Global well-posedness of 2D Hyperbolic perturbation of the Navier-Stokes system in a thin strip

Abstract

In this paper, we study a hyperbolic version of the Navier-Stokes equations, obtained by using the approximation by relaxation of the Euler system, evolving in a thin strip domain. The formal limit of these equations is a hyperbolic Prandtl type equation, our goal is to prove the existence and uniqueness of a global solution to these equations for analytic initial data in the tangential variable, under a uniform smallness assumption. Then we justify the limit from the anisotropic hyperbolic Navier-Stokes system to the hydrostatic hyperbolic Navier-Stokes system with small analytic data.

Gevrey well-posedness of the hyperbolic Prandtl equations

“…This text is motivated by the recent work of Aarach [1] and Paicu-Zhang [36], where the authors investigated the global well-posedness of hydrostatic Navier-Stokes equations of hyperbolic version. For the classical hydrostatic Navier-Stokes or Euler equations, it is far from well-explored, and there are only few works.…”

“…We mention here the very recent work [13] of Gérard-Varet-Masmoudi-Vicol on the local Gevrey well-posedness of 2D parabolic hydrostatic Navier-Stokes equation under convex assumption. The convex assumption is removed for the hyperbolic version of hydrostatic Navier-Stokes equation by the recent Paicu-Zhang's work [36] where they established the global well-posedness in Gevrey class 2, improving the earlier Aarach's work [1] in the analytic setting. This shows the hyperbolic feature may acts as stabilizing factor for hydrostatic Navier-Stokes equation.…”

3D hyperbolic Navier-Stokes equations in a thin strip: global well-posedness and hydrostatic limit in Gevrey space

“…Compared with the local theory, the global in time property is far from being well investigated. Here, we mention Xin-Zhang's work [51] on global weak solutions and some recent papers [1,23,36,[41][42][43]50] on global analytic or Gevrey solutions. Note the above results are obtained mainly in the twodimensional setting so that the global well-posedness of the three-dimensional case remains open.…”

“…Remark 1.7. The proof given in this paper is based on a direct energy method that is substantially different from the elegant and subtle arguments used in [1,42] that involve the Littlewood-Paley decomposition.…”

“…Furthermore, the global well-posedness property of the hydrostatic Navier-Stokes equations (1.2) was investigated by Paicu-Zhang-Zhang [43] in analytic function space. In addition, the global well-posedness theory of the hyperbolic version of 2D hydrostatic Navier-Stokes equations (1.5) was established recently by Aarach [1] and Paicu-Zhang [42] in analytic and Gevrey function spaces respectively.…”