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 $2-\theta$ for any $\theta>0$. This result is almost optimal by the ill-posedness result proved by G\'{e}rard-Varet and Dormy, who construct a class of solution with the growth like $e^{\sqrt{k}t}$ for the linearized Prandtl equation around a non-monotonic shear flow.