Gevrey Well-Posedness of the Hyperbolic Prandtl Equations

Abstract

We study the hyperbolic version of the Prandtl system derived from the hyperbolic Navier-Stokes system with no-slip boundary condition. Compared to the classical Prandtl system, the quasi-linear terms in the hyperbolic Prandtl equation leads to an additional instability mechanism. To overcome the loss of derivatives in all directions in the quasi-linear term, we introduce a new auxiliary function for the well-posedness of the system in an anisotropic Gevrey space which is Gevrey class 3 2 in the tangential variable and is analytic in the normal variable.

3D Hyperbolic Navier-Stokes Equations in a Thin Strip: Global Well-Posedness and Hydrostatic Limit in Gevrey Space

“…This generalizes the classical result of Sammartino and Caflisch [48] in the analytic framework. Similar well-posedness properties of hyperbolic Prandtl equations in Gevrey class were proven in [27].…”

Gevrey Well-Posedness of Quasi-Linear Hyperbolic Prandtl Equations

“…by removing the quasi-linear term η∂ t ((u•∂ x )u+v∂ y u) in (1.1). For this, the local well-posedness of (1.3) in Gevrey 2 space is obtained by [21] which is the same Gevrey space for the classical Prandtl equation. However, the Gevrey index 2 may not be optimal for the well-posedness of (1.3).…”