Fast Singular Oscillating Limits and Global Regularity for the 3D Primitive Equations of Geophysics

Anatoli Babin; Alex Mahalov; Basil Nicolaenko

ESAIM: Mathematical Modelling and Numerical Analysis (2010)

  • Volume: 34, Issue: 2, page 201-222
  • ISSN: 0764-583X

Abstract

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Fast singular oscillating limits of the three-dimensional "primitive" equations of geophysical fluid flows are analyzed. We prove existence on infinite time intervals of regular solutions to the 3D "primitive" Navier-Stokes equations for strong stratification (large stratification parameter N). This uniform existence is proven for periodic or stress-free boundary conditions for all domain aspect ratios, including the case of three wave resonances which yield nonlinear " 2 1 2 dimensional" limit equations for N → +∞; smoothness assumptions are the same as for local existence theorems, that is initial data in Hα, α ≥ 3/4. The global existence is proven using techniques of the Littlewood-Paley dyadic decomposition. Infinite time regularity for solutions of the 3D "primitive" Navier-Stokes equations is obtained by bootstrapping from global regularity of the limit resonant equations and convergence theorems.

How to cite

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Babin, Anatoli, Mahalov, Alex, and Nicolaenko, Basil. "Fast Singular Oscillating Limits and Global Regularity for the 3D Primitive Equations of Geophysics." ESAIM: Mathematical Modelling and Numerical Analysis 34.2 (2010): 201-222. <http://eudml.org/doc/197472>.

@article{Babin2010,
abstract = { Fast singular oscillating limits of the three-dimensional "primitive" equations of geophysical fluid flows are analyzed. We prove existence on infinite time intervals of regular solutions to the 3D "primitive" Navier-Stokes equations for strong stratification (large stratification parameter N). This uniform existence is proven for periodic or stress-free boundary conditions for all domain aspect ratios, including the case of three wave resonances which yield nonlinear "$2\frac\{1\}\{2\}$ dimensional" limit equations for N → +∞; smoothness assumptions are the same as for local existence theorems, that is initial data in Hα, α ≥ 3/4. The global existence is proven using techniques of the Littlewood-Paley dyadic decomposition. Infinite time regularity for solutions of the 3D "primitive" Navier-Stokes equations is obtained by bootstrapping from global regularity of the limit resonant equations and convergence theorems. },
author = {Babin, Anatoli, Mahalov, Alex, Nicolaenko, Basil},
journal = {ESAIM: Mathematical Modelling and Numerical Analysis},
keywords = {Fast singular oscillating limits; three-dimensional Navier-Stokes equations; primitive equations for geophysical fluid flows.; three-dimensional primitive equations; primitive Navier-Stokes equations; infinite time regularity; fast singular oscillating limits; geophysical flows; infinite time intervals; regular solutions; strong stratification; global existence; Littlewood-Paley dyadic decomposition; limit resonance equations; convergence},
language = {eng},
month = {3},
number = {2},
pages = {201-222},
publisher = {EDP Sciences},
title = {Fast Singular Oscillating Limits and Global Regularity for the 3D Primitive Equations of Geophysics},
url = {http://eudml.org/doc/197472},
volume = {34},
year = {2010},
}

TY - JOUR
AU - Babin, Anatoli
AU - Mahalov, Alex
AU - Nicolaenko, Basil
TI - Fast Singular Oscillating Limits and Global Regularity for the 3D Primitive Equations of Geophysics
JO - ESAIM: Mathematical Modelling and Numerical Analysis
DA - 2010/3//
PB - EDP Sciences
VL - 34
IS - 2
SP - 201
EP - 222
AB - Fast singular oscillating limits of the three-dimensional "primitive" equations of geophysical fluid flows are analyzed. We prove existence on infinite time intervals of regular solutions to the 3D "primitive" Navier-Stokes equations for strong stratification (large stratification parameter N). This uniform existence is proven for periodic or stress-free boundary conditions for all domain aspect ratios, including the case of three wave resonances which yield nonlinear "$2\frac{1}{2}$ dimensional" limit equations for N → +∞; smoothness assumptions are the same as for local existence theorems, that is initial data in Hα, α ≥ 3/4. The global existence is proven using techniques of the Littlewood-Paley dyadic decomposition. Infinite time regularity for solutions of the 3D "primitive" Navier-Stokes equations is obtained by bootstrapping from global regularity of the limit resonant equations and convergence theorems.
LA - eng
KW - Fast singular oscillating limits; three-dimensional Navier-Stokes equations; primitive equations for geophysical fluid flows.; three-dimensional primitive equations; primitive Navier-Stokes equations; infinite time regularity; fast singular oscillating limits; geophysical flows; infinite time intervals; regular solutions; strong stratification; global existence; Littlewood-Paley dyadic decomposition; limit resonance equations; convergence
UR - http://eudml.org/doc/197472
ER -

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