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}
}
  • Jinkai Li, E. Titi
  • Published 26 June 2017
  • Mathematics
  • Journal de Mathématiques Pures et Appliquées
View PDF on arXiv
33 Citations
Highly Influential Citations
8
Background Citations
11
Methods Citations
10
Results Citations
6

33 Citations

Citation Type

Citation Type

The primitive equations approximation of the anisotropic horizontally viscous 3D Navier-Stokes equations
Rigorous derivation of the full primitive equations by scaled Boussinesq equations
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
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
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 $L^\infty_H L^q_{x_3}(\Torus^3)$
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
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
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
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
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
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…
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References

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Global well-posedness of the three-dimensional viscous primitive equations of large scale ocean and atmosphere dynamics
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
TLDR
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
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…
On the equations of the large-scale ocean
As a preliminary step towards understanding the dynamics of the ocean and the impact of the ocean on the global climate system and weather prediction, the authors study the mathematical formulations…
Global well-posedness of strong solutions to a tropical climate model
In this paper, we consider the Cauchy problem to the TROPIC CLIMATE MODEL derived by Frierson-Majda-Pauluis in [Comm. Math. Sci, Vol. 2 (2004)] which is a coupled system of the barotropic and the…
Global Well–Posedness of the 3D Primitive Equations with Partial Vertical Turbulence Mixing Heat Diffusion
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
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
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
A Tropical Atmosphere Model with Moisture: Global Well-posedness and Relaxation Limit
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.;…
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