Remark on the ill-posedness of the hyperbolic Prandtl system

The paper is devoted to the ill-posedness of the linearized hyperbolic Prandtl system around a shear flow. As in [7] for the classical Prandtl system, the hyperbolic Prandtl system with initial data that does not satisfy monotonicity condition is ill posed at least in a Sobolev space. As a byproduct, we deduce that the optimal Gevrey index for the well-poseness of the hyperbolic Prandtl system is 2.