Well-posedness of the linearized Prandtl equation around a non-monotonic shear flow

In this paper, we prove the well-posedness of the linearized Prandtl equation around a non-monotonic shear flow in Gevrey class 2−θ for any θ>0. This result is almost optimal by the ill-posedness result proved by Gérard-Varet and Dormy, who construct a class of solution with the growth like ekt for the linearized Prandtl equation around a non-monotonic shear flow.