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

2021/06/01 by Jinkai Li, Li, Jinkai, Edriss S. Titi +3 · 5 citations
Engineering · Mathematics · #35Q30 #35Q86 #76D05 #86A05 #86A10 #Analysis of PDEs (math.AP) #Computational Fluid Dynamics and Aerodynamics #FOS: Mathematics #FOS: Physical sciences #Fluid Dynamics and Turbulent Flows #Mathematical Physics (math-ph) #Navier-Stokes equation solutions

paper · pdf · doi:10.48550/arxiv.2106.00201

openalex publication_date 2021/06/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

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 viscosity regime. Setting ε >0 to be the small aspect ratio of the vertical to the horizontal scales of the domain, we investigate the case when the horizontal and vertical viscosities in the incompressible three-dimensional Navier-Stokes equations are of orders O(1) and O(ε^α), respectively, with α>2, for which the limiting system is the primitive equations with only horizontal viscosity as ε tends to zero. In particular we show that for "well prepared" initial data the solutions of the scaled incompressible three-dimensional Navier-Stokes equations converge strongly, in any finite interval of time, to the corresponding solutions of the anisotropic primitive equations with only horizontal viscosities, as ε tends to zero, and that the convergence rate is of order O(ε^\fracβ2), where β=min\α-2,2\. Note that this result is different from the case α=2 studied in [Li, J.; Titi, E.S.: The primitive equations as the small aspect ratio limit of the Navier-Stokes equations: Rigorous justification of the hydrostatic approximation, J. Math. Pures Appl., 124 \rm(2019), 30--58], where the limiting system is the primitive equations with full viscosities and the convergence is globally in time and its rate of order O(ε).

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• On the Profile of Singularity Formation for the Incompressible Hydrostatic Boussinesq system
• Rigorous justification of the hydrostatic-incompressible approximation for weakly stratified isothermal flow
• Well-posedness and ill-posedness of the primitive equations with fractional horizontal dissipation
• Long-term behaviour of the primitive equations with wind-driven boundary conditions: Convergence to the Ekman spiral
• The primitive equations for the ocean and atmosphere in anisotropic spaces

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