# The primitive equations as the small aspect ratio limit of the Navier–Stokes equations: Rigorous justification of the hydrostatic approximation

@article{Li2019ThePE, title={The primitive equations as the small aspect ratio limit of the Navier–Stokes equations: Rigorous justification of the hydrostatic approximation}, author={Jinkai Li and Edriss S. Titi}, journal={Journal de Math{\'e}matiques Pures et Appliqu{\'e}es}, year={2019} }

## 33 Citations

The primitive equations approximation of the anisotropic horizontally viscous 3D Navier-Stokes equations

- MathematicsJournal of Differential Equations
- 2022

Rigorous derivation of the full primitive equations by scaled Boussinesq equations

- Mathematics
- 2021

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…

Rigorous derivation of the primitive equations with full viscosity and full diffusion by scaled Boussinesq equations

- Mathematics
- 2021

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…

Recent Advances Concerning Certain Class of Geophysical Flows

- Environmental Science, Mathematics
- 2016

Author(s): Li, Jinkai; Titi, Edriss S | Abstract: This paper is devoted to reviewing several recent developments concerning certain class of geophysical models, including the primitive equations…

Justification of the Hydrostatic Approximation of the Primitive Equations in Anisotropic Space $L^\infty_H L^q_{x_3}(\Torus^3)$

- Mathematics
- 2021

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…

An Approach to the Primitive Equations for Oceanic and Atmospheric Dynamics by Evolution Equations

- Mathematics
- 2020

The primitive equations for oceanic and atmospheric dynamics are a fundamental model for many geophysical flows. In this chapter we present a summary of an approach to these equations based on the…

On the rigorous mathematical derivation for the viscous primitive equations with density stratification

- Mathematics
- 2022

In this paper, we rigorously derive the governed equations describing the motion of stable stratiﬁed ﬂuid, from the mathematical point of view. Specially, we prove that the scaled Boussinesq…

The hydrostatic approximation of the Boussinesq equations with rotation in a thin domain

- Mathematics
- 2022

Abstract. In this paper, we improve the global existence result in [9] slightly. More precisely, the global existence of strong solutions to the primitive equations with only horizontal viscosity and…

The hydrostatic approximation for the primitive equations by the scaled Navier–Stokes equations under the no-slip boundary condition

- MathematicsJournal of Evolution Equations
- 2020

In this paper we justify the hydrostatic approximation of the primitive equations in the maximal $L^p$-$L^q$-setting in the three-dimensional layer domain $\Omega = \Torus^2 \times (-1, 1)$ under the…

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

- MathematicsNonlinearity
- 2020

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…

## References

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In this paper we prove the global existence and uniqueness (regularity) of strong solutions to the three-dimensional viscous primitive equations, which model large scale ocean and atmosphere…

Mathematical Justification of the Hydrostatic Approximation in the Primitive Equations of Geophysical Fluid Dynamics

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A convergence and existence theorem is proved for this asymptotic model of the time-dependent incompressible Navier-Stokes equations by means of anisotropic estimates and a new time-compactness criterium.

Recent Advances Concerning Certain Class of Geophysical Flows

- Environmental Science, Mathematics
- 2016

Author(s): Li, Jinkai; Titi, Edriss S | Abstract: This paper is devoted to reviewing several recent developments concerning certain class of geophysical models, including the primitive equations…

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Global Well–Posedness of the 3D Primitive Equations with Partial Vertical Turbulence Mixing Heat Diffusion

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The three–dimensional incompressible viscous Boussinesq equations, under the assumption of hydrostatic balance, govern the large scale dynamics of atmospheric and oceanic motion, and are commonly…

Finite-Time Blowup for the Inviscid Primitive Equations of Oceanic and Atmospheric Dynamics

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In an earlier work we have shown the global (for all initial data and all time) well-posedness of strong solutions to the three-dimensional viscous primitive equations of large scale oceanic and…

Global Well-posedness of the 3D Primitive Equations with Only Horizontal Viscosity and Diffusion

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In this paper, we consider the initial-boundary value problem of the 3D primitive equations for planetary oceanic and atmospheric dynamics with only horizontal eddy viscosity in the horizontal…

Strong solutions to the 3D primitive equations with only horizontal dissipation: near $H^1$ initial data

- Mathematics
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A Tropical Atmosphere Model with Moisture: Global Well-posedness and Relaxation Limit

- Mathematics
- 2015

In this paper, we consider a nonlinear interaction system between the barotropic mode and the first baroclinic mode of the tropical atmosphere with moisture; that was derived in [Frierson, D.M.W.;…