TY - JOUR
T1 - The three limits of the hydrostatic approximation
AU - Furukawa, Ken
AU - Giga, Yoshikazu
AU - Hieber, Matthias
AU - Hussein, Amru
AU - Kashiwabara, Takahito
AU - Wrona, Marc
PY - 2025/4/17
Y1 - 2025/4/17
N2 - Abstract
The primitive equations are derived from the 3D Navier–Stokes equations by the hydrostatic approximation. Formally, assuming an ‐thin domain and anisotropic viscosities with vertical viscosity where , one obtains the primitive equations with full viscosity as . Here, we take two more limit equations into consideration: For the 2D Navier–Stokes equations are obtained. For the primitive equations with only horizontal viscosity as . Thus, there are three possible limits of the hydrostatic approximation depending on the assumption on the vertical viscosity. The latter convergence has been proven recently by Li, Titi, and Yuan using energy estimates. Here, we consider more generally and show how maximal regularity methods and quadratic inequalities can be an efficient approach to the same end for . The flexibility of our methods is also illustrated by the convergence for and to the 2D Navier–Stokes equations.
AB - Abstract
The primitive equations are derived from the 3D Navier–Stokes equations by the hydrostatic approximation. Formally, assuming an ‐thin domain and anisotropic viscosities with vertical viscosity where , one obtains the primitive equations with full viscosity as . Here, we take two more limit equations into consideration: For the 2D Navier–Stokes equations are obtained. For the primitive equations with only horizontal viscosity as . Thus, there are three possible limits of the hydrostatic approximation depending on the assumption on the vertical viscosity. The latter convergence has been proven recently by Li, Titi, and Yuan using energy estimates. Here, we consider more generally and show how maximal regularity methods and quadratic inequalities can be an efficient approach to the same end for . The flexibility of our methods is also illustrated by the convergence for and to the 2D Navier–Stokes equations.
U2 - 10.1112/jlms.70130
DO - 10.1112/jlms.70130
M3 - 学術論文
SN - 0024-6107
VL - 111
JO - Journal of the London Mathematical Society
JF - Journal of the London Mathematical Society
IS - 4
ER -