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 for the classical Prandtl system (see Gérard–Varet and Dormy, J. Amer. Math. Soc., <bold>23</bold> (2010), no. 2, 591–609), 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, it is deduced that the optimal Gevrey index for the well-poseness of the hyperbolic Prandtl system is 2.