On the mathematical theory of boundary layer for an unsteady flow of incompressible fluid

Examples of Singular Limits in Hydrodynamics

“…This is due to problems related to the well-posedness of the Prandtl system. Indeed, under some monotonicity condition on the initial data, Oleinik proved the local existence for the Prandtl system [140,141] (see also [142]). These solutions can be extended as global weak solutions [173].…”

On the Inviscid Limit Problem of the Vorticity Equations for Viscous Incompressible Flows in the Half‐Plane

“…The local solvability of the Prandtl equations is proved by [29,24] under some assumptions on the monotonicity of the data, and by [2,32] for the analytic initial data. The analyticity condition is in fact required only in the tangential direction [17].…”

The Inviscid Limit and Boundary Layers for Navier-Stokes Flows

“…The solvability of (3.20) in this class has been established by Oleinik and her co-workers, especially for the case Ω 1 = {0 < x 1 < L}. ( See [121,122,123]. The reader is also referred to [124] for more details and references.)…”

“…The Crocco transformation has been a basic tool in the classical works [122,123,124,151]. Recently, an alternative, independent, approach has been presented in [4,112], where the crucial part of the proof is based on a direct energy method but for new dependent variables.…”

“…One brief remark here is that the Prandtl equations are well posed only under strong conditions on the flow, such as when boundary and the data have some degree of analyticity [5,128,95,23,84] or the data is monotonic in the normal direction to the boundary [122,124,83]. The most classical result verifying (3.9) is [128] in the analytic functional framework, after the pioneering work of [5,6].…”