Global hydrostatic approximation of the hyperbolic Navier-Stokes system with small Gevrey class 2 data In memory of Professor Geneviève Raugel

We investigate the hydrostatic approximation of a hyperbolic version of Navier-Stokes equations,which is obtained by using the Cattaneo type law instead of the Fourier law, evolving in a thin strip R ×(0, ε).The formal limit of these equations is a hyperbolic Prandtl type equation. We first prove the global existence of solutions to these equations under a uniform smallness assumption on the data in the Gevrey class 2. Then we justify the limit globally-in-time from the anisotropic hyperbolic Navier-Stokes system to the hyperbolic Prandtl system with such Gevrey class 2 data. Compared with Paicu et al.(2020) for the hydrostatic approximation of the 2-D classical Navier-Stokes system with analytic data, here the initial data belongs to the Gevrey class 2,which is very sophisticated even for the well-posedness of the classical Prandtl system(see Dietert and GerardVaret(2019) and Wang et al.(2021)); furthermore, the estimate of the pressure term in the hyperbolic Prandtl system gives rise to additional difficulties.