L-infinity instability of Prandtl's layers

PDE Seminar
Tuesday, April 10, 2018 - 3:00pm for 1 hour (actually 50 minutes)
Skiles 006
Toan Nguyen – Penn State University – nguyen@math.psu.eduhttp://www.personal.psu.edu/ttn12/
Yao Yao
In 1904, Prandtl introduced his famous boundary layer theory to describe the behavior of solutions of incompressible Navier Stokes equations near a boundary in the inviscid limit. His Ansatz was that the solution of Navier Stokes can be described as a solution of Euler, plus a boundary layer corrector, plus a vanishing error term in $L^\infty$. In this talk, I will present a recent joint work with E. Grenier (ENS Lyon), proving that, for a class of regular solutions of Navier Stokes equations, namely for shear profiles that are unstable to Rayleigh equations, this Prandtl's Ansatz is false. In addition, for shear profiles that are monotone and stable to Rayleigh equations, the Prandtl's asymptotic expansions are invalid.