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| Examples of Boundary Layers Associated with theIncompressible Navier-Stokes Equations |
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Citation： |
Xiaoming WANG.Examples of Boundary Layers Associated with theIncompressible Navier-Stokes Equations[J].Chinese Annals of Mathematics B,2010,31(5):781~792 |
| Page view： 1692
Net amount： 1167 |
Authors： |
Xiaoming WANG; |
Foundation： |
the National Science Foundation, the 111 Project from the Ministry of Education
of China at Fudan University and the COFRS award from Florida State University. |
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| Abstract： |
The author surveys a few examples of boundary layers for which the Prandtl
boundary layer theory can be rigorously validated. All of them are associated with the incompressible
Navier-Stokes equations for Newtonian fluids equipped with various Dirichlet
boundary conditions (specified velocity). These examples include a family of (nonlinear
3D) plane parallel flows, a family of (nonlinear) parallel pipe flows, as well as flows with
uniform injection and suction at the boundary. We also identify a key ingredient in establishing
the validity of the Prandtl type theory, i.e., a spectral constraint on the approximate
solution to the Navier-Stokes system constructed by combining the inviscid solution and
the solution to the Prandtl type system. This is an additional difficulty besides the wellknown
issue related to the well-posedness of the Prandtl type system. It seems that the
main obstruction to the verification of the spectral constraint condition is the possible
separation of boundary layers. A common theme of these examples is the inhibition of
separation of boundary layers either via suppressing the velocity normal to the boundary
or by injection and suction at the boundary so that the spectral constraint can be verified.
A meta theorem is then presented which covers all the cases considered here. |
Keywords： |
Boundary layer, Navier-Stokes system, Prandtl theory, Corrector, Inviscid
limit, Spectral constraint, Nonlinear plane parallel channel flow, Nonlinear
pipe flow, Injection and suction |
Classification： |
76D10, 76D05, 35Q30, 76D09 |
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