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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 $varepsilon $ with initial data $u_0=(v_0,w_0)in B^{2-2/p}_{q,p}$, $1/q+1/ple 1$ if $qge 2$ and $4/3q+2/3ple 1$ if $qle 2$, converges as $varepsilon to 0$ with convergence rate $mathcal{O} (varepsilon )$ 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.
In this paper, we provide rigorous justification of the hydrostatic approximation and the derivation of primitive equations as the small aspect ratio limit of the incompressible three-dimensional Navier-Stokes equations in the anisotropic horizontal
The primitive equations of large-scale ocean dynamics form the fundamental model in geophysical flows. It is well-known that the primitive equations can be formally derived by hydrostatic balance. On the other hand, the mathematically rigorous deriva
The primitive equations are fundamental models in geophysical fluid dynamics and derived from the scaled Navier-Stokes equations. In the primitive equations, the evolution equation to the vertical velocity is replaced by the so-called hydrostatic app
We prove a weak stability result for the three-dimensional homogeneous incompressible Navier-Stokes system. More precisely, we investigate the following problem : if a sequence $(u_{0, n})_{nin N}$ of initial data, bounded in some scaling invariant s
In this article, we establish sufficient conditions for the regularity of solutions of Navier-Stokes equations based on one of the nine entries of the gradient tensor. We improve the recently results of C.S. Cao, E.S. Titi (Arch. Rational Mech.Anal.