Rigorous justification of the hydrostatic approximation for the primitive equations by scaled Navier-Stokes equations

Abstract

Consider the anisotropic Navier-Stokes equations as well as the primitive equations. It is shown that the horizontal velocity of the solution to the anisotropic Navier-Stokes equations in a cylindrical domain of height $\epsilon$ with initial data $u_0=(v_0,w_0)\in B_{q,p}^{2-2/p}$, $1/q+1/p\le 1$ if $q\ge 2$ and $4/3q+2/3p\le 1$ if $q\le 2$, converges as $\epsilon \to 0$ with convergence rate $\mathcal{O}(\epsilon )$ to the horizontal velocity of the solution to the primitive equations with initial data $v_0$ with respect to the maximal-$L^p$-$L^q$-regularity norm. Since the difference of the corresponding vertical velocities remains bounded with respect to that norm, the convergence result yields a rigorous justification of the hydrostatic approximation in the primitive equations in this setting. It generalizes in particular a result by Li and Titi for the $L^2$-$L^2$-setting. The approach presented here does not rely on second order energy estimates but on maximal $L^p$-$L^q$-estimates for the heat equation.

arXiv comment10 pages

References in corpus (4)

• Global Strong Well-posedness of the Three Dimensional Primitive equations in $L^p$-spaces
• Global Strong $L^p$ Well-Posedness of the 3D Primitive Equations with Heat and Salinity Diffusion
• Analyticity of solutions to the primitive equations
• The Hydrostatic Stokes Semigroup and Well-Posedness of the Primitive Equations on Spaces of Bounded Functions

Cited by in corpus (5)

• On the hydrostatic limit of stably stratified fluids with isopycnal diffusivity
• Global well-posedness of $z$-weak solutions to the primitive equations without vertical diffusivity
• Relaxing the sharp density stratification and columnar motion assumptions in layered shallow water systems
• The primitive equations approximation of the anisotropic horizontally viscous Navier-Stokes equations
• Justification of the Hydrostatic Approximation of the Primitive Equations in Anisotropic Space $L^\infty_H L^q_{x_3}(\Torus^3)$