Abstract
Abstract
In this paper, we study the hydrostatic approximation for the Navier-Stokes system in a thin domain. When we have convex initial data with Gevrey regularity of optimal index
$\frac {3}{2}$
in the x variable and Sobolev regularity in the y variable, we justify the limit from the anisotropic Navier-Stokes system to the hydrostatic Navier-Stokes/Prandtl system. Due to our method in the paper being independent of
$\varepsilon $
, by the same argument, we also obtain the well-posedness of the hydrostatic Navier-Stokes/Prandtl system in the optimal Gevrey space. Our results improve upon the Gevrey index of
$\frac {9}{8}$
found in [15, 35].
Funder
National Natural Science Foundation of China
Publisher
Cambridge University Press (CUP)