51. Ill-posedness of the Prandtl equations in Sobolev spaces around a shear flow with general decay
- Author
-
Cheng-Jie Liu and Tong Yang
- Subjects
Applied Mathematics ,General Mathematics ,010102 general mathematics ,Mathematical analysis ,Prandtl number ,Mathematics::Analysis of PDEs ,01 natural sciences ,Physics::Fluid Dynamics ,010101 applied mathematics ,Sobolev space ,symbols.namesake ,Inviscid flow ,symbols ,0101 mathematics ,Exponential decay ,Shear flow ,Approximate solution ,Ill posedness ,Mathematics ,Variable (mathematics) - Abstract
Motivated by the paper Gerard-Varet and Dormy (2010) [6] [JAMS, 2010] about the linear ill-posedness for the Prandtl equations around a shear flow with exponential decay in normal variable, and the recent study of well-posedness on the Prandtl equations in Sobolev spaces, this paper aims to extend the result in [6] to the case when the shear flow has general decay. The key observation is to construct an approximate solution that captures the initial layer to the linearized problem motivated by the precise formulation of solutions to the inviscid Prandtl equations.
- Published
- 2017