Gevrey-Class-3 Regularity of the Linearised Hyperbolic Prandtl System on a Strip

In the present paper, we address a physically-meaningful extension of the linearised Prandtl equations around a shear flow. Without any structural assumption, it is well-known that the optimal regularity of Prandtl is given by the class Gevrey 2 along the horizontal direction. The goal of this paper is to overcome this barrier, by dealing with the linearisation of the so-called hyperbolic Prandtl equations in a strip domain. We prove that the local well-posedness around a general shear flow \(U_{\textrm{sh}}\in W^{3, \infty }(0,1)\) holds true, with solutions that are Gevrey class 3 in the horizontal direction.