorticity is assumed to be zero near the boundary). This large gradient is in fact the source of viscous instability (for instance, Grenier 2000, or Grenier-Guo-Nguyen ’16).

On the other hand, for Navier-Stokes with so-called Navier conditions: {D(u)n \cdot \tau \sim u\cdot \tau} on the boundary (the shear stress is propositional to the tangential velocity), Iftimie-Sueur ’11 proved that the boundary layer ansatz reads

\displaystyle u^\mathrm{app}(t,x) \approx u^\mathrm{Euler}(t,x) + \sqrt \nu u^\mathrm{Prandtl}(t,x_1,x_2/\sqrt{\nu}) .

One observes that in this case, the Prandtl’s layers have smaller amplitude, and most importantly {\nabla u^\mathrm{app}} remains bounded. This boundedness allows Iftimie-Sueur ’11 to justify the inviscid limit; see also Masmoudi-Rousset ’12 for a strong compactness approach.

In this paper, we study the inviscid limit under the Navier boundary conditions on the oscillating boundary. One of our main motivations is that the Navier condition on the highly oscillating boundary becomes the classical no-slip condition (the case where the inviscid limit problem is unsolved) in the limit of {\epsilon \rightarrow 0}. That is, the Navier condition is indeed a good approximation of the no-slip condition, when there is roughness on the boundary! This way, we wish to obtain the inviscid limit, avoiding the instability of Prandtl’s layers. In addition, it appears natural for {\nu \ll \epsilon}, for otherwise the Prandtl’s layers would come into play.

Indeed, we prove in this paper the following theorem:

Theorem 1 (Rough statement) For {\alpha>0} and {\epsilon^{N} \lesssim \nu \lesssim \epsilon^7}, {N} arbitrarily large, the Navier-Stokes with viscosity {\nu} on {\Omega^\epsilon} satisfies the asymptotic expansion

\displaystyle u^{\nu,\epsilon} \approx u^\mathrm{Euler}(t,x) + \epsilon^\alpha u^\mathrm{BL} (t,x,{x}/{\epsilon}) + \mathcal{O}_{L^\infty}(\epsilon^\alpha)

on {[0,T]\times \Omega^\epsilon}, for any {T>0}, in the limit of {\epsilon \rightarrow0}. In particular, the inviscid limit holds.

The key difficulty is again the large gradient {\nabla u^\mathrm{app} \sim \epsilon^{\alpha -1}}, due to the appearance of boundary layers. Our first crucial observation is that we are able to construct boundary layers, which are essentially inviscid, whose vorticity is zero, leaving a small error in the momentum equation. This allows us to work with the vorticity equation, which reads

\displaystyle \omega_t + (u^\mathrm{app} + v)\cdot \nabla \omega + v \cdot \nabla \omega^\mathrm{app} = \nu \Delta \omega + R^\mathrm{app} .

Since (inviscid) boundary layers have zero vorticity, {\nabla \omega^\mathrm{app} } is bounded, and hence standard energy estimates yield

\displaystyle \frac{d}{dt} \|\omega\|_{L^2} + \nu \| \nabla \omega \|_{L^2} \lesssim \| \nabla \omega^\mathrm{app}\|_{L^\infty} \int |v \omega| \; dx + \mathrm{B.C} .

As one roughly expects {\|v \| \lesssim \|\omega\|} (for instance, energy estimates for velocity yield {\frac{d}{dt}\| v\|_{L^2} \lesssim \| \omega \|_{L^2} + \mathrm{B.C.}}), the above would yield at once a stability estimate for vorticity (!). However, making this rigorous turns out to be a delicate issue, mostly due to the large vorticity on the boundary! Indeed, the boundary condition for vorticity roughly reads

\displaystyle \omega \approx \kappa_\epsilon v\cdot \tau

where {\kappa_\epsilon \sim \epsilon^{\alpha-1}}, denoting the curvature of the oscillating boundary. The above energy estimate for vorticity fails due to the large vorticity on the boundary.

Our second crucial contribution is to design a weighted energy estimate for vorticity, and thereby derive a similar stability estimate. Finally, some delicate potential estimates are derived for the type of rough domains that we are considering, in particular to obtain a close sup norm on velocity. This paper also raises an interesting open problem: to establish the inviscid limit for the case when {\alpha=0}.

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