A note on Prandtl boundary layers

Abstract

This note concerns nonlinear ill-posedness of the Prandtl equation and an invalidity of asymptotic boundary layer expansions of incompressible fluid flows near a solid boundary. Our analysis is built upon recent remarkable linear illposedness results established by Gérard-Varet and Dormy and an analysis by Guo and Tice. We show that the asymptotic boundary layer expansion is not valid for nonmonotonic shear layer flows in Sobolev spaces. We also introduce a notion of weak well-posedness and prove that the nonlinear Prandtl equation is not well-posed in this sense near nonstationary and nonmonotonic shear flows. On the other hand, we are able to verify that Oleinik's monotonic solutions are well-posed.

Prandtl Boundary Layer Expansions of Steady Navier–Stokes Flows Over a Moving Plate

“…It is widely known that the mathematical study of Prandtl boundary layers and the inviscid limit problem is challenging due to its characteristic nature at the boundary (that is, v = 0 at y = 0) and the instability of generic boundary layers ( [6,7,8,9]). Here, for steady flows, we are able to justify the Prandtl boundary layer theory.…”

The Inviscid Limit and Boundary Layers for Navier-Stokes Flows

“…The result of [48] on the ill posedness for the linearized Prandtl equations is strengthened in [61,51], where it is shown that the solutions to the non-linear Prandtl equations cannot be Lipschitz continuous with respect to the initial data in Sobolev spaces. Moreover, it is proved by [61] that in the Sobolev framework one cannot expect a natural estimate of the asymptotic boundary layer expansion for the Navier-Stokes flows, if the leading term in the boundary layer is a non-monotonic shear layer flow as in [48].…”

“…Moreover, it is proved by [61] that in the Sobolev framework one cannot expect a natural estimate of the asymptotic boundary layer expansion for the Navier-Stokes flows, if the leading term in the boundary layer is a non-monotonic shear layer flow as in [48].…”

“…This result indicates the invalidity of Prandtl expansion in the Sobolev framework, no matter whether the boundary layer profiles possess a good shape, although in this case there is still no rigorous proof about the nonlinear instability in Sobolev classes. On the other hand, as it will be mentioned in Section 3.4.2, the invalidity of the asymptotic estimate (3.9) can be also derived from the high-frequency instability of the shear profile in the Prandtl equations, which is proved in [61]. Note that, however, the invalidity of (3.9) observed in [59] and [61] relies on the assumption that the boundary layer is formed already at the initial time (and thus, the initial data for the Navier-Stokes flows is also assumed to depend on the viscosity coefficient, more precisely, on the fast variable…”

On the free surface Navier-Stokes equation in the inviscid limit

“…• even when one can construct it, the approximate solution can be unstable [15], [20] Therefore, for the Navier-Stokes equation with Dirichlet boundary condition, the justification of the inviscid limit is known only in the analytic framework [32]. Nevertheless, we point out that the above approach is efficient even for general quasilinear hyperbolic-parabolic systems (thus also for compressible fluids and MHD equations) when the boundary is non-characteristic (this happens for example with injection or succion boundary conditions), in this case the size of the boundary layer is ε (in the ansatz (2.1), V depends on z/ε) or in dimension one.…”