Gevrey well-posedness of the hyperbolic Prandtl equations

Abstract

We study the 2D and 3D Prandtl equations of degenerate hyperbolic type, and establish without any structural assumption the Gevrey well-posedness with Gevrey index ≤ 2. Compared with the classical parabolic Prandtl equations, the loss of the derivatives, caused by the hyperbolic feature coupled with the degeneracy, can't be overcame by virtue of the classical cancellation mechanism that developed for the parabolic counterpart. Inspired by the abstract Cauchy-Kowalewski theorem and by virtue of the hyperbolic feature, we give in this text a straightforward proof, basing on an elementary L 2 energy estimate. In particular our argument does not involve the cancellation mechanism used efficiently for the classical Prandtl equations.

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-Caflisch [48] in the analytic framework. Similar well-posedness properties of hyperbolic Prandtl equations in Gevrey class were proven in [27].…”

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

“…More precisely, solutions corresponding to a frequency k in x will behave like e √ |k|t which restricts well-posedness theory to the Gevrey 2 case. For a hyperbolic equation, this is expected and actually proven for (3.1) in [19] (see also [26]).…”

“…Additionally, we give a detailed discussion on possible improved well-posedness results for the nonlinear hyperbolic Prandtl system (1.5). In particular, our work shows that one cannot rely on further simplifications of (1.5) in order to achieve existence results beyond the expected Gevrey 2 class (see, e.g., [19] and [26]). We refer to Sect.…”

“…Standard considerations of the (linearized as well as non-linear) hyperbolic Prandtl system yield to a well-posedness within Gevrey-class 2, at best (see e.g. [19]). We show that a suitable cancellation mechanism is inherent to (1.1), which is similar to the one presented in [11].…”