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Examples of Boundary Layers Associated with the Incompressible Navier-Stokes Equations
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.
作 者： Xiaoming WANG 作者單位： Department of Mathematics, Florida State University, Tallahassee, FL 32306, USA 刊 名： 數學年刊B輯（英文版） ISTIC SCI 英文刊名： CHINESE ANNALS OF MATHEMATICS,SERIES B 年，卷(期)： 2010 31(5) 分類號： O1 關鍵詞： Boundary layer Navier-Stokes system Prandtl theory Corrector Inviscid limit Spectral constraint Nonlinear plane parallel channel flow Nonlinear pipe flow Injection and suction